# Higher-dimensional Nonlinear Thermodynamic Formalism

Luis Barreira  and  Carllos Holanda  Departamento de Matemática, Instituto
Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We introduce a higher-dimensional version of the nonlinear thermodynamic
formalism introduced by Buzzi and Leplaideur, in which a potential is replaced
by a family of potentials. In particular, we establish a corresponding
variational principle and we discuss the existence, characterization, and
number of equilibrium measures for this higher-dimensional version.

###### Key words and phrases:

Nonlinear thermodynamic formalism, equilibrium measures

###### 2010 Mathematics Subject Classification:

Primary: 28D20, 37D35.

Partially supported by FCT/Portugal through UID/MAT/04459/2019. C.H. was
supported by FCT/Portugal through the grant PD/BD/135523/2018.

##  1\. Introduction

Recently, Buzzi and Leplaideur [BL20] introduced a variation of the
thermodynamic formalism, which they called _nonlinear thermodynamic
formalism_. Roughly speaking, this amounts to compute the topological pressure
replacing Birkhoff sums by images of them under a given function (that may be
nonlinear and thus the name). Our main aim is twofold:

  1. (1)

to introduce a higher-dimensional version of their notion of topological
pressure, replacing a potential by a family of potentials, and to establish a
corresponding variational principle;

  2. (2)

to discuss the existence, characterization, and number of equilibrium
measures, with special attention to the new phenomena that occur in this
higher-dimensional version.

We also give a characterization of the nonlinear pressure as a Carathéodory
dimension, which allows us to extend the notion to noncompact sets.

The most basic notion of the mathematical thermodynamic formalism is
topological pressure. It was introduced by Ruelle [Rue73] for expansive maps
and by Walters [Wal76] in the general case. For a continuous map $T\colon X\to
X$ on a compact metric space, the _topological pressure_ of a continuous
function $\varphi\colon X\to\mathbb{R}$ is defined by

$P(\varphi)=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log\sup_{E}\sum_{x\in E}\exp
S_{n}\varphi(x),$ (1)

with the supremum taken over all $(n,\varepsilon)$-separated sets $E$ and
where $S_{n}\varphi=\sum_{k=0}^{n-1}\varphi\circ T^{k}$. An important relation
between the topological pressure and the Kolmogorov–Sinai entropy is given by
the variational principle

$P(\varphi)=\sup_{\mu}\left(h_{\mu}(T)+\int_{X}\varphi\,d\mu\right),$ (2)

with the supremum taken over all $T$-invariant probability measures $\mu$ on
$X$ and where $h_{\mu}(T)$ denotes the entropy with respect to $\mu$. This was
established by Ruelle [Rue73] for expansive maps and by Walters [Wal76] in the
general case. The theory is now a broad and active independent field of study
with many connections to other areas of mathematics. We refer the reader to
the books [Bar11, Bow75, KH95, Kel98, PP90, Pes97, Rue78, Wal82] for many
developments.

Building on work on the Curie–Weiss mean-field theory in [LW17], the nonlinear
topological pressure was introduced in [BL20] as a generalization of (1) as
follows (more precisely, we give an equivalent formulation using separated
sets instead of covers). Given a continuous function
$F\colon\mathbb{R}\to\mathbb{R}$, the _nonlinear topological pressure_ of a
continuous function $\varphi\colon X\to\mathbb{R}$ is given by

$P_{F}(\varphi)=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log\sup_{E}\sum_{x\in
E}\exp\left[nF\left(\frac{S_{n}\varphi(x)}{n}\right)\right],$ (3)

with the supremum taken over all $(n,\varepsilon)$-separated sets $E$. Note
that for $F(x)=x$ we recover the classical topological pressure. Buzzi and
Leplaideur also established a version of the variational principle in (2).
Namely, under an additional assumption of abundance of ergodic measures (see
Section 2.1 for the definition), they proved that

$P_{F}(\varphi)=\sup_{\mu}\left(h_{\mu}(T)+F\biggl{(}\int_{X}\varphi\,d\mu\biggr{)}\right),$
(4)

with the supremum taken over all $T$-invariant probability measures $\mu$ on
$X$. In addition, they characterized the equilibrium measures of this
thermodynamic formalism, that is, the invariant probability measures at which
the supremum in (4) is attained, and they showed that a new type of phase
transition can occur. Namely, one may have more than one equilibrium measure,
although we still have a central limit theorem (see also [Lep15, Tha80]).

As described above, our main aim in the paper is to understand whether and how
the results in [BL20] extend to the higher-dimensional case. This corresponds
to replace the functions $F$ and $\varphi$ in (3), respectively, by a
continuous function $F\colon\mathbb{R}^{d}\to\mathbb{R}$ and by a family
$\Phi=\\{\varphi_{1},\ldots,\varphi_{d}\\}$ of continuous functions
$\varphi_{i}\colon X\to\mathbb{R}$ for $i=1,\ldots,d$. The _nonlinear
topological pressure_ of $\Phi$ is then defined by

$P_{F}(\Phi)=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log\sup_{E}\sum_{x\in
E}\exp\left[nF\biggl{(}\frac{S_{n}\varphi_{1}(x)}{n},\ldots,\frac{S_{n}\varphi_{d}(x)}{n}\biggr{)}\right],$

with the supremum taken over all $(n,\varepsilon)$-separated sets $E$.
Whenever possible, we follow a similar approach to obtain a variational
principle and to discuss the existence, characterization, and number of
equilibrium measures. We note that our setup can also be seen as a
generalization of related work on the Curie–Weiss–Potts model in [LW20].

In particular, assuming that the pair $(T,\Phi)$ has an abundance of ergodic
measures, we establish the variational principle

$P_{F}(\Phi)=\sup_{\mu}\left(h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\right),$
(5)

with the supremum taken over all $T$-invariant probability measures $\mu$ on
$X$. As in [BL20], for a certain class of pairs $(T,\Phi)$ we also
characterize the equilibrium measures, that is, the invariant probability
measures at which the supremum in (5) is attained. To a moderate extent, our
approach also uses some multifractal analysis results taken from [BSS02].
Consider the sets

$L(\Phi)=\biggl{\\{}\biggl{(}\int_{X}\varphi_{1}\,d\mu,\ldots,\int_{X}\varphi_{d}\,d\mu\biggr{)}:\mu\textrm{
is $T$-invariant}\biggr{\\}}\subset\mathbb{R}^{d}$

and

$C_{z}(\Phi)=\biggl{\\{}x\in
X:\biggl{(}\lim_{n\to\infty}\frac{S_{n}\varphi_{1}(x)}{n},\ldots,\lim_{n\to\infty}\frac{S_{n}\varphi_{d}(x)}{n}\biggr{)}=z\biggr{\\}}.$

We reduce the problem of finding equilibrium measures to the one of finding
maximizers of the function $E\colon L(\Phi)\to\mathbb{R}$ defined by

$E(z)=h(z)+F(z),$

where $h\colon L(\Phi)\to\mathbb{R}$ is the topological entropy of
$T|_{C_{z}(\Phi)}$. In fact, we show that for each
$z\in\operatorname{int}L(\Phi)$ maximizing $E$ there exists a unique
equilibrium measure $\nu_{z}$. This is actually a classical equilibrium
measure for a certain function $\psi_{z}$ that depends on the family of
functions $\Phi$. In addition, we give conditions for the uniqueness of the
equilibrium measures, both for $d=1$ and for $d>1$ (see Theorems 9 and 10).

In contrast to the case of $d=1$, we show that there might exist points at the
boundary of $L(\Phi)$ maximizing the function $E$. For these points we do not
know if there is a characterization of equilibrium measures as classical
equilibrium measures. Furthermore, we show that for $d>1$ there exist analytic
pairs $(T,\Phi)$ with uncountably many equilibrium measures, in strong
contrast to what happens for $d=1$ where any analytic pair $(T,\varphi)$ has
only finitely many equilibrium measures.

##  2\. Nonlinear topological pressure

###  2.1. Basic notions

We first recall the notion of nonlinear topological pressure introduced by
Buzzi and Leplaideur in [BL20] as an extension of the classical topological
pressure. Let $T\colon X\to X$ be a continuous map on a compact metric space
$X=(X,d)$. For each $n\in\mathbb{N}$ we consider the distance

$d_{n}(x,y)=\max\bigl{\\{}d(T^{k}(x),T^{k}(y)):k=0,\ldots,n-1\bigr{\\}}.$

Take $n\in\mathbb{N}$ and $\varepsilon>0$. A set $C\subset X$ is said to be an
_$(n,\varepsilon)$ -cover_ of $X$ if $\bigcup_{x\in C}B_{n}(x,\varepsilon)=X$,
where

$B_{n}(x,\varepsilon)=\bigl{\\{}y\in X:d_{n}(y,x)<\varepsilon\bigr{\\}}.$

Given a continuous function $F\colon\mathbb{R}\to\mathbb{R}$, the _nonlinear
topological pressure_ of a continuous function $\varphi\colon X\to\mathbb{R}$
is defined by

$P_{F}(\varphi)=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log\inf_{C}\sum_{x\in
C}\exp\left[nF\biggl{(}\frac{S_{n}\varphi(x)}{n}\biggr{)}\right],$

where $S_{n}\varphi=\sum_{k=0}^{n-1}\varphi\circ T^{k}$, with the infimum
taken over all $(n,\varepsilon)$-covers $C$.

Let $\mathcal{M}$ be the set of $T$-invariant probability measures on $X$.
Following [BL20], we say that the pair $(T,\varphi)$ has an _abundance of
ergodic measures_ if for each $\mu\in\mathcal{M}$ and $\varepsilon>0$ there
exists an ergodic measure $\nu\in\mathcal{M}$ such that

$h_{\nu}(T)\geq
h_{\mu}(T)-\varepsilon\quad\text{and}\quad\biggl{\lvert}\int_{X}\varphi\,d\nu-\int_{X}\varphi\,d\mu\biggr{\rvert}<\varepsilon.$

Assuming that $(T,\varphi)$ has an abundance of ergodic measures, they
obtained the variational principle

$P_{F}(\varphi)=\sup_{\mu\in\mathcal{M}}\biggl{\\{}h_{\mu}(T)+F\biggl{(}\int_{X}\varphi\,d\mu\biggr{)}\biggr{\\}}.$

We say that $\nu\in\mathcal{M}$ is an _equilibrium measure for $(F,\varphi)$_
with respect to $T$ if

$P_{F}(\varphi)=h_{\nu}(T)+F\biggl{(}\int_{X}\varphi\,d\nu\biggr{)}.$

In this paper we consider a higher-dimensional generalization of the nonlinear
topological pressure. Given $n\in\mathbb{N}$ and $\varepsilon>0$, a set
$E\subset X$ is said to be $(n,\varepsilon)$-_separated_ if
$d_{n}(x,y)>\varepsilon$ for every $x,y\in E$ with $x\neq y$. Since $X$ is
compact, any $(n,\varepsilon)$-separated set has finite cardinality. Let
$F\colon\mathbb{R}^{d}\to\mathbb{R}$ be a continuous function and let
$\Phi=\\{\varphi_{1},\ldots,\varphi_{d}\\}$ be a family of continuous
functions $\varphi_{i}\colon X\to\mathbb{R}$ for $i=1,\ldots,d$. The
_nonlinear topological pressure_ of the family $\Phi$ is defined by

$P_{F}(\Phi)=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log\sup_{E}\sum_{x\in
E}\exp\left[nF\biggl{(}\frac{S_{n}\varphi_{1}(x)}{n},\ldots,\frac{S_{n}\varphi_{d}(x)}{n}\biggr{)}\right],$

with the supremum taken over all $(n,\varepsilon)$-separated sets $E$. One can
easily verify that the function

$\varepsilon\mapsto\limsup_{n\to\infty}\frac{1}{n}\log\sup_{E}\sum_{x\in
E}\exp\left[nF\biggl{(}\frac{S_{n}\varphi_{1}(x)}{n},\ldots,\frac{S_{n}\varphi_{d}(x)}{n}\biggr{)}\right]$

is nondecreasing and so $P_{F}(\Phi)$ is well defined.

