diff --git "a/4NE1T4oBgHgl3EQfAgLJ/content/tmp_files/load_file.txt" "b/4NE1T4oBgHgl3EQfAgLJ/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/4NE1T4oBgHgl3EQfAgLJ/content/tmp_files/load_file.txt" @@ -0,0 +1,961 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf,len=960 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='02841v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='DS] 7 Jan 2023 LEVEL-2 LARGE DEVIATION PRINCIPLE FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES HIROKI TAKAHASI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We consider level-2 large deviations for the one-sided countable full shift without assuming the existence of Bowen’s Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' To deal with non- compact closed sets, we provide a sufficient condition in terms of inducing which ensures the exponential tightness of a sequence of Borel probability measures constructed from periodic configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Under this condition we establish the level-2 Large Deviation Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We apply our results to the continued fraction expansion of real numbers in [0, 1) generated by the R´enyi map, and obtain the level-2 Large Deviation Principle, as well as a weighted equidistribution of a set of quadratic irrationals to equilibrium states of the R´enyi map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Introduction Dynamical systems (iterated maps) equipped with finite Markov partitions are represented as finite Markov shifts, and the construction of relevant invariant mea- sures and the investigation of their statistical properties are done on the symbolic level, with adaptations of ideas in statistical mechanics (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [4, 5, 29, 32, 39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This thermodynamic formalism initiated in the 60s has been successfully extended to maps with infinite Markov partitions and shift spaces with countably infinite number of states (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [1, 6, 10, 11, 18, 30, 31, 41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This paper is concerned with level-2 large deviations for such countable Markov shifts, and its application to a dynamical system related to number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The theory of large deviations aims to characterize limit behaviors of probability measures in terms of rate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let X be a topological space, and let M(X ) denote the space of Borel probability measures on X endowed with the weak* topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We say a sequence {˜µn}∞ n=1 in M(X ) satisfies the Large Deviation Prin- ciple (LDP) if there exists a lower semicontinuous function I : X → [0, ∞] such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1) lim inf n→∞ 1 n log ˜µn(G) ≥ − inf G I for any open set G ⊂ X , and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) lim sup n→∞ 1 n log ˜µn(C) ≤ − inf C I for any closed set C ⊂ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We call x ∈ X a minimizer if I(x) = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The set of minimizers is a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The LDP means that in the limit n → ∞ the measure ˜µn assigns all but Date: January 10, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 37A44, 37A50, 37A60, 60F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Keywords: thermodynamic formalism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Gibbs state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Large Deviation Principle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' periodic points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' equidistribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 1 2 HIROKI TAKAHASI exponentially small mass to the set of minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The function I is called a rate function, and called a good rate function if its level set {x ∈ X : I(x) ≤ α} is compact for any α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If X is a metric space and {˜µn}∞ n=1 satisfies the LDP, the rate function is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The setup in our mind is that X is the space of Borel probability measures on a topological space X on which a Borel map σ: X → X acts, and each ˜µn ∈ M(X ) is given in terms of empirical measures δn x = (1/n) �n−1 k=0 δσkx, where δσkx ∈ X denotes the unit point mass at σkx ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We refer to the LDP in this setup as level-2 [8, Chapter 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For topologically mixing finite Markov shifts together with H¨older continuous potentials, the level-2 LDP for empirical distributions and that for sequences con- structed from empirical measures on periodic orbits were established in [15, 22, 33] and [16] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A key ingredient in these classical cases is the existence of Bowen’s Gibbs states [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' With the aid of Bowen’s Gibbs states, one can deduce the lower bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1) by combining Birkhoff’s and Shannon-McMillan-Breiman’s theorems, and the upper bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) by modifying the standard proof of the variational principle [40] (see [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For countable Markov shifts, Bowen’s Gibbs states were constructed under the assumption of a good regularity of potentials and a strong connectivity of transition matrices defining the shift spaces (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [1, 6, 11, 18, 30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Several level-2 LDPs were established in [34] under the existence of Bowen’s Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' It has been realized that not all dynamically relevant invariant probability mea- sures correspond to Bowen’s Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' One of the best known examples is the absolutely continuous invariant probability measure of an interval map of Manneville-Pomeau type, with finitely many branches and a neutral fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Such a measure still retains a weak form of Bowen’s Gibbs state [41], and has the weak Gibbsian property in statistical mechanics sense [9, 17, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For a ther- modynamic formalism and level-2 large deviations for a class of this map, see [12, 28, 41] and [25, 26] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' With these historical developments and the abundance of interesting dynamical systems modeled by countable Markov shifts without Bowen’s Gibbs states (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [13, 43]), it is important to establish the level-2 LDP for countable Markov shifts without assuming the existence of Bowen’s Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A main new difficulty for countable Markov shifts is a treatment of non-compact closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We say a sequence {˜µn}∞ n=1 of Borel probability measures on a non- compact space X is exponentially tight if for any L > 0 there exists a compact set K ⊂ X such that lim sup n→∞ 1 n log ˜µn(X \\ K) ≤ −L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If {˜µn}∞ n=1 is exponentially tight, then the upper bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) for any compact closed set implies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) for any closed set which is not necessarily compact, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [7] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The proof of the level-2 LDPs in [34] relies on the existence of Bowen’s Gibbs states in order to verify the exponential tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Our strategy for countable Markov shifts without Bowen’s Gibbs states is to use inducing to verify the exponential tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Inducing is a familiar procedure in ergodic theory originally considered in works by Kakutani, Rohlin and others, and LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 3 was used in the construction of absolutely continuous invariant measures or Gibbs- equilibrium states (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [2, 3, 23, 24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' An inducing scheme we use here is given by the first return map to an a priori fixed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In terms of this inducing, we will formulate a sufficient condition which ensures the exponential tightness for the