We also describe briefly a characterization of the nonlinear topological
pressure using $(n,\varepsilon)$-covers. Let

$\mathcal{W}_{n}(C)=\sum_{x\in
C}\exp\left[nF\biggl{(}\frac{S_{n}\varphi_{1}(x)}{n},\ldots,\frac{S_{n}\varphi_{d}(x)}{n}\biggr{)}\right].$

Following closely arguments in [Bar11], one can show that

$P_{F}(\Phi)=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log\inf_{C}\mathcal{W}_{n}(C)=\lim_{\varepsilon\to
0}\liminf_{n\to\infty}\frac{1}{n}\log\inf_{C}\mathcal{W}_{n}(C),$ (6)

with the infimum taken over all $(n,\varepsilon)$-covers $C$ of $X$.

###  2.2. Extension to noncompact sets

Based on work of Pesin and Pitskel’ in [PP84], we give a characterization of
the nonlinear topological pressure as a Carathéodory dimension. This allows
one to extend the notion to noncompact sets.

We continue to consider a continuous map $T\colon X\to X$ on a compact metric
space. Given a finite open cover $\mathcal{U}$ of $X$, for each
$n\in\mathbb{N}$ let $\mathcal{X}_{n}$ be the set of strings
$U=(U_{1},\ldots,U_{n})$ with $U_{i}\in\mathcal{U}$ for $i=1,\ldots,n$. We
write $l(U)=n$ and we define

$X(U)=\bigl{\\{}x\in X:T^{k-1}\in U_{k}\text{ for }k=1,\ldots,l(U)\bigr{\\}}.$

We say that $\Gamma\subset\bigcup_{n\in\mathbb{N}}\mathcal{X}_{n}$ _covers_ a
set $Z\subset X$ if $Z\subset\bigcup_{U\in\Gamma}X(U)$.

Given a family of continuous functions
$\Phi=\\{\varphi_{1},\ldots,\varphi_{d}\\}$, for each $n\in\mathbb{N}$ we
define $S_{n}\Phi=(S_{n}\varphi_{1},\ldots,S_{n}\varphi_{d})$. Moreover, given
a function $F\colon\mathbb{R}^{d}\to\mathbb{R}$, for each
$U\in\mathcal{X}_{n}$ let

$F_{\Phi}(U)=\begin{cases}\sup_{X(U)}nF\left(\frac{1}{n}S_{n}\Phi\right)&\text{if
}X(U)\neq\emptyset,\\\ -\infty&\text{if }X(U)=\emptyset.\end{cases}$

Finally, given a set $Z\subset X$ and a number $\alpha\in\mathbb{R}$, we
define

$M_{Z}(\alpha,\Phi,\mathcal{U})=\lim_{n\to\infty}\inf_{\Gamma}\sum_{U\in\Gamma}\exp(-\alpha
l(U)+F_{\Phi}(U)),$

with the infimum taken over all $\Gamma\subset\bigcup_{k\geq
n}\mathcal{X}_{k}$ covering $Z$ and with the convention that
$\exp(-\infty)=0$. One can easily verify that the map $\alpha\mapsto
M_{Z}(\alpha,\Phi,\mathcal{U})$ goes from $+\infty$ to zero at a unique
$\alpha\in\mathbb{R}$ and so one can define

$P_{F}(Z,\Phi,\mathcal{U})=\inf\bigl{\\{}\alpha\in\mathbb{R}:M_{Z}(\alpha,\Phi,\mathcal{U})=0\bigr{\\}}.$

One can proceed as in the proof of Theorem 2.2.1 in [Bar11] to show that the
limit $P_{F}(Z,\Phi)=\lim_{\operatorname{diam}\mathcal{U}\to
0}P_{F}(Z,\Phi,\mathcal{U})$ exists.

######  Theorem 1.

We have $P_{F}(\Phi)=P_{F}(X,\Phi)$.

###### Proof.

The proof is obtained modifying arguments in Section 4.2.3 of [Bar11] and so
we only give a brief sketch. Given a finite open cover $\mathcal{U}$ of $X$,
we define

$Z_{n}(\Phi,\mathcal{U})=\inf_{\Gamma}\sum_{U\in\Gamma}\exp F_{\Phi}(U),$

with the infimum taken over all $\Gamma\subset\mathcal{X}_{n}$ covering $X$.
Given $\Gamma_{1}\subset\mathcal{X}_{n_{1}}$ and
$\Gamma_{2}\subset\mathcal{X}_{n_{2}}$, let

$\Gamma^{\prime}=\bigl{\\{}UV:U\in\Gamma_{1}\text{ and
}V\in\Gamma_{2}\bigr{\\}}.$

Note that if $\Gamma_{1}$ and $\Gamma_{2}$ cover $X$, then $\Gamma^{\prime}$
also covers $X$. Moreover,

$F_{\Phi}(UV)\leq F_{\Phi}(U)+F_{\Phi}(V)$

for each $UV\in\Gamma^{\prime}$. We have

$\begin{split}Z_{n_{1}+n_{2}}(\Phi,\mathcal{U})&\leq\sum_{UV\in\Gamma^{\prime}}\exp
F_{\Phi}(UV)\\\ &\leq\sum_{U\in\Gamma_{1}}\exp
F_{\Phi}(U)\sum_{V\in\Gamma_{2}}\exp F_{\Phi}(V)\end{split}$

and so

$Z_{n_{1}+n_{2}}(\Phi,\mathcal{U})\leq
Z_{n_{1}}(\Phi,\mathcal{U})Z_{n_{2}}(\Phi,\mathcal{U}).$

Therefore, one can define

$Z(\Phi,\mathcal{U})=\lim_{n\to\infty}\frac{1}{n}\log
Z_{n}(\Phi,\mathcal{U}).$

Finally, it follows as in Lemmas 2.2.5 and 2.2.6 in [Bar11] that

$\lim_{\operatorname{diam}\mathcal{U}\to 0}Z(\Phi,\mathcal{U})=P_{F}(X,\Phi)$

and

$P_{F}(\Phi)=\lim_{\operatorname{diam}\mathcal{U}\to 0}Z(\Phi,\mathcal{U}).$

This yields the desired result. ∎

##  3\. Variational Principle

In this section we establish a variational principle for the nonlinear
topological pressure.

Let $T\colon X\to X$ be a continuous map on a compact metric space and let
$\Phi=\\{\varphi_{1},\ldots,\varphi_{d}\\}$ be a family of continuous
functions. We say that the pair $(T,\Phi)$ has an _abundance of ergodic
measures_ if for each $\mu\in\mathcal{M}$ and $\varepsilon>0$ there exists an
ergodic measure $\nu\in\mathcal{M}$ such that $h_{\nu}(T)\geq
h_{\mu}(T)-\varepsilon$ and

$\biggl{\lvert}\int_{X}\varphi_{i}\,d\nu-\int_{X}\varphi_{i}\,d\mu\biggr{\rvert}<\varepsilon\quad\text{for}\
i=1,\ldots,d.$

Moreover, we say that $T$ has _entropy density of ergodic measures_ if for
every $\mu\in\mathcal{M}$ there exist ergodic measures $\nu_{n}\in\mathcal{M}$
for $n\in\mathbb{N}$ such that $\nu_{n}\to\mu$ in the weak∗ topology and
$h_{\nu_{n}}(T)\to h_{\mu}(T)$ when $n\to\infty$. Note that if $T$ has entropy
density of ergodic measures, then the pair $(T,\Phi)$ has an abundance of
ergodic measures for any family of continuous functions $\Phi$.

In order to give plenty examples of abundance of ergodic measures we first
recall a few notions. Given $\delta>0$, we say that $T$ has _weak
specification at scale $\delta$_ if there exists $\tau\in\mathbb{N}$ such that
for every $(x_{1},n_{1}),\ldots,(x_{k},n_{k})\in X\times\mathbb{N}$ there are
$y\in X$ and times $\tau_{1},\ldots,\tau_{k-1}\in\mathbb{N}$ such that
$\tau_{i}\leq\tau$ and

$d_{n_{i}}(T^{s_{i-1}+\tau_{i-1}}(y),x_{i})<\delta\quad\text{for}\
i=1,\ldots,k,$

where $s_{i}=\sum_{i=1}^{i}n_{i}+\sum_{i=1}^{i-1}\tau_{i}$ with
$n_{0}=\tau_{0}=0$. When one can take $\tau_{i}=\tau$ for $i=1,\ldots,k-1$, we
say that $T$ has _specification at scale $\delta$_. Finally, we say that $T$
has _weak specification_ if it has weak specification at every scale $\delta$
and, analogously, we say that $T$ has _specification_ if it has specification
at every scale $\delta$.

It was shown in [CLT20] that a continuous map $T\colon X\to X$ on a compact
metric space with the weak specification property such that the entropy map
$\mu\mapsto h_{\mu}(T)$ is upper semicontinuous, has entropy density of
ergodic measures. In particular, this implies that the pair $(T,\Phi)$ has
abundance of ergodic measures for any family of continuous functions $\Phi$.
Some examples of maps with abundance of ergodic measures include expansive
maps with specification or with weak specification, topologically transitive
locally maximal hyperbolic sets for diffeomorphisms, and transitive
topological Markov chains.

The following theorem establishes a variational principle for the nonlinear
topological pressure.

######  Theorem 2.

Let $T\colon X\to X$ be a continuous map on a compact metric space and let
$\Phi=\\{\varphi_{1},\ldots,\varphi_{d}\\}$ be a family of continuous
functions. Given a continuous function $F\colon\mathbb{R}^{d}\to\mathbb{R}$,
if the pair $(T,\Phi)$ has an abundance of ergodic measures, then

$P_{F}(\Phi)=\sup_{\mu\in\mathcal{M}}\biggl{\\{}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\\}},$
(7)

where
$\int_{X}\Phi\,d\mu=\big{(}\int_{X}\varphi_{1}\,d\mu,\ldots,\int_{X}\varphi_{d}\,d\mu\big{)}$.

###### Proof.

To the possible extent we follow arguments in [BL20] for a single function. We
divide the proof into two lemmas.

######  Lemma 1.

We have

$P_{F}(\Phi)\geq\sup_{\mu\in\mathcal{M}}\biggl{\\{}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\\}}.$

###### Proof of the lemma.

Given $r>0$, since $X$ is compact there exist $\delta,\varepsilon>0$ such that

$|\varphi_{i}(x)-\varphi_{i}(y)|<\delta/2\quad\text{whenever}\
d(x,y)<\varepsilon$

for $i=1,\ldots,d$ and

$|F(v)-F(w)|<r\quad\text{whenever}\ \|v-w\|<\delta.$

For definiteness we shall take the $\ell^{\infty}$ norm on $\mathbb{R}^{d}$.
Now let $\mu\in\mathcal{M}$ be an ergodic measure. By Birkhoff’s ergodic
theorem and the Shannon–McMillan–Breiman theorem, together with Egorov’s
theorem, there exist a set $A\subset X$ of measure $\mu(A)>1-r$ and
$N\in\mathbb{N}$ such that

$\biggl{\lvert}\frac{S_{n}\varphi_{i}(x)}{n}-\int_{X}\varphi_{i}\,d\mu\biggr{\rvert}<\delta/2$
(8)

for all $i=1,\ldots,d$ and

$\biggl{\lvert}\frac{1}{n}\log\mu(B_{n}(x,2\varepsilon))+h_{\mu}(T)\biggr{\rvert}<r,$
(9)

for $x\in A$ and $n>N$.

Now let $C$ be an $(n,\varepsilon)$-cover with $\mathcal{W}_{n}(C)$ minimal
and let $D\subset C$ be a minimal $(n,\varepsilon)$-cover of $A$. For each
$x\in D$, the ball $B_{n}(x,\varepsilon)$ intersects $A$ at some point $y$
(otherwise one could discard the point $x$ in $D$). Note that

$d(T^{k}(x),T^{k}(y))<\varepsilon\quad\text{for}\ k=0,\ldots,n-1.$

Hence, it follows from (8) that

$\begin{split}\biggl{\lvert}\frac{S_{n}\varphi_{i}(x)}{n}-\int_{X}\varphi_{i}\,d\mu\biggr{\rvert}&\leq\frac{1}{n}\lvert
S_{n}\varphi_{i}(x)-S_{n}\varphi_{i}(y)\rvert\\\
&\phantom{\leq}+\biggl{\lvert}\frac{S_{n}\varphi_{i}(y)}{n}-\int_{X}\varphi_{i}\,d\mu\biggr{\rvert}<\delta/2+\delta/2=\delta\end{split}$

for $i=1,\ldots,d$ and so

$\biggl{\lvert}F\biggl{(}\frac{S_{n}\varphi_{1}(x)}{n},\ldots,\frac{S_{n}\varphi_{d}(x)}{n}\biggr{)}-F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\rvert}<r.$

Moreover, $B_{n}(x,\varepsilon)\subset B_{n}(y,2\varepsilon)$ and so it
follows from (9) that

$1-r<\mu(A)\leq|D|\max_{x\in
D}\mu(B_{n}(x,\varepsilon))\leq|D|e^{-n(h_{\mu}(T)-r)},$

where $|D|$ denotes the cardinality of $D$. Therefore,

$\begin{split}\mathcal{W}_{n}(C)&\geq|D|\exp\bigg{[}nF\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}-r\bigg{]}\\\
&\geq(1-r)\exp[n(h_{\mu}(T)-r)]\exp\bigg{[}nF\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}-r\bigg{]}\end{split}$

for any sufficiently large $n\in\mathbb{N}$. It follows from (6) that

$P_{F}(\Phi)\geq h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}-2r.$

Finally, by the arbitrariness of $r>0$ we obtain

$P_{F}(\Phi)\geq h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}.$ (10)

Now we consider an arbitrary measure $\nu\in\mathcal{M}$. Since $(T,\Phi)$ has
an abundance of ergodic measures and $F$ is continuous, for each
$\varepsilon>0$ there exists an ergodic measure $\mu\in\mathcal{M}$ such that

$\biggl{\lvert}F\biggl{(}\int_{X}\Phi\,d\nu\biggr{)}-F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\rvert}<\varepsilon\quad\text{and}\quad
h_{\mu}(T)\geq h_{\nu}(T)-\varepsilon.$

By (10) we have

$P_{F}(\Phi)\geq h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\geq
h_{\nu}(T)+F\biggl{(}\int_{X}\Phi\,d\nu\biggr{)}-2\varepsilon$ (11)

and the desired result follows from the arbitrariness of $\varepsilon$. ∎

We also obtain the reverse inequality.

######  Lemma 2.

We have

$P_{F}(\Phi)\leq\sup_{\mu\in\mathcal{M}}\biggl{\\{}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\\}}.$

###### Proof of the lemma.

Given $r>0$, take $\varepsilon>0$ such that

$\limsup_{n\to\infty}\frac{1}{n}\log\inf_{C}\mathcal{W}_{n}(C)>P_{F}(\Phi)-r,$

with the infimum taken over all $(n,\varepsilon)$-covers $C$. Since each
$(n,\varepsilon)$-separated set $E_{n}$ is an $(n,\varepsilon)$-cover, we have

$\limsup_{n\to\infty}\frac{1}{n}\log\mathcal{W}_{n}(E_{n})>P_{F}(\Phi)-r$

and there exists a diverging subsequence $(n_{k})_{k\in\mathbb{N}}$ such that

$\mathcal{W}_{n_{k}}(E_{n_{k}})\geq\exp[n_{k}(P_{F}(\Phi)-2r)]\quad\text{for}\
k\in\mathbb{N}.$ (12)

We cover the compact set $\Phi(X)$ by balls $B(z_{i},r_{i})$ for
$i=1,\ldots,L$ such that $|F(z)-F(z_{i})|<r$ for all $z\in B(z_{i},r_{i})$ and
$i=1,\ldots,L$. Now let

$\Lambda^{i}_{k}=\biggl{\\{}x\in
E_{n_{k}}:\biggl{(}\frac{S_{n_{k}}\varphi_{1}(x)}{n_{k}},\ldots,\frac{S_{n_{k}}\varphi_{d}(x)}{n_{k}}\biggr{)}\in
B(z_{i},r_{i})\biggr{\\}}.$

Note that

$\mathcal{W}_{n_{k}}(E_{n_{k}})\leq\sum_{i=1}^{L}\mathcal{W}_{n_{k}}(\Lambda^{i}_{k})\leq
L\mathcal{W}_{n_{k}}(\Lambda^{i}_{k})\quad\text{for some
$i\in\\{1,\ldots,L\\}$}$

and so it follows from (12) that

$\begin{split}\exp[n_{k}(P_{F}(\Phi)-2r)]&\leq\mathcal{W}_{n_{k}}(E_{n_{k}})\\\
&\leq L\mathcal{W}_{n_{k}}(\Lambda^{i}_{k})\leq
L|\Lambda^{i}_{k}|\exp[n_{k}(F(z_{i})+r)].\end{split}$

This implies that

$|\Lambda^{i}_{k}|\geq\exp[n_{k}(P_{F}(\Phi)-F(z_{i})-4r)]$ (13)

for any sufficiently large $k$. Proceeding as in the proof of the variational
principle in [Mis76], we also consider the measures

$\mu^{i}_{k}=\frac{1}{|\Lambda^{i}_{k}|}\sum_{x\in\Lambda^{i}_{k}}\delta_{x}\quad\text{and}\quad\nu^{i}_{k}=\frac{1}{n_{k}}\sum_{j=0}^{n_{k}-1}\mu^{i}_{k}T^{-j}.$

Without loss of generality, one can assume that $\nu^{i}_{k}$ converges to a
$T$-invariant measure $\mu^{i}$ in the weak∗ topology satisfying

$h_{\mu^{i}}(T)\geq\limsup_{n_{k}\to\infty}\frac{1}{n_{k}}\log|\Lambda^{i}_{k}|.$
(14)

By the definition of $\nu^{i}_{k}$ we have

$\begin{split}\int_{X}\Phi\,d\mu^{i}&=\lim_{k\to\infty}\int_{X}\Phi\,d\nu^{i}_{k}\\\
&=\lim_{k\to\infty}\biggl{(}\int_{X}\frac{S_{n_{k}}\varphi_{1}}{n_{k}}\,d\mu^{i}_{k},\ldots,\int_{X}\frac{S_{n_{k}}\varphi_{d}}{n_{k}}\,d\mu^{i}_{k}\biggr{)}\in\overline{B(z_{i},r_{i})}.\end{split}$

Hence, by (13) and (14) we obtain

$\begin{split}h_{\mu^{i}}(T)+F\biggl{(}\int_{X}\Phi\,d\mu^{i}\biggr{)}&\geq
P_{F}(\Phi)-F(z_{i})-4r+F(z_{i})-r\\\ &=P_{F}(\Phi)-5r.\end{split}$

The desired result follows from the arbitrariness of $r$. ∎

Lemmas 1 and 2 establish the statement in the theorem. ∎

For a general continuous map $T$, we obtain a variational principle for an
arbitrary convex function $F$.

######  Theorem 3.

Let $T\colon X\to X$ be a continuous map on a compact metric space and let
$\Phi=\\{\varphi_{1},\ldots,\varphi_{d}\\}$ be a family of continuous
functions. If $F\colon\mathbb{R}^{d}\to\mathbb{R}$ is a convex continuous
function, then identity (7) holds.

###### Proof.

It follows from the proof of Lemma 1 (see (10)) that

$P_{F}(\Phi)\geq h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}$

for every ergodic measure $\mu\in\mathcal{M}$. Now let $\nu\in\mathcal{M}$ be
an arbitrary measure and consider its ergodic decomposition with respect to
$T$. It is described by a probability measure $\tau$ on $\mathcal{M}$ that is
concentrated on the subset of ergodic measures $\mathcal{M}_{\rm erg}$. We
recall that for every bounded measurable function $\psi\colon X\to\mathbb{R}$
we have

$\int_{X}\psi\,d\nu=\int_{\mathcal{M}}\biggl{(}\int_{X}\psi\,d\mu\biggr{)}\,d\tau(\mu).$

For a convex function $F$ one can use Jensen’s inequality to obtain

$\begin{split}F\biggl{(}\int_{X}\Phi\,d\nu\biggr{)}&=F\biggl{(}\int_{\mathcal{M}}\biggl{(}\int_{X}\varphi_{1}\,d\mu\biggr{)}d\tau(\mu),\ldots,\int_{\mathcal{M}}\biggl{(}\int_{X}\varphi_{d}\,d\mu\biggr{)}d\tau(\mu)\biggr{)}\\\
&\leq\int_{\mathcal{M}}F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}d\tau(\mu).\end{split}$

Hence,

$h_{\nu}(T)+F\biggl{(}\int_{X}\Phi\,d\nu\biggr{)}\leq\int_{\mathcal{M}}\biggl{[}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{]}d\tau(\mu)\leq
P_{F}(\Phi).$ (15)

The desired result follows now readily from Lemma 2 whose proof does not
require an abundance of ergodic measures. ∎

We also obtain a variational principle over the ergodic measures.

######  Corollary 4.

Let $T\colon X\to X$ be a continuous map on a compact metric space, let
$\Phi=\\{\varphi_{1},\ldots,\varphi_{d}\\}$ be a family of continuous
functions, and let $F\colon\mathbb{R}^{d}\to\mathbb{R}$ be a continuous
function. If the pair $(T,\Phi)$ has an abundance of ergodic measures or $F$
is convex, then

$P_{F}(\Phi)=\sup_{\mu\in\mathcal{M}_{\rm
erg}}\biggl{\\{}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\\}}.$
(16)

###### Proof.

Since $\mathcal{M}_{\rm erg}\subset\mathcal{M}$, we have

$\sup_{\mu\in\mathcal{M}}\biggl{\\{}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\\}}\geq\sup_{\mu\in\mathcal{M}_{\rm
erg}}\biggl{\\{}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\\}}.$

Now we establish the reverse inequality. Assume that the pair $(T,\Phi)$ has
an abundance of ergodic measures. It follows from (11) that for each
$\nu\in\mathcal{M}$ and $\varepsilon>0$, there exists an ergodic measure
$\mu\in\mathcal{M}$ such that

$h_{\mu}(T)+F\bigg{(}\int_{X}\Phi\,d\mu\bigg{)}\geq
h_{\nu}(T)+F\bigg{(}\int_{X}\Phi d\nu\bigg{)}-\varepsilon.$

Since $\varepsilon$ is arbitrary, this readily implies that

$\sup_{\mu\in\mathcal{M}}\biggl{\\{}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\\}}\leq\sup_{\nu\in\mathcal{M}_{\rm
erg}}\biggl{\\{}h_{\nu}(T)+F\biggl{(}\int_{X}\Phi\,d\nu\biggr{)}\biggr{\\}}.$

Finally, it follows from Theorem 2 that identity (16) holds.

Now assume that $F$ is convex. It follows from (15) that

$h_{\nu}(T)+F\biggl{(}\int_{X}\Phi\,d\nu\biggr{)}\leq\sup_{\mu\in\mathcal{M}_{\rm
erg}}\biggl{\\{}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\\}}$

for each $\nu\in\mathcal{M}$. Therefore,

$\sup_{\nu\in\mathcal{M}}\biggl{\\{}h_{\nu}(T)+F\biggl{(}\int_{X}\Phi\,d\nu\biggr{)}\biggr{\\}}\leq\sup_{\mu\in\mathcal{M}_{\rm
erg}}\biggl{\\{}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\\}}$

and applying Theorem 3 we also obtain identity (16). ∎

##  4\. Equilibrium measures: existence and characterization

In this section we consider the problem of characterizing the equilibrium
measures of the nonlinear topological pressure.