original system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A key concept is that of local Gibbs states introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Statements of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Throughout the rest of this paper, let N denote the discrete set of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let X denote the one-sided infinite Cartesian product topological space of N, called a countable full shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The topology of X has a base that consists of cylinders [p1 · · · pn] = {x = (xn)∞ n=1 ∈ X : xk = pk for every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , n}}, where n ≥ 1 and p1 · · ·pn ∈ Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This topology is metrizable with the metric d(x, y) = exp (− inf{n ≥ 1: xn ̸= yn}) where exp(−∞) = 0 by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let σ denote the left shift acting on X: (σx)n = xn+1 for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be a function, called a potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We say φ is acceptable if it is uniformly continuous and satisfies sup p∈N � sup [p] φ − inf [p] φ � < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We say φ is locally H¨older continuous if there exist C > 0 and α ∈ (0, 1] such that for any p ∈ N and all x, y ∈ [p], |φ(x) − φ(y)| ≤ Cd(x, y)α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Clearly, if φ is locally H¨older continuous then it is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each n ≥ 1 we write Snφ for the Birkhoff sum �n−1 k=0 φ ◦ σk, and introduce a pressure (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3) P(φ) = lim n→∞ 1 n log � p1···pn∈Nn sup [p1···pn] exp Snφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This limit exists by the sub-additivity [4, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='18], which is never −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We consider a sequence {˜µn}∞ n=1 of Borel probability measures on M(X) given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) ˜µn = 1 Zn(φ) � x∈En exp Snφ(x)δδn x, where En = {x ∈ X : σnx = x}, and δδn x denotes the unit point mass at δn x, and Zn(φ) the normalizing constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In dynamical systems terms, En is the set of periodic points of period n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In statistical mechanics terms, the measure ˜µn is closely related to the canonical ensemble subject to a periodic boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' An inducing scheme consists of a subset X∗ of X of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5) X∗ = X \\ � p∈N∩[1,p∗−1] [p], 4 HIROKI TAKAHASI where p∗ ≥ 2, and a function R: X∗ → N ∪ {∞} given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6) R(x) = inf{n ≥ 1: σnx ∈ X∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Given an inducing scheme (X∗, R) we define an induced map (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) τ : X∗ ∩ ∞ � k=1 σ−kX∗ �→ σR(x)x ∈ X∗, and an inducing domain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8) Σ = ∞ � n=0 τ −n � X∗ ∩ ∞ � k=1 σ−kX∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In other words, τ is the first return map to X∗ and Σ is the domain on which τ can be iterated infinitely many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We call (Σ, τ|Σ) an induced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Given a potential φ: X → R, we introduce a parametrized family of induced potentials Φγ : Σ → R (γ ∈ R) by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='9) Φγ(x) = SR(x)φ(x) − γR(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' As shown in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1, the induced system has a countably infinite partition that conjugates the system to the countable full shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The local H¨older continuity of the induced potential Φγ and its pressure P(Φγ) are well-defined in terms of this conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Our main result is stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Theorem A (the level-2 Large Deviation Principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume there exists an induced system for which the induced potentials Φγ, γ ∈ R are locally H¨older continuous, and there exists γ0 ∈ R such that P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then {˜µn}∞ n=1 is exponentially tight and satisfies the LDP with the good rate function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let us define the rate function in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let M(X, σ) denote the set of σ- invariant elements of M(X) and let Mφ(X, σ) = {µ ∈ M(X, σ): � φdµ > −∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Define Fφ : M(X) → [−∞, 0] by Fφ(µ) = � h(µ) + � φdµ − P(φ) if µ ∈ Mφ(X, σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' −∞ otherwise, where h(µ) ∈ [0, ∞] denotes the measure-theoretic entropy of µ with respect to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since φ is acceptable and P(φ) < ∞, sup φ < ∞ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each µ ∈ Mφ(X, σ), � φdµ is finite [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='9] and we have h(µ) + � φdµ ≤ P(φ) < ∞, and so h(µ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If φ is acceptable, then we have P(φ) = sup � h(µ) + � φdµ: µ ∈ Mφ(X, σ) � , known as the variational principle [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A measure µ ∈ Mφ(X, σ) which attains this supremum is called an equilibrium state for the potential φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The rate function Iφ : M(X) → [0, ∞] in Theorem A is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='10) Iφ(µ) = − inf G∋µ sup G Fφ, LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 5 where the infimum is taken over all open subsets G of M(X) containing µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the entropy is not upper semicontinuous on M(X, σ), Iφ may not be equal to −Fφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the sequence {˜µn}∞ n=1 in Theorem A is exponentially tight, it is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By Prohorov’s theorem, it has a limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the rate function Iφ in Theorem A is the good rate function, there exists at least one minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If the minimizer is unique, we obtain a “level-2 weighted equidistribution of elements of �∞ n=1 En toward minimizers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Theorem B (level-2 weighted equidistribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume there exists an induced system for which the induced potentials Φγ, γ ∈ R are locally H¨older continuous, and there exists γ0 ∈ R such that P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume that the minimizer of the rate function Iφ is unique, denoted by µmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any bounded continuous function ˜ϕ: M(X) → R, lim n→∞ 1 Zn(φ) � x∈En exp Snφ(x) ˜ϕ(δn x) = ˜ϕ(µmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Under the assumption of Theorem A, minimizers are not always unique, and not always an equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A sufficient condition was given in [35] which ensures that minimizers are equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Taking various continuous functions ˜ϕ in Theorem B, we obtain convergences of various time averages over the elements of En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let C(X) denote the set of real-valued bounded continuous functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Corollary (Inspired by Olsen [21, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Under the assumption of Theo- rem B, assume moreover the minimizer is unique, denoted by µmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (a) For all ϕ, ψ ∈ C(X), lim n→∞ 1 Zn(φ) � x∈En exp Snφ(x) 1 n2Snϕ(x)Snψ(x) = � ϕdµmin � ψdµmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (b) For ϕ, ψ ∈ C(X) with inf ψ > 0, lim n→∞ 1 Zn(φ) � x∈En exp Snφ(x)Snϕ(x) Snψ(x) = � ϕdµmin � ψdµmin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (c) For π1, π2 ∈ C(X) and a bounded continuous function f : R → R, lim n→∞ 1 Zn(φ) � x∈En exp Snφ(x) 1 n2 n−1 � k1,k2=0 f(π1(σk1x) + π2(σk2x)) = � fd(µmin ◦ π−1 1 ⊗ µmin ◦ π−1 2 ), where ⊗ denotes the convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Apply Theorem B to the bounded continuous functions µ ∈ M(X) �→ � ϕdµ � ψdµ, µ ∈ M(X) �→ � ϕdµ/ � ψdµ, µ ∈ M(X) �→ � fd(µ ◦ π−1 1 ⊗ µ ◦ π−1 2 ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 6 HIROKI TAKAHASI 1 0 1/2 2/3 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The