###  4.1. Existence of equilibrium measures

In view of Theorem 2, we say that $\nu\in\mathcal{M}$ is an _equilibrium
measure for $(F,\Phi)$_ with respect to $T$ if

$P_{F}(\varphi)=h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}.$

We first formulate a result on the existence of equilibrium measures.

######  Theorem 5.

Let $T\colon X\to X$ be a continuous map on a compact metric space such that
the map $\mu\mapsto h_{\mu}(T)$ is upper semicontinuous, let
$\Phi=\\{\varphi_{1},\ldots,\varphi_{d}\\}$ be a family of continuous
functions, and let $F\colon\mathbb{R}^{d}\to\mathbb{R}$ be a continuous
function. If the pair $(T,\Phi)$ has an abundance of ergodic measures or $F$
is convex, then there exists at least one equilibrium measure for $(F,\Phi)$.

###### Proof.

Since the map $\mu\mapsto h_{\mu}(T)$ is upper semicontinuous, $F$ is
continuous and the map $\mu\mapsto\int_{X}\psi\,d\mu$ is continuous for each
continuous function $\psi\colon X\to\mathbb{R}$, we conclude that $\mu\mapsto
h_{\mu}(T)+F(\int_{X}\Phi\,d\mu)$ is upper semicontinuous. Together with the
compactness of $\mathcal{M}$, this guarantees that there exists a measure
$\mu_{\Phi}\in\mathcal{M}$ such that

$\sup_{\mu\in\mathcal{M}}\biggl{\\{}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\\}}=h_{\mu_{\Phi}}(T)+F\biggl{(}\int_{X}\Phi\,d\mu_{\Phi}\biggr{)}.$

Hence, it follows from the variational principles in Theorems 2 and 3 that
$\mu_{\Phi}$ is an equilibrium measure for $(F,\Phi)$. ∎

In some cases one can pass to the one-dimensional setting of the nonlinear
thermodynamic formalism.

######  Example 1.

Consider the function $F\colon\mathbb{R}^{d}\to\mathbb{R}$ defined by

$F(z_{1},\ldots,z_{d})=f(\alpha_{1}z_{1}+\cdots+\alpha_{d}z_{d}),$

where $f\colon\mathbb{R}\to\mathbb{R}$ is a continuous function and
$\alpha_{j}\in\mathbb{R}$ for $j=1,\ldots,d$. Then

$F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}=F\biggl{(}\int_{X}\varphi_{1}\,d\mu,\ldots,\int_{X}\varphi_{d}\,d\mu\biggr{)}=f\biggl{(}\int_{X}\varphi\,d\mu\biggr{)}$

for every $\mu\in\mathcal{M}$, where

$\varphi=\alpha_{1}\varphi_{1}+\cdots+\alpha_{d}\varphi_{d}.$

Moreover, $P_{F}(\Phi)=P_{f}(\varphi)$ and this implies that $(F,\Phi)$ and
$(f,\varphi)$ have the same equilibrium measures. In other words, for a
function $F$ as above the study of equilibrium measures can be reduced to the
case when $d=1$.

Of course, in general the continuous function $F$ can be much more
complicated. For instance, the Potts model involves the study of the
topological pressure for
$F(z_{1},\ldots,z_{2})=(z_{1}^{2}+\cdots+z_{d}^{2})^{1/2}$.

For the following example, we recall the notion of cohomology. We say that a
function $\varphi\colon X\to\mathbb{R}$ is _cohomologous_ to a function
$\psi\colon X\to\mathbb{R}$ (with respect to $T$) if there exists a measurable
bounded function $q\colon X\to\mathbb{R}$ such that

$\varphi(x)=\psi(x)+q(T(x))-q(x)\quad\text{for}\ x\in X.$

######  Example 2 (Reduction of dimension via cohomology).

Let $T$ be a continuous map on a compact metric space and let
$\Phi=\\{\varphi_{1},\ldots,\varphi_{d}\\}$ be a family of continuous
functions such that the pair $(T,\Phi)$ has an abundance of ergodic measures.
Let $F\colon\mathbb{R}^{d}\to\mathbb{R}$ be a continuous function and assume
that $\varphi_{1}$ is cohomologous to $\varphi_{d}$. This implies that
$\int_{X}\varphi_{1}\,d\mu=\int_{X}\varphi_{d}\,d\mu$ for every
$\mu\in\mathcal{M}$. Therefore,

$\begin{split}F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}&=F\biggl{(}\int_{X}\varphi_{1}\,d\mu,\int_{X}\varphi_{2}\,d\mu,\ldots,\int_{X}\varphi_{1}\,d\mu\biggr{)}\\\
&=G\biggl{(}\int_{X}\varphi_{1}\,d\mu,\ldots,\int_{X}\varphi_{d-1}\,d\mu\biggr{)}\end{split}$

for every $\mu\in\mathcal{M}$, where

$G(z_{1},\ldots,z_{d-1})=F(z_{1},z_{2},\ldots,z_{d-1},z_{1})$

for each $(z_{1},\ldots,z_{d})\in\mathbb{R}^{d}$. The cohomology assumption
also implies that

$\|S_{n}\varphi_{1}-S_{n}\varphi_{d}\|_{\infty}/n\to 0\quad\text{when}\
n\to\infty.$

Together with the continuity of $F$, this implies that
$P_{F}(\Phi)=P_{G}(\Psi)$, where
$\Psi=\\{\varphi_{1},\ldots,\varphi_{d-1}\\}$. Hence, the pairs $(F,\Phi)$ and
$(G,\Psi)$ have the same equilibrium measures. More generally, one could
consider any cohomology relation between two or more functions in $\Phi$ or
even more cohomology relations in order to reduce further the dimension of the
problem.

######  Example 3 (Reduction to the classical case via cohomology).

Let $T$ be a continuous map on a compact metric space and let
$\Phi=\\{\varphi_{1},\varphi_{2}\\}$ be a pair of continuous functions such
that $(T,\Phi)$ has an abundance of ergodic measures. Moreover, assume that
$\varphi_{1}$ is cohomologous to $\varphi_{2}$ and consider the function
$F\colon\mathbb{R}^{2}\to\mathbb{R}$ given by
$F(z_{1},z_{2})=(z_{1}^{3}+z_{2}^{3})^{1/3}$. This implies that
$\int_{X}\varphi_{1}\,d\mu=\int_{X}\varphi_{2}\,d\mu$ for every
$\mu\in\mathcal{M}$ and so

$\begin{split}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}&=h_{\mu}(T)+F\biggl{(}\int_{X}\varphi_{1}\,d\mu,\int_{X}\varphi_{1}\,d\mu\biggr{)}\\\
&=h_{\mu}(T)+\int_{X}2^{1/3}\varphi_{1}\,d\mu\end{split}$

for every $\mu\in\mathcal{M}$. Letting $\psi=2^{1/3}\varphi_{1}$, it follows
from the definitions that $P_{F}(\Phi)=P(\psi)$, where $P$ denotes the
classical topological pressure. Hence, $\nu$ is an equilibrium measure for
$(F,\Phi)$ if and only if $\nu$ is an equilibrium measure for $\psi$.

Recall that a continuous function $\varphi\colon X\to\mathbb{R}$ is said to
have the _Bowen property_ if for every $K>0$ there exists $\varepsilon>0$ such
that whenever

$d(T^{k}(x),T^{k}(y))<\varepsilon\quad\text{for}\ k=0,1,\ldots,n-1$

we have $|S_{n}\varphi(x)-S_{n}\varphi(y)|\leq K$. If $T\colon X\to X$ is an
expansive map with specification and $\varphi_{1}$ (or $\varphi_{2}$) is a
continuous function with the Bowen property, then there exists a unique
equilibrium measure $\mu_{\psi}$ for $\psi$ (see [Bow74]). Therefore,
$\mu_{\psi}$ is also the unique equilibrium measure for $(F,\Phi)$.

We observe that this example can be easily generalized to the case when

$F(z_{1},\ldots,z_{d})^{n}=H_{n}(z_{1},\ldots,z_{d}),$

where $H_{n}$ is a homogeneous polynomial of degree $n$, together with some
cohomology relations between the functions in $\Phi$.

###  4.2. Characterization of equilibrium measures

Now we consider the problem of characterizing the equilibrium measures. Given
a pair $(T,\Phi)$, we consider the set

$L(\Phi)=\biggl{\\{}\int_{X}\Phi\,d\mu:\mu\in\mathcal{M}\biggr{\\}}.$

Since the map $\mu\mapsto\int_{X}\psi\,d\mu$ is continuous for each continuous
function $\psi\colon X\to\mathbb{R}$ and $\mathcal{M}$ is compact and
connected, the set $L(\Phi)$ is a compact and connected subset of
$\mathbb{R}^{d}$. For each $z\in\mathbb{R}^{d}$, we also consider the level
sets

$\mathcal{M}(z)=\biggl{\\{}\mu\in\mathcal{M}:\int_{X}\Phi\,d\mu=z\biggr{\\}}$

and

$C_{z}(\Phi)=\biggl{\\{}x\in
X:\lim_{n\to\infty}\frac{S_{n}\Phi(x)}{n}=z\biggr{\\}}.$ (17)

Following closely [BL20], we say that the pair $(T,\Phi)$ is $C^{r}$-_regular_
(for some $2\leq r\leq\omega$, where $\omega$ refers to the analytic case) if
the following holds:

  1. (1)

each function in $\operatorname{span}\\{\varphi_{1},\ldots,\varphi_{d},1\\}$
has a unique equilibrium measure with respect to the classical topological
pressure;

  2. (2)

for each $z\in\operatorname{int}L(\Phi)$ the map $q\mapsto P(\langle
q,\Phi-z\rangle)$ is of class $C^{r}$, is strictly convex, and its second
derivative is a positive definite bilinear form for each $q\in\mathbb{R}^{d}$,
where $\langle\cdot,\cdot\rangle$ is the usual inner product;

  3. (3)

the entropy map $\mu\mapsto h_{\mu}(T)$ is upper semicontinuous.

Examples of $C^{r}$-regular pairs $(T,\Phi)$ include topologically mixing
subshifts of finite type, $C^{1+\varepsilon}$ expanding maps, and
$C^{1+\varepsilon}$ diffeomorphisms with a locally maximal hyperbolic set,
with $\Phi$ composed of Hölder continuous functions. Finally, we say that the
family of functions $\Phi=\\{\varphi_{1},\ldots,\varphi_{d}\\}$ is
_cohomologous_ to a constant $c=(c_{1},\ldots,c_{d})$ if $\varphi_{i}$ is
cohomologous $c_{i}$ for $i=1,\ldots,d$. Then $L(\Phi)=\\{c\\}$ and so
$\operatorname{int}L(\Phi)=\emptyset$.

The following theorem is our main result. Given a function
$F\colon\mathbb{R}^{d}\to\mathbb{R}$, we consider the set

$K(F,\Phi)=\biggl{\\{}\int_{X}\Phi\,d\mu:\mu\text{ is an equilibrium measure
for }(F,\Phi)\biggr{\\}}\subset L(\Phi).$

We also consider the function $h\colon L(\Phi)\to\mathbb{R}$ defined by

$h(z)=\sup\bigl{\\{}h_{\mu}(T):\mu\in\mathcal{M}(z)\bigr{\\}}.$ (18)

######  Theorem 6.

Let $T\colon X\to X$ be a continuous map on a compact metric space and let
$\Phi=\\{\varphi_{1},\ldots,\varphi_{d}\\}$ be a family of continuous
functions such that the pair $(T,\Phi)$ is $C^{r}$ regular for some $r\geq 2$.
For each function $F\colon\mathbb{R}^{d}\to\mathbb{R}$ of class $C^{r}$, the
following properties hold:

  1. (1)

$K(F,\Phi)$ is a nonempty compact set;

  2. (2)

$K(F,\Phi)$ is the set of maximizers of the function $z\mapsto h(z)+F(z)$;

  3. (3)

if $K(F,\Phi)\subset\operatorname{int}L(\Phi)$, then the equilibrium measures
for $(F,\Phi)$ are the elements of $\\{\nu_{z}:z\in K(F,\Phi)\\}$, where each
$\nu_{z}\in\mathcal{M}$ is an ergodic measure that is the unique equilibrium
measure for some function
$\psi_{z}\in\operatorname{span}\\{\varphi_{1},\ldots,\varphi_{d},1\\}$.

###### Proof.

We divide the proof into steps.

######  Lemma 3.

$K(F,\Phi)$ is a nonempty compact subset of $L(\Phi)$.

###### Proof of the lemma.

Let $(z_{n})_{n\in\mathbb{N}}$ be a sequence in $K(F,\Phi)$ converging to a
point $z\in L(\Phi)$. For each $n\in\mathbb{N}$ there exists an equilibrium
measure $\mu_{n}\in\mathcal{M}$ for $(F,\Phi)$ such that
$z_{n}=\int_{X}\Phi\,d\mu_{n}$. Passing eventually to a subsequence, we may
assume that there exists $\mu\in\mathcal{M}$ such that $\mu_{n}\to\mu$ when
$n\to\infty$ in the weak∗ topology. Since the map $\mu\mapsto h_{\mu}(T)$ is
upper semicontinuous, we obtain

$P_{F}(\Phi)=\limsup_{n\to\infty}\bigg{[}h_{\mu_{n}}(T)+F\biggl{(}\int_{X}\Phi\,d\mu_{n}\biggr{)}\bigg{]}\leq
h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)},$

which implies that $\mu$ is an equilibrium measure for $(F,\Phi)$. Since
$z=\int_{X}\Phi\,d\mu$, we conclude that $z\in K(F,\Phi)$. Hence, $K(F,\Phi)$
is a closed and bounded subset of $\mathbb{R}^{d}$. Theorem 5 guarantees that
$K(F,\Phi)$ is nonempty. ∎

######  Lemma 4.

For each $z\in\operatorname{int}L(\Phi)$ there exists an ergodic measure
$\nu_{z}\in\mathcal{M}$ such that $\int_{X}\Phi\,d\nu_{z}=z$. In fact,
$\nu_{z}$ is the unique equilibrium measure for some function
$\psi_{z}\in\operatorname{span}\\{\varphi_{1},\ldots,\varphi_{d},1\\}$.

###### Proof of the lemma.

By the proof of Theorem 8 in [BSS02], for each $z\in\operatorname{int}L(\Phi)$
there exists $q(z)\in\mathbb{R}^{d}$ and an ergodic measure
$\nu_{z}\in\mathcal{M}(z)$ such that $\nu_{z}$ is an equilibrium measure for
the function

$\psi_{z}=\langle q(z),\Phi-z\rangle-h(T|_{C_{z}(\Phi)}),$

where $h$ denotes the topological entropy. Since
$\psi_{z}\in\operatorname{span}\\{\varphi_{1},\ldots,\varphi_{d},1\\}$, we
conclude that $\nu_{z}$ is the unique equilibrium measure for $\psi_{z}$. ∎

######  Lemma 5.

For each $z\in L(\Phi)$ there exists $\mu\in\mathcal{M}(z)$ with
$h(z)=h_{\mu}(T)$. Moreover, when $z\in\operatorname{int}L(\Phi)$ this measure
is unique and coincides with $\nu_{z}$.

###### Proof of the lemma.

Take $z\in L(\Phi)$. By the definition of $L(\Phi)$, there exists
$\mu\in\mathcal{M}$ such that $\int_{X}\Phi\,d\mu=z$, that is,
$\mathcal{M}(z)\neq\emptyset$. By the compactness of $\mathcal{M}(z)$ and the
upper semicontinuity of the map $\mu\mapsto h_{\mu}(T)$, there exists
$\mu\in\mathcal{M}(z)$ maximizing the metric entropy.

Now take $z\in\operatorname{int}L(\Phi)$. By Lemma 4, there exists
$\nu_{z}\in\mathcal{M}$ such that $\int_{X}\Phi\,d\nu_{z}=z$, where $\nu_{z}$
is the unique equilibrium for $\psi_{z}$. Let $\mu\in\mathcal{M}(z)$ be a
measure maximizing the metric entropy. Since
$\int_{X}\Phi\,d\mu=\int_{X}\Phi\,d\nu_{z}$, one can verify that
$\int_{X}\psi_{z}\,d\mu=\int_{X}\psi_{z}\,d\nu_{z}$. Then

$h_{\mu}(T)+\int_{X}\psi_{z}\,d\mu\geq
h_{\nu_{z}}(T)+\int_{X}\psi_{z}\,d\nu_{z}=P(\psi_{z}),$

which implies that $\mu$ is also an equilibrium measure for $\psi_{z}$ (for
the classical topological pressure). Since $\psi_{z}$ has a unique equilibrium
measure, we conclude that $\mu=\nu_{z}$. ∎

######  Lemma 6.

$z\in K(F,\Phi)$ if and only if $z$ maximizes the function $E$ defined by
$E(z)=h(z)+F(z)$.

###### Proof of the lemma.

First assume that $z\in L(\Phi)$ maximizes the function $E$. By Lemma 5, there
exists $\mu\in\mathcal{M}(z)$ such that $h(z)=h_{\mu}(T)$ and so

$h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}=h(z)+F(z)=\sup_{\mu\in\mathcal{M}}\biggl{\\{}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\\}}.$

This implies that $\mu$ is an equilibrium measure for $(F,\Phi)$ and so $z\in
K(F,\Phi)$.

Now assume that $z\in K(F,\Phi)$. Then there exists an equilibrium measure
$\mu$ for $(F,\Phi)$ such that $z=\int_{X}\Phi\,d\mu$ and so

$\begin{split}E(z)&=h(z)+F(z)\\\ &\geq
h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\\\
&=\sup_{\mu\in\mathcal{M}}\biggl{\\{}h_{\mu}(T)+F\biggl{(}\int_{X}\Phi\,d\mu\biggr{)}\biggr{\\}}.\end{split}$

This shows that $z$ maximizes $E$. ∎

Lemmas 3 and 6 give items (1) and (2) in the theorem. Now we establish item
(3). For each $z\in K(F,\Phi)$ there exists an equilibrium measure $\mu$ for
$(F,\Phi)$ such that $\int_{X}\Phi\,d\mu=z$. When
$K(F,\Phi)\subset\operatorname{int}L(\Phi)$, it follows from Lemmas 4 and 5
that $\mu$ is the unique measure with $\int_{X}\Phi\,d\mu=z$ and that
$\mu=\nu_{z}$, where $\nu_{z}$ is ergodic and is the unique equilibrium
measure for some function $\psi_{z}$. ∎

##  5\. Number of equilibrium measures

In this section we consider the problem of how many equilibrium measures a
$C^{r}$ regular system has.

###  5.1. Preliminary results

We start with some auxiliary results about the function $h$ in (18). Note that

$h(z)=\sup\left\\{h_{\mu}(T):\int_{X}\Phi\,d\mu=z\text{ with
$\mu\in\mathcal{M}$}\right\\}.$

######  Proposition 7.

Let $(T,\Phi)$ be a $C^{r}$ regular pair for some $r\geq 2$. Then the function
$h\colon L(\Phi)\to\mathbb{R}$ is upper semicontinuous, concave and finite.

###### Proof.

Take $z\in L(\Phi)$ and consider a sequence $(z_{n})_{n\in\mathbb{N}}$ in
$L(\Phi)$ such that $z_{n}\to z$ when $n\to\infty$. By Lemma 5, eventually
passing to a subsequence one can assume that for each $n\in\mathbb{N}$ there
exists $\mu_{n}\in\mathcal{M}(z_{n})$ such that $h(z_{n})=h_{\mu_{n}}(T)$ and
$\mu_{n}\to\mu$ when $n\to\infty$ for some $\mu\in\mathcal{M}$ in the weak∗
topology. We also have

$\int_{X}\Phi\,d\mu=\lim_{n\to\infty}\int_{X}\Phi\,d\mu_{n}=\lim_{n\to\infty}z_{n}=z$

and so $\mu\in\mathcal{M}(z)$. Moreover, since $\mu\mapsto h_{\mu}(T)$ is
upper semicontinuous, we obtain

$\limsup_{n\to\infty}h(z_{n})=\limsup_{n\to\infty}h_{\mu_{n}}(T)\leq
h_{\mu}(T)\leq h(z)$

and so $h$ is upper semicontinuous on $L(\Phi)$.

Now we prove the concavity property. Take $z_{1},z_{2}\in L(\Phi)$ and
$\mu_{1}\in\mathcal{M}(z_{1})$, $\mu_{2}\in\mathcal{M}(z_{2})$ such that
$h(z_{1})=h_{\mu_{1}}(T)$ and $h(z_{2})=h_{\mu_{2}}(T)$. Since the entropy map
is affine, for each $t\in[0,1]$ we have

$\begin{split}h(tz_{1}+(1-t)z_{2})\geq
h_{t\mu_{1}+(1-t)\mu_{2}}(T)&=th_{\mu_{1}}(T)+(1-t)h_{\mu_{2}}(T)\\\
&=th(z_{1})+(1-t)h(z_{2}).\end{split}$

The upper semicontinuity of $h$ on $L(\Phi)$ together with the compactness of
$L(\Phi)$ and the fact that $\mathcal{M}(z)\neq\emptyset$ for each $z\in
L(\Phi)$, guarantee that $h$ is finite on $L(\Phi)$. ∎

As pointed out in the recent work [Wol20], in strong contrast to what happens
for $d=1$, the function $z\mapsto h(z)$ need not be continuous on $L(\Phi)$.

######  Proposition 8.

If the pair $(T,\Phi)$ is $C^{r}$ regular, then the function
$h|_{\operatorname{int}L(\Phi)}$ is $C^{r-1}$. Moreover, if $(T,\Phi)$ is
$C^{\omega}$ regular, then $h|_{\operatorname{int}L(\Phi)}$ is analytic.

###### Proof.

It follows from Theorem 12 in [BSS02] that if $(T,\Phi)$ is $C^{r}$ regular,
then the map $\operatorname{int}L(\Phi)\ni z\mapsto h(T|_{C_{z}(\Phi)})$ is of
class $C^{r-1}$, and that if the pair is $C^{\omega}$ regular, then this map
is analytic. Since $h(z)=h(T|_{C_{z}(\Phi)})$ for every
$z\in\operatorname{int}L(\Phi)$, we obtain the desired statement. ∎

For $d=1$, Corollary 1.11 in [BL20] says that if the pair $(T,\Phi)$ is
$C^{\omega}$ and $F$ is analytic on $\operatorname{int}L(\Phi)$, then the set
$K(F,\Phi)$ is finite. In particular, there exist finitely many equilibrium
measures. The next example shows that this may not hold for $d>1$.

######  Example 4 (Infinite number of equilibrium measures).

Let $(T,\Phi)$ be a two-dimensional $C^{\omega}$ regular pair. Proposition 8
says that the function $h|_{\operatorname{int}L(\Phi)}$ is analytic. Now we
consider the function $F\colon\operatorname{int}L(\Phi)\to\mathbb{R}$ given by

$F(z_{1},z_{2})=-h(z_{1},z_{2})-z_{1}^{2}z_{2}^{2}.$

Since $h$ is analytic on $\operatorname{int}L(\Phi)$, the same happens to $F$.
Notice that

$E(z_{1},z_{2})=h(z_{1},z_{2})+F(z_{1},z_{2})=-z_{1}^{2}z_{2}^{2}.$

One can easily verify that the set of critical points of $E$ in
$\operatorname{int}L(\Phi)$ is

$\\{(x,0):x\in\mathbb{R}\\}\cup\\{(0,y):y\in\mathbb{R}\\}\cap\operatorname{int}L(\Phi).$

In fact, all critical points maximize $E$ on $\operatorname{int}L(\Phi)$. Now
assume that $(0,0)$ belongs to $\operatorname{int}L(\Phi)$. Then any open
neighborhood of $(0,0)$ in $\operatorname{int}L(\Phi)$ contains an uncountable
number of critical points. In particular, by Theorem 6, there exist infinitely
many equilibrium measures for $(F,\Phi)$.