graph of the R´enyi map T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Our results can be applied to dynamical systems modeled by the countable full shift without Bowen’s Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The assumption in Theo- rem A can be verified, for example, for the infinite Manneville-Pomeau map [13, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2], and the two-dimensional conformal maps in [43, Section 5] related to number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Minimizers of the associated rate functions are not unique, and so Theorem B does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Further applications of different taste will be given in our forthcoming paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A prime example to which our results apply is the R´enyi map T : [0, 1) → [0, 1) given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11) T(ξ) = 1 1 − ξ − � 1 1 − ξ � , where ⌊·⌋ denotes the floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The graph of T is obtained by reversing the graph of the well-known Gauss map ξ ∈ (0, 1] → 1/ξ − ⌊1/ξ⌋ ∈ [0, 1) around the axis {ξ = 1/2}, as shown in FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The map T leaves invariant the absolutely continuous infinite measure dx/x, and x = 0 is its neutral fixed point:T(0) = 0, T ′(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The asymptotic distribution of typical orbits, in the Lebesgue measure sense, are concentrated on this neutral fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The iteration of T generates an infinite continued fraction expansion of each number ξ ∈ [0, 1) of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12) ξ = 1 − 1 d1(ξ) − 1 d2(ξ) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , where dn(ξ) = ⌊1/(1 − T n−1(ξ))⌋ + 1 ≥ 2 for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Using the infinite Markov partition {Jp}p∈N, Jp = [1 − 1/p, 1 − 1/(p + 1)) of [0, 1), one can represent T as the left shift acting on X [13, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The map (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='13) π: (xn)∞ n=1 ∈ X �→ π((xn)∞ n=1) ∈ ∞ � n=1 T −n+1(J(xn)) ⊂ [0, 1) is a well-defined homeomorphism onto its image satisfying T ◦ π = π ◦ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We consider the potential φ = − log |T ′ ◦ π|, where T ′ denotes the derivative of T which is one-sided at boundary points of the Markov partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From the mean LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 7 value theorem applied to the inverse branches of T, for any p ≥ 1 and all ξ, η ∈ Jp we have log |T ′(ξ)| |T ′(η)| ≤ 2|T(ξ) − T(η)| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In particular, φ is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since sup[p] eφ is comparable to p−2, P(βφ) < ∞ holds if and only if � p∈N p−2β is finite, which is equivalent to β > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' It is easy to see that for n ≥ 1, T n maps [0, 1/(n + 1)) diffeomorphically onto [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The mean value theorem implies limn→∞ sup[0,1/(n+1)) |(T n)′| = ∞, while |(T n)′(0)| = 1 for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' It follows that for any β > 1/2 there is no Bowen’s Gibbs state for the potential βφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Meanwhile, it is known [13] that for β > 1/2, the equilibrium state for βφ is unique, which we denote by µβφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For 1/2 < β < 1, µβφ has positive entropy and fully supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For β ≥ 1, µβφ is the unit point mass at π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let I denote the set of irrational numbers in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The set En corresponds to the set of numbers in I ∪ {0} for which the continued fraction (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12) is periodic of period n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' As in the proof of [14, Theorem 28], one can show that any number in �∞ n=1{ξ ∈ I: T n(ξ) = ξ} is a quadratic irrational, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', an irrational root of a quadratic polynomial with integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Conversely, any quadratic irrational in I has an eventually periodic continued fraction of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12), see [20, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' An induced system as in Theorem A is obtained from the first return map to the interval (1/2, 1) not containing the neutral fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From Theorems A and B we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For ξ ∈ [0, 1) and n ≥ 1, let δn ξ denote the empirical measure (1/n) �n−1 k=0 δT k(ξ) on [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any 1/2 < β ≤ 1, the sequence of Borel probability measures on M(π(X)) given by 1 Zn(βφ) � ξ∈I∪{0} T n(ξ)=ξ |(T n)′(ξ)|−βδδn ξ for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' satisfies the LDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The minimizer is unique and it is the unit point mass at µβφ◦π−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Moreover, for any bounded continuous function ˜ϕ: M(π(X)) → R we have lim n→∞ 1 Zn(βφ) � ξ∈I∪{0} T n(ξ)=ξ |(T n)′(ξ)|−β ˜ϕ(δn ξ ) = ˜ϕ(µβφ ◦ π−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The rest of this paper consists of two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 2 we verify the exponential tightness of the sequence in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) under the assumption of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 3 we complete proofs of all the theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We close with a remark on possible generalizations of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Exponential tightness The aim of this section is to verify the exponential tightness of the sequence in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1 we start with a symbolic representation of the induced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2 we introduce the notion of local Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3 we prove a main technical estimate assuming the existence of a local Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Using this 8 HIROKI TAKAHASI estimate, we verify the exponential tightness in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5 we show that the assumption of Theorem A implies the existence of a local Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Symbolic representation of the induced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For a set S and an integer j ≥ 1, let Sj denote the set of words of elements of S of word length j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We introduce an empty word ∅ and set S0 = {∅}, a∅ = a = ∅a, a∅b = ab for a, b ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We set W(S) = � j≥1 Sj, N0 = N ∪ {0} and W0(S) = W(S) ∪ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let (X∗, R) be an inducing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let N∗ = N ∩ [p∗, ∞) and N∗ = N ∩ [1, p∗ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each p ∈ N∗ and ω ∈ W0(N∗), the set � q∈N∗[pωq] is mapped by the induced map τ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) bijectively onto X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the domain X∗∩�∞ k=1 σ−kX∗ of definition of τ is partitioned into countably infinite sets of this form, the induced system τ|Σ is represented as the countable full shift over the infinite alphabet (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1) A = � � q∈N∗ [pωq]: p ∈ N∗ and ω ∈ W0(N∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' To make this statement into a rigorous one, we endow A with the discrete topology, and consider the countable full shift AN = {z = (zn)∞ n=1: zn ∈ A for every n ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We use bold letters to denote elements of W(A), and a double square bracket [[·]] to denote cylinders in AN: the n-cylinder (n ≥ 1) spanned by a = a1 · · · an ∈ An is [[a]] = {z = (zk)∞ k=1 ∈ AN : zk = ak for every k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , n}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By definition, for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , n} we have ak = � q∈N∗[pkωjkq] where pk ∈ N∗, jk ∈ N0, ωjk ∈ Njk ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let ∥a∥ denote the word length of p1ωj1p2ωj2 · · · pnωjn in W(N), namely ∥a∥ = n + j1 + · · · + jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let [a] denote the corresponding ∥a∥-cylinder in X, namely [a] = [p1ωj1p2ωj2 · · · pnωjn] ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' It is easy to check that a coding map Π: AN → Σ given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) Π: (zn)∞ n=1 ∈ AN �→ Π((zn)∞ n=1) ∈ ∞ � n=1 [z1 · · · zn] ⊂ Σ is well-defined, and is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let θ denote the left shift acting on AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Clearly we have Π ◦ θ = τ|Σ ◦ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The following notation will be frequently used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For a = a1 · · · an ∈ An as above with [a] = [p1ωj1p2ωj2 · · · pnωjn] and q ∈ N∗, let aq = p1ωj1p2ωj2 · · · pnωjnq ∈ W(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let (Σ, τ|Σ) be an induced system and let Π be the coding map in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 9 (a) For every a ∈ W(A), Π[[a]] = Σ ∩ � q∈N∗ [aq].