Example 4 also shows that one can have $K(F,\Phi)\cap\partial
L(\Phi)\neq\emptyset$. In fact, if $(0,0)\in\operatorname{int}L(\Phi)$, then
$\partial L(\Phi)$ intersects the two axes at least four times. This justifies
requiring that $K(F,\Phi)\subset\operatorname{int}L(\Phi)$ in the last item of
Theorem 6.

###  5.2. Equilibrium measures I

For $d=1$, it was shown in [BL20] that no point on $\partial L(\Phi)$
maximizes the function $E=h+F$. By Lemma 6, this implies that
$K(F,\Phi)\subset\operatorname{int}L(\Phi)$. It is also shown that
$h^{\prime\prime}(z)<0$ for every $z\in\operatorname{int}L(\Phi)$ and so
$h\colon L(\Phi)\to\mathbb{R}$ is a strictly concave function. Note that for
$d=1$ we have $L(\varphi)=[A,B]$, where
$A=\inf_{\mu\in\mathcal{M}}\int\varphi\,d\mu$ and
$B=\sup_{\mu\in\mathcal{M}}\int\varphi\,d\mu$.

The next result is a criterion for uniqueness of equilibrium measures.

######  Theorem 9.

Let $(T,\varphi)$ be a $C^{r}$ regular pair and let
$F\colon[A,B]\to\mathbb{R}$ be a concave $C^{r}$ function. Then there exists a
unique equilibrium measure for $(F,\varphi)$. Moreover, the equilibrium
measure is ergodic.

###### Proof.

Since $F$ is concave and $h$ is strictly concave, the function $E=h+F$ is
strictly concave. This implies that $E$ has at most one maximizer in $(A,B)$.
Since there is no maximizer of $E$ on $\partial L(\varphi)=\\{A,B\\}$ and
$K(F,\varphi)\neq\emptyset$, we conclude that there exists a unique point
$z^{*}\in(A,B)$ maximizing $E$. Hence, it follows from Lemma 6 that
$K(F,\varphi)=\\{z^{*}\\}$. By Theorem 4.3 in [BL20] together with Lemma 4, we
conclude that there exists a unique equilibrium measure for $(F,\varphi)$ and
that this measure is ergodic. ∎

The following example illustrates various possibilities.

######  Example 5.

Let $\Sigma=\\{-1,1\\}$ and let
$T\colon\Sigma^{\mathbb{Z}}\to\Sigma^{\mathbb{Z}}$ be the two-sided shift. We
consider the function $\varphi\colon\Sigma\to\mathbb{R}$ defined by
$\varphi(\cdots\omega_{-1}\omega_{0}\omega_{1}\cdots)=\omega_{0}$. Then
$L(\varphi)=[-1,1]$ and the entropy function $h\colon L(\varphi)\to\mathbb{R}$
is given by

$h(z)=-\frac{1-z}{2}\log\left(\frac{1-z}{2}\right)-\frac{1+z}{2}\log\left(\frac{1+z}{2}\right).$
(19)

For the function $F\colon L(\Phi)\to\mathbb{R}$ defined by
$F(z)=\alpha/(z^{2}-2)$, where $\alpha\in\mathbb{R}$, we have

$F^{\prime\prime}(z)=2\alpha(3z^{2}+2)/(z^{2}-2)^{3}.$

Notice that for $\alpha>0$ we have $F^{\prime\prime}<0$ on
$\operatorname{int}L(\varphi)$. Since $F\equiv 0$ for $\alpha=0$, the function
$F$ is concave on $\operatorname{int}L(\varphi)$ whenever $\alpha\geq 0$.
Hence, by Theorem 9 there exists a unique equilibrium measure $\nu_{z^{*}}$
for $(F,\varphi)$, where $z^{*}=0$ (see Figure 1). For $\alpha<0$, the number
of equilibrium measures may vary. For instance, for $\alpha=-1$ there is one
equilibrium measure, while for $\alpha=-1.7$ there are two equilibrium
measures.

Figure 1. Graph of $E$ for $\alpha=1$.

Theorem 9 also shows that in order to have finitely many equilibrium measures
it is not necessary that the pair $(T,\varphi)$ is $C^{\omega}$ and that the
function $F$ is analytic. We give an example in the nonanalytic $C^{\infty}$
case.

######  Example 6.

Consider the pair $(T,\varphi)$ in Example 5 and let
$F\colon\mathbb{R}\to\mathbb{R}$ be the function given by

$F(z)=\begin{cases}3\exp(-1/z)&\text{if }z>0,\\\ 0&\text{if }z\leq
0.\end{cases}$

One can show that $F$ is $C^{\infty}$ but not analytic. For $-1\leq z\leq 0$,
we have

$E=h+F=h+0=h.$

It follows from (19) that $E$ has a local maximum $y_{1}=1$ at $z_{1}^{*}=0$.
For $0<z\leq 1$, one can verify that $E$ has a local maximum $y_{2}\approx
1.33$ at $z_{2}^{*}\approx 0.75$. Since $y_{1}<y_{2}$, the function $E$ has a
unique global maximum at
$z_{2}^{*}\in(0,1)\subset\operatorname{int}L(\varphi)$ (see Figure 2). By
Theorem 6, we conclude that $\nu_{z_{2}^{*}}$ is the unique equilibrium
measure for $(F,\varphi)$.

Figure 2. Nonanalytic $C^{\infty}$ case

###  5.3. Equilibrium measures II

The following statement is a version of the existence result in Theorem 9 for
$d>1$.

######  Theorem 10.

Let $(T,\Phi)$ be a $C^{r}$ regular pair and let $F\colon
L(\Phi)\to\mathbb{R}$ be a strictly concave $C^{r}$ function such that $E=h+F$
attains its maximum on $\operatorname{int}L(\Phi)$. Then there exists a unique
equilibrium measure for $(F,\Phi)$. Moreover, the equilibrium measure is
ergodic.

###### Proof.

By Proposition 7, the map $z\mapsto h(z)$ is upper semicontinuous on
$L(\Phi)$. Since $F$ is $C^{r}$ on $L(\Phi)$, we conclude that $z\mapsto E(z)$
is upper semicontinuous on $L(\Phi)$. Together with the compactness of
$L(\Phi)$, this guarantees the existence of at least one point in $L(\Phi)$
maximizing the function $E$. On the other hand, by Propositions 7 and 8 and
the strict concavity of $F$, the function $E$ is strictly concave on $L(\Phi)$
and $C^{r-1}$ on $\operatorname{int}L(\Phi)$. The concavity property of $E$
implies that there is one maximizer in $\operatorname{int}L(\Phi)$. Since
$K(F,\Phi)\subset\operatorname{int}L(\Phi)$, there exists a unique point
$z^{*}\in\operatorname{int}L(\Phi)$ such that $E(z^{*})$ is a maximal value.
It follows from Theorem 6 that $K(F,\Phi)=\\{z^{*}\\}$, that is, $\nu_{z^{*}}$
is the unique equilibrium measure for $(F,\Phi)$. Moreover, by Lemma 4,
$\nu_{z^{*}}$ is an ergodic measure. ∎

In Example 5, we have $h|_{\partial L(\varphi)}\equiv 0$. It turns out that
this behavior at the boundary of $L(\varphi)$ is typical for some $C^{r}$
regular systems, even for $d>1$. Let $\mathcal{H}_{\theta}$ be the space of
Hölder continuous functions with Hölder exponent $\theta>0$. The following
result is a particular case of Theorem 14 in [BSS02].

######  Theorem 11.

Let $T$ be a subshift of finite type, a $C^{1+\varepsilon}$ diffeomorphism
with a hypebolic set, or a $C^{1+\varepsilon}$ map with a repeller, that is
assumed to be topologically mixing. Then there exists a residual set
$\mathcal{O}\subset(\mathcal{H}_{\theta})^{d}$ such that for each
$\Phi\in\mathcal{O}$ we have

$h|_{\partial L(\Phi)}\equiv 0\quad\text{and}\quad
L(\Phi)=\overline{\operatorname{int}L(\Phi)}.$ (20)

We also note that in Example 5 with $\alpha>0$, the function $F$ satisfies

$F|_{\operatorname{int}L(\varphi)}>\max_{z\in\partial L(\varphi)}F(z),$ (21)

where $F_{\operatorname{int}L(\varphi)}$ is the restriction to
$\operatorname{int}L(\varphi)$. This is an example of a more general situation
in which $E=h+F$ attains its maximum on $\operatorname{int}L(\Phi)$:

$\max_{z\in\operatorname{int}L(\Phi)}E(z)>\max_{z\in\partial L(\Phi)}E(z).$
(22)

Note that this condition may depend not only on $F$, but also on the family of
functions $\Phi$.

A similar idea works for typical $C^{r}$ regular systems in the sense that
they belong to the residual set $\mathcal{O}$ in Theorem 11. Let $(T,\Phi)$ be
a $C^{r}$ regular pair satisfying (20). In particular,
$\operatorname{int}L(\Phi)\neq\emptyset$. Now let
$F\colon\mathbb{R}^{d}\to\mathbb{R}$ be a function satisfying (21) with
$\varphi$ replaced by $\Phi$. Since $h\geq 0$, we have

$\begin{split}\max_{z\in\partial L(\Phi)}E(z)&\leq\max_{z\in\partial
L(\Phi)}h(z)+\max_{z\in\partial L(\Phi)}F(z)\\\ &\leq
h|_{\operatorname{int}L(\Phi)}+\max_{z\in\partial L(\Phi)}F(z)\\\
&<h|_{\operatorname{int}L(\Phi)}+F|_{\operatorname{int}L(\Phi)}=E_{\operatorname{int}L(\Phi)},\end{split}$

which implies that property (22) holds. Therefore,
$K(F,\Phi)\subset\operatorname{int}L(\Phi)$ and so one can apply item (3) of
Theorem 6.

For $d=1$ and $C^{r}$ regular systems, the function $h$ in (18) is strictly
concave. The next examples illustrate that this may still happen for $d>1$,
but unfortunately we are not able to describe for which $C^{r}$ regular
systems the function $h$ is strictly concave.

######  Example 7 (A system with $\operatorname{int}L(\Phi)=\emptyset$).

Let $T\colon X\to X$ be the two-sided shift with $X=\\{1,2\\}^{\mathbb{Z}}$.
Moreover, consider the characteristic functions $\varphi_{1}=\chi_{C_{1}}$ and
$\varphi_{2}=\chi_{C_{2}}$, where $C_{i}$ is the set of all sequences

$(\cdots\omega_{-1}\omega_{0}\omega_{1}\cdots)\in X$

with $\omega_{0}=i$. For each $\mu\in\mathcal{M}$, we have
$\int_{X}\varphi_{i}=\mu(C_{i})$ for $i=1,2$ and so

$L(\Phi)=\bigl{\\{}(\mu(C_{1}),\mu(C_{2})):\mu\in\mathcal{M}\bigr{\\}}.$

By Theorem 8 in [BSS02], we have

$\begin{split}h(z_{1},z_{2})&=\max_{\mu\in\mathcal{M}}\bigl{\\{}h_{\mu}(T):(\mu(C_{1}),\mu(C_{2}))=(z_{1},z_{2})\bigr{\\}}\\\
&=-z_{1}\log z_{1}-z_{2}\log z_{2}.\end{split}$

Since $\mu(C_{1})+\mu(C_{2})=1$ for each $\mu\in\mathcal{M}$, we obtain

$L(\Phi)=\bigl{\\{}(z_{1},z_{2})\in[0,1]\times[0,1]:z_{1}+z_{2}=1\bigr{\\}}$

and so $\operatorname{int}L(\Phi)=\emptyset$. Moreover,

$h(z)=-z\log z-(1-z)\log(1-z)\quad\text{for $z\in[0,1]$}.$

One can easily verify that $(z_{1},z_{2})\mapsto h(z_{1},z_{2})$ is strictly
concave on $L(\Phi)$.