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (b) If a, b ∈ W(A) satisfy ∥a∥ = ∥b∥, then a = b or [[a]] ∩ [[b]] = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If n ≥ 1, a ∈ An then �n−1 k=0 R ◦ τ k equals ∥a∥ on Π[[a]], which implies (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If ∥a∥ = ∥b∥ and a ̸= b then (a) implies Π[[a]] ∩ Π[[b]] = ∅, and so [[a]] ∩ [[b]] = ∅, which verifies (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Local Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R satisfy P(φ) < ∞ and let (Σ, τ|Σ) be an induced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A Borel probability measure λφ on AN is called a local Gibbs state for the potential φ associated with (Σ, τ|Σ), if there exist constants C ≥ 1, γ0 ∈ R such that for any a ∈ W(A) and any x ∈ Π[[a]] we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3) C−1 ≤ λφ[[a]] exp � S∥a∥φ(x) − γ0∥a∥ � ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We do not require the θ-invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If the context is clear, we simply call λφ a local Gibbs state (associated with (Σ, τ|Σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If λφ is a local Gibbs state, then for any a ∈ W(A), the λφ-measure of the cylinder [[a]] in AN is given (up to multiplicative constants) by the Birkhoff sum of φ along the orbit of x of length ∥a∥ and the word length ∥a∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the word length of a as a word in W(A) does not appear in the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3), λφ well captures part of the original dynamics (X, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If λφ is a local Gibbs state, the Borel probability measure λφ ◦ Π−1 on Σ is τ|Σ-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' These two measures are related as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R satisfy P(φ) < ∞, let (Σ, τ|Σ) be an induced system and let λφ be a local Gibbs state associated with (Σ, τ|Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any a ∈ W(A) we have λφ[[a]] = λφ ◦ Π−1(Σ ∩ [a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We write νφ for λφ ◦ Π−1, and {R = n} for {z ∈ Σ: R(z) = n} for each n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We have νφ{R = n} > 0 for every n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let νφ|{R=n} denote the restriction of νφ to {R = n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The measure µφ = ∞ � n=1 n−1 � k=0 νφ|{R=n} ◦ σ−k is a finite measure if and only if � Rdνφ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since {R = n} is disjoint from �n−1 k=1 σ−k(Σ) for n ≥ 2, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) µφ|Σ = ∞ � n=1 νφ|{R=n} = νφ = λφ ◦ Π−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any a ∈ W(A) we have Π[[a]] ⊂ Σ, and so (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5) λφ[[a]] = µφΠ[[a]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 10 HIROKI TAKAHASI By µφ|Σ = νφ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1(a), for any a ∈ W(A) we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6) µφΠ[[a]] = νφΠ[[a]] = � q∈N∗ νφ(Σ ∩ [aq]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let j ≥ 1 be such that a ∈ Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then σ∥a∥ and τ j coincide on Σ ∩ [a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since � q∈N∗(Σ∩[aq]) ⊂ (τ|Σ)−j(� q∈N∗[q]) = ∅ by τ(Σ) ⊂ Σ, we have νφ(� q∈N∗ Σ∩[aq]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Combining this with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6) we obtain λφ[[a]] = � q∈N∗ νφ(Σ ∩ [aq]) + � q∈N∗ νφ(Σ ∩ [aq]) = νφ(Σ ∩ [a]), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Exponential decay on partition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The next proposition pro- vides a main technical estimate under the existence of a local Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume there exist an induced system (Σ, τ|Σ) and an associated local Gibbs state λφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' There exist δ′ ∈ (0, 1/5] and n0 ≥ 1 such that if δ ∈ (0, δ′] and {Ni}∞ i=1 is a non-decreasing integer sequence such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) max N∗ ≤ N1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8) ∞ � k=Ni+1 � a∈A Π[[a]]⊂[k] λφ[[a]] ≤ δ2i for every i ≥ 1, then for every n ≥ n0 and every m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , n} we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='9) � x∈En δn x (X\\Γ)=m/n exp Snφ(x) ≤ eγ0n2nn(4δ)m 1 − 4δ , where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='10) Γ = {x = (xi)∞ i=1 ∈ X : xi ≤ Ni for every i ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3 asserts that contributions of elements of En to Zn(φ) whose orbit escape from the compact set Γ exactly m times within period n is exponentially small in m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Similar estimates were obtained in [34] under the existence of Bowen’s Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since λφ is a local Gibbs state, there exist constants C ≥ 1 and γ0 ∈ R such that for any a ∈ W(A) and any x ∈ Π[[a]] we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11) C−1 ≤ λφ[[a]] exp � S∥a∥φ(x) − γ0∥a∥ � ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For the rest of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3, we shall use the notation a ≪ b for two positive reals a, b to indicate that a/b is bounded from infinity by a constant which depends only on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If a ≪ b and b ≪ a, we shall write a ≍ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 11 The first inequality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11) will be used to bound a partial sum of Zn(φ) from above by a sum of λφ-measures of cylinders in AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Further, we will bound this sum using the product property which is a consequence of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11): (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12) λφ[[ab]] ≍ λφ[[a]]λφ[[b]] for a, b ∈ W(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let δ ∈ (0, 1/5] and let {Ni}∞ i=1 be a non-decreasing integer sequence satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let Γ = Γ({Ni}∞ i=1) be the compact subset of X given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If x ∈ En \\ �n−1 i=0 σ−i(Σ) then xi ≤ max N∗ = N1 ≤ Ni for 1 ≤ i ≤ n, and so δn x(Γ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By this and the periodicity of elements of En, for 1 ≤ m ≤ n we have � x∈En δn x (X\\Γ)=m/n exp Snφ(x) = � x∈En∩�n−1 i=0 σ−i(Σ) δn x (X\\Γ)=m/n exp Snφ(x) ≤ n � x∈En∩Σ δn x (X\\Γ)=m/n exp Snφ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='13) To bound the last sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='13), we decompose the set {x ∈ En ∩ Σ: δn x(X \\ Γ) = m/n} into subsets sharing the same itinerary up to time n, and estimate a contribution from each subset separately, and finally unify all these estimates counting the total number of possible itineraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Define a function r: X \\ Γ → N by r(x) = min{i ≥ 1: xi > Ni}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) we have Σ ⊂ X \\ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence, for each x ∈ Σ there are infinitely many i ≥ 0 with σix /∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By an itinerary of x ∈ Σ we mean two sequences {nj(x)}∞ j=1, {rj(x)}∞ j=1 in N0 given by the recursion formulas n1(x) = min{i ≥ 0: σix /∈ Γ}, and rj(x) = r(σnj(x)x), nj+1(x) = min{i ≥ nj(x) + rj(x): σix /∈ Γ} for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let x ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (a) {i ≥ 0: σix /∈ Γ} = �∞ j=1[nj(x), nj(x) + rj(x) − 1] ∩ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (b) xnj(x)+rj(x) ∈ N∗ for every j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since {Ni}∞ i=1 is non-decreasing, if x /∈ Γ then σix /∈ Γ for 0 ≤ i ≤ r(x) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This implies (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since σnj(x)x = xnj(x)+1xnj(x)+2 · · · and σnj(x)x /∈ Γ with r(σnj(x)x) = rj(x), we obtain xnj(x)+rj(x) > Nrj(x) ≥ N1, which together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) yields xnj(x)+rj(x) ∈ N∗ as in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ For each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , m} and n1 · · · nj ∈ Nj 0, r1 · · · rj ∈ Nj with n1 < · · · < nj ≤ n, we put (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='14) ∆ r1···rj n1···nj = {x ∈ En ∩ Σ: (ni(x), ri(x)) = (ni, ri) for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , j}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If δ > 0 is sufficiently small, then for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , m}, n1 · · · nj ∈ Nj 0 and r1 · · · rj ∈ Nj such that ∆ r1···rj n1···nj ̸= ∅ we have � x∈∆ r1···rj n1···nj exp Snφ(x) ≤ eγ0nδr1+···+rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 12 HIROKI TAKAHASI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We start with the case j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We introduce two sets of induced words B0 = {b ∈ W(A): ∥b∥ = n − n1 − r1 + 1, Π[[b]] ⊂ ∪∞ k=Nr1+1[k]} and D0 = {d ∈ W(A): ∥d∥ = n1 + r1 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each x ∈ ∆r1 n1 we have xn+1 = x1 ∈ N∗ by the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='14), and xn1+r1 ∈ N∗ by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence there exist d ∈ D0 and b ∈ B0 such that [d] = [x1 · · ·xn1+r1−1] and [b] = [xn1+r1 · · · xn], and so x ∈ Π[[db]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12), exp Snφ(x) ≪ eγ0nλφ[[db]] ≪ eγ0nλφ[[d]]λφ[[b]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Summing this inequality over all x ∈ ∆r1 n1 and then using � b∈B0 λφ[[b]] ≤ δ2r1 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8) and � b∈D0 λφ[[d]] ≤ 1 which follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1(b), we obtain � x∈∆r1 n1 exp Snφ(x) ≪eγ0n � d∈D0 λφ[[d]] � b∈B0 λφ[[b]] ≤ eγ0nδ2r1 ≤ eγ0nδr1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='15) provided δ is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In case m = 1 we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' To proceed, suppose m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' let j, j+1 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , m} and let n1 · · · njnj+1 ∈ Nj+1 0 , r1 · · · rjrj+1 ∈ Nj+1 be such that ∆ r1···rjrj+1 n1···njnj+1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Define Aj = {a ∈ W(A): ∥a∥ = nj + rj, Π[[a]] ∩ ∆ r1···rj+1 n1···nj+1 ̸= ∅}, Bj = {b ∈ W(A): ∥b∥ = n − nj+1 − rj+1 + 1, Π[[b]] ⊂ ∪∞ k=Nrj+1+1[k]}, Cj = {c ∈ W(A): ∥c∥ = n − nj − rj} , Dj = {d ∈ W(A): ∥d∥ = nj+1 + rj+1 − 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let a ∈ Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each c ∈ Cj we have ∥ac∥ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since X is the full shift, Π[[ac]] contains a unique element of En ∩ Σ which we denote by ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then we have ac ∈ Π[[a]] ∩ ∆ r1···rj n1···nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12), exp Snφ(ac) ≫ eγ0nλφ[[a]]λφ[[c]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This implies � x∈Π[[a]]∩∆ r1···rj n1···nj exp Snφ(x) ≫ eγ0nλφ[[a]] � c∈Cj λφ[[c]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='16) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2, for each c ∈ Cj we have λφ[[c]] = λφ ◦ Π−1(Σ ∩ [c]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the sets Σ ∩ [c], c ∈ Cj are pairwise disjoint and their union equals Σ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='17) � c∈Cj λφ[[c]] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='16) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='17) yields � x∈Π[[a]]∩∆ r1···rj n1···nj exp Snφ(x) ≫ eγ0nλφ[[a]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='18) Similarly to the case j = 1, for each x ∈ Π[[a]]∩∆ r1···rj+1 n1···nj+1 we have xn+1 = x1 ∈ N∗ by the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='14) and xnj+1+rj+1 ∈ N∗ by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence there exist LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 13 d ∈ Dj, b ∈ Bj such that [d] = [x1 · · · xnj+1+rj+1−1] and [b] = [xnj+1+rj+1 · · ·xn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We have [[d]] ⊂ [[a]] and x ∈ Π[[db]], and by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12), exp Snφ(x) ≪ eγ0nλφ[[d]]λφ[[b]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Summing this inequality over all x ∈ Π[[a]]∩∆ r1···rj+1 n1···nj+1, and then using � b∈Bj λφ[[b]] ≤ δ2rj+1 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8) and � d∈Dj [[d]]⊂[[a]] λφ[[d]] ≤ λφ[[a]] from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1(b), we obtain � x∈Π[[a]]∩∆ r1···rj+1 n1···nj+1 exp Snφ(x) ≪ eγ0n � d∈Dj [[d]]⊂[[a]] λφ[[d]] � b∈Bj λφ[[b]] ≤ eγ0nλφ[[a]]δ2rj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='19) The two estimates in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='18) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='19) yield � x∈Π[[a]]∩∆ r1···rj+1 n1···nj+1 exp Snφ(x) � x∈Π[[a]]∩∆ r1···rj n1···nj exp Snφ(x) ≤ δrj+1, provided δ is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Rearranging this inequality and summing the result over all a ∈ Aj yields � x∈∆ r1···rj+1 n1···nj+1 exp Snφ(x) ≤ δrj+1 � x∈∆ r1···rj n1···nj exp Snφ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Applying this inequality recursively and combining the final result with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='15) yields the desired inequality in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ For integers L ≥ m and s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , m}, we denote by KL,s the set of elements (n1 · · · ns, r1 · · · rs) of Ns 0×Ns such that 0 ≤ n1 < · · · < ns ≤ n and r1+· · ·+rs = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The number of ways of locating n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ns in [0, n] does not exceed ( n s ), and for each location (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ns) the number of all feasible combinations of (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , rs) with r1 +· · ·+rs = L is bounded by the number of ways of dividing L objects into s groups, not exceeding � L+s−1 s−1 � ≤ 2L+s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This yields #KL,s ≤ ( n s ) � L+s−1 s−1 � ≤ 2n2L+s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Clearly, for each x ∈ En ∩ Σ satisfying δn x(X \\ Γ) = m/n there exist L ≥ m, s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , m} and (n1 · · · ns, r1 · · · rs) ∈ KL,s such that x ∈ ∆r1···rs n1···ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If δ ∈ (0, 1/5] is sufficiently small, then together with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5 we obtain � x∈En∩Σ δn x (X\\Γ)=m/n exp Snφ(x) ≤ m � s=1 ∞ � L=m � (n1···ns,r1···rs)∈KL,s � x∈∆r1···rs n1···ns exp Snφ(x) ≤ 2n m � s=1 ∞ � L=m 2L+s−1δL ≤ 2n ∞ � L=m (4δ)L = 2n(4δ)m 1 − 4δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From this and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='13), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='9) follows and the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 14 HIROKI TAKAHASI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Verifying exponential tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We now use Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3 to show the desired exponential tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable such that P(φ) < ∞ and let (Σ, τ|Σ) be an induced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If there exists a local Gibbs state for the potential φ associated with (Σ, τ|Σ), then {˜µn}∞ n=1 is exponentially tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The argument below is an adaptation of the proof of Sanov’s theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [7]) to our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each integer ℓ ≥ 1, we fix δℓ ∈ (0, 1/5] such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='20) 1 1 − 4δℓ ∞ � m=0 e2ℓ2m(4δℓ)m ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3 and fix a non-decreasing integer sequence {Ni}∞ i=1 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8) with δ = δℓ hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We define a compact subset Γℓ = Γ({Ni}∞ i=1) of X by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='10), and set Kℓ = � ν ∈ M(X): ν(Γℓ) ≥ 1 − 1 ℓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since M(X) is a Polish space and Γℓ is a closed set, the weak* convergence µk → µ for a sequence {µk}∞ k=1 in Kℓ implies lim supk→∞ µk(Γℓ) ≤ µ(Γℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence, Kℓ is a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For an integer L ≥ 1 we define KL = ∞ � ℓ=L Kℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This set is tight, and by Prohorov’s theorem any sequence in KL has a limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence it is sequentially compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the weak* topology is metrizable with the bounded Lipschitz metric, KL is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By Chebyshev’s inequality, for n ≥ 1 we have � x∈En exp(ℓ2nδn x (X\\Γℓ))≥eℓn exp Snφ(x) ≤ e−2ℓn � x∈En δn x (X\\Γℓ)≥1/n exp � 2ℓ2nδn x(X \\ Γℓ) � exp Snφ(x) = e−2ℓn n � m=1 e2ℓ2m � x∈En δn x (X\\Γ)=m/n exp Snφ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Combining this inequality with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='20), we have � x∈En δn x /∈Kℓ exp Snφ(x) = � x∈En δn x (X\\Γℓ)≥ 1 ℓ exp Snφ(x) = � x∈En exp(ℓ2nδn x (X\\Γℓ))≥eℓn exp Snφ(x) ≤ 2nn 1 − 4δℓ eγ0ne−2ℓn n � m=0 e2ℓ2m(4δℓ)m ≤ 10 · 2nneγ0ne−2ℓn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 15 If L ≥ 1 is large enough, then ˜µn(M(X) \\ KL) ≤ ∞ � ℓ=L ˜µn(M(X) \\ Kℓ) = ∞ � ℓ=L 1 Zn(φ) � x∈En δn x /∈Kℓ exp Snφ(x) ≤ 10 · 2nneγ0n Zn(φ) ∞ � ℓ=L e−2ℓn ≤ 2neγ0n Zn(φ) e−Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Combining this with the equality limn→∞(1/n) log Zn(φ) = P(φ) < ∞ which fol- lows from the uniform continuity of φ, we obtain lim sup n→∞ 1 n log ˜µn(M(X) \\ KL) ≤ −L + log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since L ≥ 1 is an arbitrary large integer, {˜µn}∞ n=1 is exponentially tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Existence of a local Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The next proposition ensures the exis- tence of a local Gibbs state under the assumption of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume there exists an induced system (Σ, τ|Σ) for which the induced potentials Φγ, γ ∈ R associated with φ are locally H¨older continuous, and there exists γ0 ∈ R such that P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then there exists a local Gibbs state for the potential φ associated with (Σ, τ|Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Note that P(Φγ0 ◦ Π) = P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since AN is the countable full shift, the finiteness of P(Φγ0 ◦ Π) implies the summability of the potential Φγ0 ◦ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By [18, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5] together with the summability and the local H¨older continuity of Φγ0 ◦ Π, there exists a unique θ-invariant Bowen’s Gibbs state for the potential Φγ0 ◦ Π, which we denote by λφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' There exists C ≥ 1 such that for every m ≥ 1, any a ∈ Am and any z ∈ [[a]] we have C−1 ≤ λφ[[a]] exp � −P(Φγ0 ◦ Π)m + �m−1 k=0 Φγ0 ◦ Π(θkz) � ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For the series in the denominator of the fraction, for x ∈ Π[[a]] we have m−1 � k=0 Φγ0 ◦ Π(θkΠ−1(x)) = S�m−1 k=0 R(τ kx)φ(x) − γ0 m−1 � k=0 R(τ kx) = S∥a∥φ(x) − γ0∥a∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Substituting this and P(Φγ0 ◦ Π) = 0 into the denominator of the fraction implies that λφ is a local Gibbs state for the potential φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Under the assumption and notation of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7 and its proof, if γ0 = P(φ), λφ is θ-invariant and � Rd(λφ ◦ Π−1) < ∞, then the measure 1 � Rd(λφ ◦ Π−1) ∞ � n=0 (λφ ◦ Π−1)|{R>n} ◦ σ−n is in Mφ(X, σ), and it is an equilibrium state for the potential φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The normalized restriction of this measure to Σ is λφ ◦ Π−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 16 HIROKI TAKAHASI 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proofs of the main results In this section we complete the proofs of all the theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2, we prove lower and upper bounds for certain fundamental open and closed subsets of M(X) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3, we combine these bounds and the exponential tightness verified in Section 2 to complete the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4 we complete the proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In view of applications, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5 we give a sufficient condition for the vanishing of the pressure of the induced potential that is assumed in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Using this, we complete the proof of Theorem C in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lower bound for fundamental open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We introduce notations in this and the next two subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let Cu(X) denote the set of real-valued bounded uniformly continuous functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For an integer ℓ ≥ 1 we define Cu(X)ℓ = {⃗ϕ = (ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ϕℓ): ϕj ∈ Cu(X) for every j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ℓ}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For ⃗ϕ = (ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ϕℓ) ∈ Cu(X)ℓ, ⃗α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , αℓ) ∈ Rℓ and µ ∈ M(X), the expression � ⃗ϕdµ > ⃗α indicates that � ϕjdµ > αj holds for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The meaning of � ⃗ϕdµ ≥ ⃗α is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Put ∥⃗α∥ = max1≤j≤ℓ |αj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For ε ∈ R we write ⃗ε = (ε, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ε) ∈ Rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For n ≥ 1 and p1 · · · pn ∈ Nn, let p1 · · · pn denote the element of En that is contained in [p1 · · · pn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let ℓ ≥ 1, ⃗ϕ ∈ Cu(X)ℓ and ⃗α ∈ Rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let G ⊂ M(X) be an open set of the form G = � µ ∈ M(X): � ⃗ϕdµ > ⃗α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any measure µ ∈ Mφ(X, σ) ∩ G, we have lim inf n→∞ 1 n log ˜µn(G) ≥ Fφ(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By virtue of the definition of the pressure P(φ), it suffices to show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1) lim inf n→∞ 1 n log � x∈En δn x ∈G exp Snφ(x) ≥ h(µ) + � φdµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The proof of [36, Main Theorem] works verbatim to show the next lemma that approximates non-ergodic measures with ergodic ones in a particular sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any µ ∈ Mφ(X, σ) and any ε > 0 there exists an ergodic measure µ′ ∈ Mφ(X, σ) which is supported on a compact set and satisfies |h(µ) − h(µ′)| < ε, ���� � ⃗ϕdµ − � ⃗ϕdµ′ ���� < ε and ���� � φdµ − � φdµ′ ���� < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2, it suffices to show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1) for all µ ∈ Mφ(X, σ) which is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let ε > 0 be such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) � ⃗ϕdµ > ⃗α + ⃗ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 17 From the uniform continuity of each component of ⃗ϕ and that of φ, from Birkhoff’s ergodic theorem and Shannon-McMillan-Breiman’s theorem, for any sufficiently large n ≥ 1 there is a finite subset Gn of Nn such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3) ���� 1 n log #Gn − h(µ) ���� < ε 2, and for every p1 · · · pn ∈ Gn, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) sup x∈[p1···pn] ���� � ⃗ϕdδn x − � ⃗ϕdµ ���� < ε 2 and sup x∈[p1···pn] ���� 1 nSnφ(x) − � φdµ ���� < ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) and the first inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) yield � ⃗ϕdδn p1···pn > ⃗α, and the second inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) yields (1/n)Snφ(p1 · · · pn) > � φdµ − ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Therefore � x∈En δn x ∈G exp Snφ(x) ≥ � p1···pn∈Gn exp Snφ(p1 · · · pn) ≥ #Gn exp � n � φdµ − εn 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Taking logarithms and dividing by a sufficiently large n we have 1 n log � x∈En δn x ∈G exp Snφ(x) ≥ 1 n log #Gn + � φdµ − ε 2 > h(µ) + � φdµ − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Letting n → ∞ and then ε → 0 yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Upper bound for fundamental closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We proceed to upper bounds on fundamental closed sets, which are not necessarily compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let ℓ ≥ 1, ⃗ϕ ∈ Cu(X)ℓ, ⃗α ∈ Rℓ and let C ⊂ M(X) be a non-empty closed set of the form C = � µ ∈ M(X): � ⃗ϕdµ ≥ ⃗α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any ε > 0 there exists µ ∈ Mφ(X, σ) such that � ⃗ϕdµ > ⃗α − ⃗ε and lim sup n→∞ 1 n log ˜µn(C) ≤ Fφ(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A main ingredient is the next lemma, the proof of which is analogous to the standard proof of the variational principle [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For n ≥ 1 we put Dn(φ) = sup p1···pn∈Nn sup x,y∈[p1···pn] Snφ(x) − Snφ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any ε > 0 there exists n0 ≥ 1 such that if n ≥ n0 then for any non-empty finite subset Cn of Nn satisfying δn p1···pn ∈ C for every p1 · · · pn ∈ Cn, there exists a measure µ0 ∈ Mφ(X, σ) such that log � p1···pn∈Cn sup [p1···pn] exp Snφ ≤ � h(µ0) + � φdµ0 � n+Dn(φ) and � ⃗ϕdµ0 > ⃗α−⃗ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 18 HIROKI TAKAHASI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since all components of ⃗ϕ are bounded uniformly continuous, for any ε > 0 there exists n0 ≥ 1 such that if n ≥ n0 then for any p1 · · · pn ∈ Nn satisfying δn p1···pn ∈ C, � ⃗ϕdδn x ≥ ⃗α − (1/2)⃗ε holds for any x ∈ [p1 · · · pn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In what follows we assume n ≥ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Set Λ = �∞ k=0 σ−nk(� p1···pn∈Cn[p1 · · · pn]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then σn|Λ : Λ → Λ is topologically conjugate to the left shift acting on the finite full shift space CN n = {(ˆpm)∞ m=1 : ˆpm ∈ Cn for every m ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the function ˆφ = Snφ induces a continuous potential on CN n , the variational principle [4] yields sup ˆµ∈M(Λ,σn|Λ) � h(ˆµ) + � ˆφdˆµ � = lim m→∞ 1 m log � ˆp1···ˆpm∈Cm n sup [ˆp1···ˆpm] � exp m−1 � k=0 ˆφ ◦ σnk � , where M(Λ, σn|Λ) denotes the space of σn|Λ-invariant Borel probability measures endowed with the weak* topology, and h(ˆµ) denotes the measure-theoretic entropy of ˆµ ∈ M(Λ, σn|Λ) with respect to σn|Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For the series in the right-hand side, we have � ˆp1···ˆpm∈Cm n sup [ˆp1···ˆpm] exp �m−1 � k=0 ˆφ ◦ σnk � ≥ � � p1···pn∈Cn inf [p1···pn] exp Snφ �m ≥ � exp(−Dn(φ)) � p1···pn∈Cn sup [p1···pn] exp Snφ �m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Taking logarithms of both sides, dividing by m and then letting m → ∞ gives lim m→∞ 1 m log � ˆp1···ˆpm∈Cm n sup [ˆp1···ˆpm] exp �m−1 � k=0 ˆφ ◦ σnk � ≥ log � p1···pn∈Cn sup [p1···pn] exp Snφ−Dn(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Plugging this into the previous inequality yields sup ˆµ∈M(Λ,σn|Λ) � h(ˆµ) + � ˆφdˆµ � ≥ log � p1···pn∈Cn sup [p1···pn] exp Snφ − Dn(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By the compactness of the space M(Λ, σn|Λ) and the upper semicontinuity of the map ˆµ �→ h(ˆµ) + � ˆφdˆµ on this space, the supremum is attained, say by ˆµ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The measure µ0 = (1/n) �n−1 j=0 ˆµ0 ◦ σ−j is in Mφ(X, σ) and satisfies � h(µ0) + � φdµ0 � n = sup ˆµ∈M(Λ,σn|Λ) � h(ˆµ) + � ˆφdˆµ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the support of µ0 is contained in set {x ∈ X : � ⃗ϕdδn x > ⃗α − ⃗ε/2} by the choice of n0 and the assumption n ≥ n0, we obtain � ⃗ϕdµ0 > ⃗α−⃗ε as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ Continuing the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3, we note that P(φ) < ∞ implies Zn(φ) < ∞ for every n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence it is possible to choose a finite subset Cn of the countable LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 19 set � p1 · · · pn ∈ Nn : δn p1···pn ∈ C � such that � p1···pn∈Nn δn p1···pn∈C exp Snφ(p1 · · · pn) ≤ 2 � p1···pn∈Cn exp Snφ(p1 · · · pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By this inequality and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4, there exists µ0 ∈ Mφ(X, σ) such that � ⃗ϕdµ0 > ⃗α − ⃗ε and log � x∈En δn x ∈C exp Snφ(x) = log � p1···pn∈Nn δn p1···pn∈C exp Snφ(p1 · · · pn) ≤ log � p1···pn∈Cn exp Snφ(p1 · · · pn) + log 2 ≤ log � p1···pn∈Cn sup [p1···pn] exp Snφ + log 2 ≤ � h(µ0) + � φdµ0 � n + Dn(φ) + log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since φ is acceptable, it is uniformly continuous and so Dn(φ) = o(n) (n → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Dividing both sides of the above last displayed inequality by n, letting n → ∞ and combining the result with P(φ) = limn→∞(1/n) log Zn(φ) yields the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume there exists an induced system for which the induced potentials Φγ, γ ∈ R associated with φ are locally H¨older continuous, and there exists γ0 ∈ R such that P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let G be a non-empty open subset of M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since subsets of M(X) of the form � µ ∈ M(X): � ⃗ϕdµ > ⃗α � with ℓ ≥ 1, ⃗ϕ ∈ Cu(X)ℓ, ⃗α ∈ Rℓ constitute a base of the weak* topology of M(X), G is written as the union G = � λ Gλ of sets Gλ of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each Gλ, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1 gives lim inf n→∞ 1 n log ˜µn(Gλ) ≥ sup Gλ Fφ, and hence lim inf n→∞ 1 n log ˜µn(G) ≥ sup λ sup Gλ Fφ = sup G Fφ = − inf G Iφ, as required in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let C be a compact closed subset of M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let G be an arbitrary open set containing C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since M(X) is metrizable by the bounded Lipschitz metric and C is compact, we can choose ε > 0 and finitely many closed sets C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , Cs of the form Ck = � µ ∈ M(X): � ⃗ϕkdµ ≥ ⃗αk � with ℓk ≥ 1, ⃗ϕk ∈ Cu(X)ℓk, ⃗αk ∈ Rℓk, so that C ⊂ �s k=1 Ck ⊂ �s k=1 Ck(ε) ⊂ G where Ck(ε) = {µ ∈ M(X): � ⃗ϕkdµ > ⃗αk − ⃗ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4 and Fφ ≤ −Iφ, for 1 ≤ k ≤ s we have lim sup n→∞ 1 n log ˜µn(Ck) ≤ − inf Ck(ε) Iφ + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 20 HIROKI TAKAHASI Then we have lim sup n→∞ 1 n log ˜µn(C) ≤ max 1≤k≤s � − inf Ck(ε) Iφ � + ε ≤ − inf G Iφ + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since ε > 0 is arbitrary and G is an arbitrary open set containing C, it follows that lim sup n→∞ 1 n log ˜µn(C) ≤ inf G⊃C � − inf G Iφ � = − inf C Iφ, as required in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The last equality is due to the lower semicontinuity of Iφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since {˜µn}∞ n=1 is exponentially tight by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6, the standard arguments as in [7] show the upper bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) for any non-compact closed subset of M(X), and that Iφ is a good rate function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This completes the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume there exists an induced system for which the associated induced potentials Φγ, γ ∈ R are locally H¨older continuous, and there exists γ0 ∈ R such that P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume that the minimizer of the rate function Iφ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='10) is unique, denoted by µmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let {˜µnj}∞ j=1 be an arbitrary convergent subsequence of {˜µn}∞ n=1 with the limit measure ˜µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' It suffices to show that ˜µ is the unit point mass at µmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We fix a metric which generates the weak* topology on M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the rate function Iφ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='10) is the good rate function by Theorem A, for any α > 0 the level set L(α) = {µ ∈ M(X): Iφ(µ) ≤ α} is a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let µ ∈ M(X) \\ {µmin}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By the lower semicontinuity of the rate function and Iφ(µ) > 0, it is possible to take r > 0 such that the closure of the open ball Br(µ) of radius r about µ does not intersect L(Iφ(µ)/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The weak* convergence ˜µnj → ˜µ gives ˜µ(Br(µ)) ≤ lim inf j→∞ ˜µnj(Br(µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By this and the large deviations upper bound for closed sets (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2), we have ˜µ(Br(µ)) ≤ lim sup j→∞ ˜µnj(Br(µ)) ≤ lim sup j→∞ exp � −Iφ(µ)nj 2 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence, the support of ˜µ does not contain µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since µ is an arbitrary element of M(X) \\ {µmin}, it follows that ˜µ is the