######  Example 8.

Let $T\colon X\to X$ be the two-sided shift with $X=\\{1,2,3\\}^{\mathbb{Z}}$
and let $\varphi_{1}=\chi_{C_{1}}$ and $\varphi_{2}=\chi_{C_{3}}$. Proceeding
as in Example 7, we obtain

$h(z_{1},z_{2})=-z_{1}\log z_{1}-z_{2}\log
z_{2}-(1-z_{1}-z_{2})\log(1-z_{1}-z_{2})$

and

$L(\Phi)=\bigl{\\{}(z_{1},z_{2})\in[0,1]\times[0,1]:z_{1}+z_{2}\leq
1\bigr{\\}}.$

Note that $\operatorname{int}L(\Phi)\neq\emptyset$ and that $\partial L(\Phi)$
is the set

$\bigl{(}(\mathbb{R}\times\\{0\\})\cup(\\{0\\}\times\mathbb{R})\cup\\{(z_{1},z_{2}):z_{1}+z_{2}=1\\}\bigr{)}\cap([0,1]\times[0,1]).$

Observe that for $(z_{1},z_{2})=(1/2,0)$, $(0,1/2)$, $(1/2,1/2)\in\partial
L(\Phi)$ we have $h(z_{1},z_{2})=1>0$ and so the system is not typical. On the
other hand, the map $(z_{1},z_{2})\mapsto h(z_{1},z_{2})$ is still strictly
concave on $L(\Phi)$.

Now consider the function $F(z_{1},z_{2})=\beta(z_{1}^{2}+z_{2}^{2})/2$ with
$\beta\in\mathbb{R}$. One can verify that the determinant of the Hessian
matrix of $E=h+F$ is given by

$\det
H_{E}(z_{1},z_{2})=\beta^{2}+\beta\frac{z_{1}(1-z_{1})+z_{2}(1-z_{2})}{z_{1}z_{2}(1-z_{1}-z_{2})}+\frac{1}{z_{1}z_{2}(1-z_{1}-z_{2})}.$

Since $\det H_{E}(z_{1},z_{2})>0$ for
$(z_{1},z_{2})\in\operatorname{int}L(\Phi)$ and $\beta\geq 0$, every critical
point of $E$ is nondegenerate for all $\beta\geq 0$. Hence, for each
$\beta\geq 0$, the function $E$ has at most finitely many critical points. In
addition, if condition (22) holds, then $E$ attains its maximal value only at
critical points. Hence, by Theorem 6 the pair $(F,\Phi)$ has finitely many
equilibrium measures. For a specific example, for $\beta=1$ one can verify
that condition (22) holds and so $(F,\Phi)$ has finitely many equilibrium
measures.

On the other hand, for $\beta<0$ the function $F$ is strictly concave and one
can use Theorem 10 to conclude that $(F,\Phi)$ has a unique equilibrium
measure.

###  5.4. Coincidence of equilibrium measures

The following result gives a sufficient condition so that two systems share
equilibrium measures. We say that
$\Phi_{1}=\\{\varphi_{1,1},\ldots,\varphi_{1,d}\\}$ is _cohomologous_ to
$\Phi_{2}=\\{\varphi_{2,1},\ldots,\varphi_{2,d}\\}$ if $\varphi_{1,i}$ is
cohomologous to $\varphi_{2,i}$ for $i=1,\ldots,d$. Then

$\int_{X}\Phi_{1}\,d\mu=\int_{X}\Phi_{2}\,d\mu\quad\text{for each
$\mu\in\mathcal{M}$,}$

which readily implies that $L(\Phi_{1})=L(\Phi_{2})$.

######  Proposition 12.

Let $(T,\Phi_{1})$ and $(T,\Phi_{2})$ be $C^{r}$ regular pairs such that
$\Phi_{1}$ is cohomologous to $\Phi_{2}$ and let $F_{1}\colon
L(\Phi_{1})\to\mathbb{R}$ and $F_{2}\colon L(\Phi_{2})\to\mathbb{R}$ be
$C^{r}$ functions. If
$z\in\operatorname{int}L(\Phi_{1})\cap\operatorname{int}L(\Phi_{2})$ is
simultaneously a maximizer of the functions $E_{1}=h_{1}+F_{1}$ and
$E_{2}=h_{2}+F_{2}$, then $\nu_{z}$ is an equilibrium measure for
$(F_{1},\Phi_{1})$ and $(F_{2},\Phi_{2})$.

###### Proof.

Since $\Phi_{1}$ is cohomologous to $\Phi_{2}$, we have
$L:=L(\Phi_{1})=L(\Phi_{2})$. Now take $z\in\operatorname{int}L$ and consider
the functions

$L_{1}(q)=P(\langle q,\Phi_{1}-z\rangle)-h_{1}(z)\quad\text{and}\quad
L_{2}(q)=P(\langle q,\Phi_{2}-z\rangle)-h_{2}(z),$

where $P$ denotes the classical topological pressure and where each $h_{i}$ is
the corresponding entropy function (see (18)). By the cohomology assumption,
we have

$\lim_{n\to\infty}\frac{\|S_{n}\varphi_{1,i}-S_{n}\varphi_{2,i}\|_{\infty}}{n}=0\quad\text{for
$i=1,\ldots,d$}$

and so $C_{z}(\Phi_{1})=C_{z}(\Phi_{2})$ for all $z\in\mathbb{R}^{d}$ (see
(17)). In particular, this implies that $h:=h_{1}=h_{2}$. Therefore,

$[\langle q,\Phi_{1}-z\rangle-h_{1}(z)]-[\langle q,\Phi_{2}-z\rangle-
h_{2}(z)]=\langle q,\Phi_{1}-\Phi_{2}\rangle$

for $q\in\mathbb{R}^{d}$. Again since $\Phi_{1}$ is cohomologous to
$\Phi_{2}$, we conclude that

$L_{1}(q)=L_{2}(q)\quad\text{for $q\in\mathbb{R}^{d}$.}$ (23)

On the other hand, by the proof of Theorem 8 in [BSS02] the function $q\mapsto
L_{1}(q)$ attains its minimum at a point $q_{1}(z)$ and $\nu_{1,z}$ is the
unique equilibrium measure for the function $\langle
q_{1}(z),\Phi_{1}-z\rangle-h(z)$. Similarly, $q\mapsto L_{2}(q)$ attains its
minimum at a point $q_{2}(z)$ and $\nu_{2,z}$ is the unique equilibrium
measure for the function $\langle q_{2}(z),\Phi_{2}-z\rangle-h(z)$. By (23),
one can take $q_{1}(z)=q_{2}(z)$ and so $\nu_{z}:=\nu_{1,z}=\nu_{2,z}$. The
desired result follows now from Theorem 6. ∎

A direct consequence of Proposition 12 is that if $\Phi_{1}$ is cohomologous
to $\Phi_{2}$ and the functions $E_{1}$ and $E_{2}$ attain maximal values at
the same points, then $(F_{1},\Phi_{1})$ and $(F_{2},\Phi_{2})$ have the same
equilibrium measures (in particular, this happens when $F_{1}=F_{2}$). For the
converse to hold we need stronger conditions so that the coincidence of two
equilibrium measures yields a cohomology relation.

######  Theorem 13.

Let $X$ be a topologically mixing locally maximal hyperbolic set for a
diffeomorphism $T$ and let $\Phi_{1}$ and $\Phi_{2}$ be families of Hölder
continuous functions. Moreover, let $F_{1}$ and $F_{2}$ be functions such that
$E_{1}$ and $E_{2}$ attain maximal values in $\operatorname{int}L(\Phi_{1})$
and $\operatorname{int}L(\Phi_{2})$, respectively. If $(F_{1},\Phi_{1})$ and
$(F_{2},\Phi_{2})$ have the same equilibrium measures, then for each $z_{1}$
and $z_{2}$ maximizing $E_{1}$ and $E_{2}$, respectively, there exist
$q_{1},q_{2}\in\mathbb{R}^{d}$ such that $\langle q_{1},\Phi_{1}-z_{1}\rangle$
is cohomologous to $\langle q_{2},\Phi_{2}-z_{2}\rangle$.

###### Proof.

By Theorem 6, each equilibrium measure for $(F_{i},\Phi_{i})$ is a measure
$\nu_{z_{i}}$ with $z_{i}\in K(F_{i},\Phi_{i})$ that is the unique equilibrium
measure for

$\psi_{i}=\langle q_{i}(z_{i}),\Phi_{i}-z_{i}\rangle-h_{i}(z_{i}),$

where $q_{i}(z_{i})$ is a minimizer of the function

$L(q)=P(\langle q,\Phi_{i}-z_{i}\rangle)-h_{i}(z_{i}).$

Since by hypotheses $\nu_{z_{1}}=\nu_{z_{2}}$, the function
$\psi_{1}-\psi_{2}$ is cohomologous to $P(\psi_{1})-P(\psi_{2})\in\mathbb{R}$.
But since

$L_{1}(q_{1}(z_{1}))=L_{2}(q_{2}(z_{2}))=0$

(see Lemma 2 in [BSS02]), we have $P(\psi_{1})=P(\psi_{2})$. So there exists a
continuous function $S=S(z_{1},z_{2}):X\to\mathbb{R}$ such that
$\psi_{1}-\psi_{2}=S\circ T-S$, that is,

$\begin{split}S\circ T-S&=\langle q_{1}(z_{1}),\Phi_{1}-z_{1}\rangle-\langle
q_{2}(z_{2}),\Phi_{2}-z_{2}\rangle-h_{1}(z_{1})+h_{2}(z_{2}).\end{split}$

Again since $\nu_{z_{1}}=\nu_{z_{2}}$, by Lemma 5 we have

$h_{1}(z_{1})=h_{\nu_{z_{1}}}(T)=h_{\nu_{z_{2}}}(T)=h_{2}(z_{2}).$

Hence, for each $z_{1}$ and $z_{2}$ maximizing $E_{1}$ and $E_{2}$,
respectively, there exist points
$q_{1}=q_{1}(z_{1}),q_{2}=q_{2}(z_{2})\in\mathbb{R}^{d}$ as in the statement
of the theorem. ∎

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#  Two-component model description of Bose-Einstein correlations in pp
collisions at 13 TeV measured by the CMS Collaboration at the LHC

Takuya Mizoguchi1, Seiji Matsumoto2, and Minoru Biyajima3
1National Institute of Technology, Toba College, Toba 517-8501, Japan
2School of General Education, Shinshu University, Matsumoto 390-8621, Japan
3Department of Physics, Shinshu University, Matsumoto 390-8621, Japan

###### Abstract

Using the two-component model, we analyze Bose-Einstein correlations in pp
collisions at the center-of-mass energy of 13 TeV, measured by the CMS
Collaboration at the LHC, and compare results with the $\tau$-model. We
utilize data described by the double ratios with an average pair transverse
momentum $0\leq k_{T}\leq 1.0$ GeV and six intervals described by the
reconstructed charged particle multiplicity as $N_{\rm trk}^{\rm offline}$.
The estimated ranges are 1-4 fm for the magnitude of extension of emitting
source expressed by the exponential function $\exp(-RQ)$ and 0.4-0.5 fm for
that by the Gaussian distribution $\exp(-(RQ)^{2}))$, respectively. Moreover,
we estimate the upper limits of the 3-pion BEC to test the two-component model
and investigate the role of the long-range correlation.