unit point mass at µmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This completes the proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Sufficient condition for vanishing of pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A direct check of the condition P(Φγ0) = 0 in Theorem A may be cumbersome, while checking the finiteness of induced pressures is considered to be easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In view of applications, we give a sufficient condition for the second assumption in Theorem A on the induced potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let (Σ, τ|Σ) be an induced system and let Φγ : Σ → R (γ ∈ R) be the associated family of induced potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If there exists δ ∈ R such that 0 < P(Φδ) < ∞, then there exists γ0 ∈ R such that P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let γ ∈ R and suppose P(Φγ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since ∥a∥ ≥ n for any a ∈ An and P(Φγ) is finite, for any γ′ > γ we have � a∈An sup [a] exp(S∥a∥φ − γ′∥a∥) ≤ exp(−(γ′ − γ)n) � a∈An sup [a] exp(S∥a∥φ − γ∥a∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Taking logarithms, dividing by n and letting n → ∞ yields P(Φγ′) ≤ −γ′ + γ + P(Φγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This and the assumption in the lemma together imply that both γ∞ = inf{γ ∈ R: P(Φγ) < ∞} and γ0 = inf{γ > γ∞: P(Φγ) ≥ 0} are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By the variational principle [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8], γ ∈ (γ∞, ∞) �→ P(Φγ) is convex and so continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence P(Φγ0) = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Recall that T : [0, 1) → [0, 1) denotes the R´enyi map (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For a bounded interval J ⊂ R let |J| denote its Euclidean length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In consideration of the neutral fixed point 0 of T, we set N∗ = N \\ {1}, N∗ = {1} and define an inducing scheme (X∗, R) by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6), the induced system (Σ, τ|Σ) by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8), and define an infinite alphabet A and a coding map Π by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) respectively, keeping the notation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We have Σ = (1/2, 1)∩I and A = ��∞ q=2[p1n−1q]: p ≥ 2 and n ≥ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For simplicity we will denote by C various positive constants which depend only on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each a = �∞ q=2[p1n−1q] ∈ A, we put J(a) = T −1([1/(∥a∥ + 1), 1/∥a∥)) ∩ Jp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then R equals ∥a∥ on the set Π[[a]] = π−1(J(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' There exists C ≥ 1 such that for any a ∈ W(A) we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5) C−1 ≤ |J(a)| · ∥a∥2 ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Define an induced map U : � a∈A J(a) → [0, 1) by U|J(a) = T ∥a∥|J(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Recall that φ = − log |T ′ ◦ π|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For β, γ ∈ R define Φβ,γ : Σ → R by Φβ,γ(x) = βSR(x)φ(x) − γR(x), which is the induced potential associated with βφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For all β, γ ∈ R, Φβ,γ is locally H¨older continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From the bounded distortion near the neutral fixed point [19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2], there exists C > 0 such that for any a ∈ A and all x, y ∈ [a] we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6) Φβ,γ(x) − Φβ,γ(y) ≤ Cβ|U(ξ) − U(η)| ≤ Cβ, where ξ = π(x) and η = π(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If x ̸= y then d(x, y) = e−n holds for some n ≥ 2, and there exists a1 · · ·an ∈ An such that x, y ∈ [[a1 · · · an]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since there is ρ > 1 such that inf[0,1)\\J1 |T ′| ≥ ρ, if n ≥ 3 then we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) |U(ξ) − U(η)| ≤ |Un−1(ξ) − Un−1(η)| inf�n−1 k=2 U−k(J(ak)) |(Un−2)′| ≤ ρ2−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The local H¨older continuity of Φβ,γ follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any β ∈ (1/2, 1] there exists γ ∈ R such that 0 ≤ P(Φβ,γ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 22 HIROKI TAKAHASI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6, |T(Jp)| = 1 for p ≥ 2 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5), there exists C ≥ 1 such that for a = �∞ q=2[p1n−1q] ∈ A and all β, γ ∈ R we have 1 |Jp|β sup [a] exp Φβ,γ = e−γn sup Π[a] exp(βSnφ) |Jp|β ≤ Ce−γnn−2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Summing the result over all a ∈ A, we have ∞ � n=1 � a∈A ∥a∥=n sup [a] exp Φβ,γ ≤ C ∞ � n=1 e−γnn−2β ∞ � p=2 |Jp|β ≤ C ∞ � n=1 e−γnn−2β ∞ � p=2 p−2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let β ∈ (1/2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then we have P(βφ) > 0 [13], and the above series is finite for all γ ∈ (0, P(βφ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In particular, P(Φβ,P (βφ)) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since any measure in M(X, σ) other than the unit point mass at 1∞ = 111 · · · charges Σ, the equilibrium state µβφ for the potential βφ satisfies µβφ(Σ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let ˆµβφ denote the normalized restriction of µβφ to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since τ is the first return map to Σ, ˆµβφ is τ|Σ-invariant and satisfies � Rdˆµβφ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By the variational principle for Φβ,P (βφ) and Abramov- Kac’s formula [44, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1], we obtain ∞ > P(Φβ,P (βφ)) ≥ h(ˆµβφ) + � (Φβ − P(βφ)R)dˆµβφ = (Fβφ(µβφ) + P(βφ) − P(βφ)) � Rdˆµβφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We have verified1 that 0 ≤ P(Φβ,P (βφ)) < ∞ as reqiured in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For the remaining case β = 1, we have P(φ) = 0 [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6 there is C ≥ 1 such that for n ≥ 1 and a = a1 · · · an ∈ An, C−1 ≤ ���n k=1 U−k(J(ak)) �� sup[a] exp ��n−1 k=0 Φ1,0 ◦ τ k� ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Summing this double inequalities over all a ∈ An, taking logarithms, dividing by n and then letting n → ∞ yields P(Φ1,0) = 0 as required in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6 and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7 together verify the assumption in Theorem A for the potential βφ, β ∈ (1/2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' It follows from [35] that for any β ∈ (1/2, 1], any minimizer of the rate function Iβφ is an equilibrium state for βφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the equilibrium state for βφ is unique [13], the minimizer of the rate function Iβφ is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the map π in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='13) is continuous, the assertions in Theorem C follow from Theorems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Some generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We have worked on two full shift spaces X and Σ (or AN), the latter obtained from the former via inducing (recall Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The assumption that X is the full shift has been used to construct sets of periodic points of the same period, in the proofs of exponential tightness (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5) and the lower large deviation bound (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For the induced system (Σ, τ|Σ), we have effected the thermodynamic formalism for countable Markov shifts [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 1In fact, one can show P(Φβ,P (βφ)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' See [23] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 23 The setup in this paper can be slightly generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The above-mentioned con- structions of sets of periodic points can be done even if X is replaced by a finitely primitive shift (see [18] for the definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then the induced shift space becomes finitely irreducible, for which the thermodynamic formalism works too [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This research was partially supported by the JSPS KAK- ENHI 19K21835 and 20H01811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' References [1] Aaronson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', Denker, M.' metadata={'source': 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