##  1 Introduction

This article investigates the Bose-Einstein correlations (BEC) described by
double ratios (DRs) in pp collisions at the center-of-mass energy 13 TeV,
obtained by the CMS Collaboration at the LHC [1]. The DR is defined by two
single rations (SR’s), i.e., ${\rm C_{2}^{data}}=N^{(2+:2-)}/N^{\rm(+-)}$ and
${\rm C_{2}^{MC}}=N_{\rm MC}^{(2+;2-)}/N_{\rm MC}^{(+-)}$, where $N$’s mean
the number of events in data and the Monte Carlo simulation. The suffixes
${\rm(2+:2-)}$ and ${\rm(+-)}$ mean the charge combinations. Therein, CMS
Collaboration only reports $\chi^{2}/$ndf (number of degrees of freedom)
values obtained using the $\tau$-model. Here, we analyze the DRs at an average
pair transverse momentum $0\leq k_{T}\leq 1.0$ GeV ($k_{T}=|{\bf p}_{T,1}+{\bf
p}_{T,2}|/2$), and six intervals expressed by means of constraint $a\leq
N_{\rm trk}^{\rm offline}\leq b$ as illustrated in Fig. 1. The formula used in
the CMS analysis [2] is

$\displaystyle
F_{\tau}=C[1+\lambda\cos((r_{0}Q)^{2}+\tan(\alpha_{\tau}\pi/4)(Qr)^{\alpha_{\tau}})e^{-(Qr)^{\alpha_{\tau}}}]\cdot(1+\delta
Q)$ (1)

where $\lambda$, $r_{0}$, $r$, and $\alpha_{\tau}$ are parameters introduced
in the stable distribution based on stochastic theory, namely the degree of
coherence, two interaction ranges, and the characteristic index, respectively
(see, also Refs. [3, 4]). $Q=\sqrt{-(p_{1}-p_{2})^{2}}$ is the magnitude of
the 4-momentum transfer between two pions. The last term $(1+\delta Q)$ is
named the long range correlation with the index (linear) (LRC(linear)). Our
estimated values are presented in Table 1.

Figure 1: Fit to the BEC measurements by CMS in pp collisions at 13 TeV by Eq. (1). $C_{2}^{MC}\equiv N_{\rm MC}^{(2+:2-)}/N_{\rm MC}^{(+-)}$, where $N_{\rm MC}$’s mean the numbers of the same charged and opposite charged pairs recorded in MC simulations. Table 1: Fit parameters to the CMS BEC measurements in pp collisions at 13 TeV at $0.0\leq k_{T}\leq 1.0$ GeV by Eq. (1). $\delta$ values are estimated (top to bottom): $-0.016\pm 0.004$, $0.03\pm 0.01$, $0.005\pm 0.005$, $(-1.2\pm 0.1)\times 10^{-3}$, $(1.3\pm 0.2)\times 10^{-3}$, and $0.002\pm 0.001$. $N_{\rm trk}^{\rm offline}$ |  $r_{0}$ (fm) |  $r$ (fm) | $\lambda$ | $\alpha_{\tau}$ |  $\chi^{2}$/ndf |  $\chi^{2}$(CMS)
---|---|---|---|---|---|---
$0-4$ | $0.139\pm 0.021$ | $0.93\pm 0.06$ | $0.96\pm 0.05$ | $0.781\pm 0.026$ | 195/93 | 195
$10-12$ | $0.244\pm 0.004$ | $9.08\pm 1.40$ | $2.43\pm 0.23$ | $0.420\pm 0.013$ | 140/93 | 140
$31-33$ | $0.232\pm 0.005$ | $21.8\pm 4.0$ | $3.36\pm 0.37$ | $0.377\pm 0.011$ | 135/93 | 135
$80-84$ | $0.224\pm 0.001$ | $43.7\pm 2.5$ | $4.48\pm 0.15$ | $0.351\pm 0.003$ | 899/93 | 902
$105-109$ | $0.216\pm 0.003$ | $47.0\pm 5.2$ | $4.71\pm 0.31$ | $0.352\pm 0.005$ | 282/93 | 281
$130-250$ | $0.228\pm 0.013$ | $53.3\pm 19.9$ | $5.32\pm 1.27$ | $0.353\pm 0.020$ | 84.5/93 | 84

Because estimated values of all parameters by $\tau$-model, i.e., Eq. (1),
have not been presented in Ref. [1], it is difficult to draw physical picture
through the analyses of BEC in pp collisions at 13 TeV. Thus for this aim, we
present them in Table 1. Table 1 shows that the $\chi^{2}/$ndf values obtained
from our analysis are consistent with those reported by the CMS Collaboration
[1]. In other words, through concrete figures in Table 1, we are able to
consider physical picture based on the $\tau$-model.

As indicated in Table 1, the interaction ranges of the Levy-type form
($e^{-(Qr)^{\alpha}}$) increase as the interval containing $N_{\rm trk}^{\rm
offline}$ increases. The estimated values $r=20\sim 50$ fm appear large for pp
collisions at 13 TeV.

This paper also investigates this issue from a different perspective, focusing
on the collision mechanism. Three processes occur in collisions at the LHC [5,
6, 7]: the non-diffractive dissociation, the single-diffractive dissociation,
and the double-diffractive dissociation (DD). BEC are related to the chaotic
components of particle production. Since the contribution from the DD is
Poissonian [7], there is no effect to the BEC. Thus we calculated the
following two-component model correlation function [7, 8] (see also empirical
Refs. [9, 10, 11]),

$\displaystyle{\rm CF_{II}}=1+\lambda_{1}E_{\rm BE_{1}}+\lambda_{2}E_{\rm
BE_{2}}.$ (2)

The exchange function is the Fourier transform of the space-time region
emitting bosons (mainly pion) with overlapping wave functions. For the
exchange functions $E_{\rm BE_{1}}$ and $E_{\rm BE_{2}}$, we assign the
following two functions [12],

$\displaystyle\exp(-R_{1}Q)\quad{\rm and}\quad\exp\left(-(R_{2}Q)^{2}\right)$
(3)

characterizing the exponential and Gaussian type of BEC. Thus $R_{1}$ and
$R_{2}$ mean the extensions of the sources [12]. Concerning with two kinds of
exchange functions, see also the different approach[13].

Moreover, we discuss the LRC’s below. Three decades ago, the OPAL
Collaboration [14] adopted LRC${}_{(\rm OPAL)}=c(1+\delta Q+\varepsilon
Q^{2})$ to improve the linear form LRC${}_{(\rm linear)}=C(1+\delta Q)$.
Recently we proposed the inverse power series form LRC${}_{(\rm
p.s.)}=C/[1-\alpha Q\exp(\beta Q)]$ [15], because the number of parameters
($\alpha$ and $\beta$) is the same as the LRC(OPAL)’s, and it converges to $C$
as $Q$ is large. Taking into account of those investigations and mathematical
descriptions shown in Ref. [1], i.e., the distribution of opposite charged
pion pair $N^{(+-)}=C[1+a\exp{(-bQ^{2})}]$ and so on, we propose the following
form:

$\displaystyle{\rm LRC_{(Gauss)}}=\frac{C}{1.0+\alpha\exp(-\beta Q^{2})}.$ (4)

This function converges to $C$ as $Q$ is large and behaves as
$C[1-\alpha(1-\beta Q^{2})+\cdots]$, $Q$ being small. In Table 2, we compare
our approach with formulas shown in Ref. [1].

Table 2: Comparison of our approach with formulas utilized by CMS Collaboration [1]. | formulas | cf.
---|---|---
|  ${\rm CF_{II}\times LRC}$ | 1) ${\rm CF_{II}}$ is reflecting to three kinds of multiplicity
Our | where | distributions of the ND, SD, and DD in pp collisions.
approach |  ${\rm LRC_{(Exp)}}=\dfrac{1}{1+\alpha e^{-\beta Q}}$, | 2) Through the generalization of ${\rm LRC_{(OPAL)}}$
| or |  $=1+\delta Q+\varepsilon Q^{2}$ in e+e- annihilation at Z0-pole [14],
|  ${\rm LRC_{(Gauss)}}=\dfrac{1}{1+\alpha e^{-\beta Q^{2}}}$. | we obtained ${\rm LRC_{(Exp)}}$ [15].
|  | 3) Referring to mathematical descriptions on $F_{2N}$ and
|  |  $F_{2D}$ in Ref [1], ${\rm LRC_{(Gauss)}}$ is proposed for pp collisions.
| 1) Distributions $N^{(2+:2-)}$, $N^{(+-)}$, | 1) ${\rm CF_{I}}=1+\lambda E_{\rm BE}$, where $E_{\rm BE}=\exp(-RQ)$ [1].
CMS |  $N_{\rm MC}^{(2+:2-)}$ and $N_{\rm MC}^{(+-)}$ are assumed | 2) Provided that $N_{\rm MC}^{(+-)}\cong N^{(+-)}$ and the cross term
| as follows: ${\rm CF_{I}}\cdot C(1+ae^{-bQ^{2}})$, | ($\lambda aE_{\rm BE}\cdot e^{-bQ^{2}}$) is small, we obtain
|  $C^{\prime}(1+a^{\prime}e^{-b^{\prime}Q^{2}})$, $C_{M}(1+a_{M}e^{-b_{M}Q^{2}})$ |  $\dfrac{F_{2N}}{F_{2D}}\cong\dfrac{{\rm CF_{I}}\times(1+ae^{-bQ^{2}})}{1+a_{M}e^{-b_{M}Q^{2}}}\cong{\rm CF_{II}\times LRC_{(Gauss)}}$.
| and $C_{M}^{\prime}(1+a_{M}^{\prime}e^{-b_{M}^{\prime}Q^{2}})$, respectively. | Thus, $\alpha$ and $\beta$ in Eq. (4) are approximately identified
| 2) SR’s $F_{2N}=N^{(2+:2-)}/N^{(+-)}$ and | with $a_{M}$ and $b_{M}$ describing the Monte Carlo events
|  $F_{2D}=N_{\rm MC}^{(2+:2-)}/N_{\rm MC}^{(+-)}$ are used for | in $F_{2D}$, respectively.
| analysis of data of DR by the ratio | 3) Their Monte Carlo events are calculated by
|  $F_{2N}/F_{2D}$. | PYTHIA6. See Fig. 3 in Ref. [2].
| 3) $\tau$-model is also used for data of DR. |

In the second section, we analyze the BEC at 13 TeV using Eqs. (2)–(4). In the
third section, we present our predictions for 3-pion BEC using the two-
component model. In the final section, we provide concluding remarks. Appendix
A presents an analysis of BEC at 13 TeV using the $\tau$-model with Eq. (4).
In Appendix B, we reanalyze the CMS BEC at 0.9 and 7 TeV utilizing Eq. (4),
because in previous works [7, 8], we used LRC${}_{(\rm linear)}=C(1+\delta
Q)$.

##  2 Analysis of BEC at 13 TeV using Eqs. (2)–(4).

Considering the results of the CMS BEC at 7 TeV in Ref. [7], we assume a
combination of exponential function and Gaussian distribution, as this
combination has shown the valuable role. Moreover, it is worthwhile to mention
that Shimoda et al. in Ref. [12] investigated several possible distributions
for $E_{\rm BE}$’s. Our results are presented in Fig. 2 and Table 3. We
observe extraordinary behaviors in the two intervals, $0\leq N_{\rm trk}^{\rm
offline}\leq 4$ and $10\leq N_{\rm trk}^{\rm offline}\leq 12$, of the LRC
shown in Fig. 3.

As indicated by Fig. 2 and Table 3, the two-component model with Eqs. (2)–(4)
effectively characterizes three intervals: $31\leq N_{\rm trk}^{\rm
offline}\leq 33$, $80\leq N_{\rm trk}^{\rm offline}\leq 84$, and $105\leq
N_{\rm trk}^{\rm offline}\leq 109$.

Figure 2: Fit to the BEC measurements by CMS in pp collisions at 13 TeV by Eqs. (2)–(4). Table 3: Fit parameters of the CMS measurements of BEC in pp collisions at 13 TeV ($0.0\leq k_{T}\leq 1.0$ GeV) by Eqs. (2)–(4). Three constraints are used: $\lambda_{1}\leq 1.0$, $\lambda_{2}\leq 1.0$, and $\lambda_{1}+\lambda_{2}\leq 1.0$. The $p$-values for the three intervals $31\leq N_{\rm trk}^{\rm offline}\leq 33$, $80\leq N_{\rm trk}^{\rm offline}\leq 84$, and $130\leq N_{\rm trk}^{\rm offline}\leq 250$ are 73.0 %, 77.3 % and 85.3 %, respectively. $C$ (top to bottom): $0.980\pm 0.004$, $1.031\pm 0.001$, $1.007\pm 0.001$, $1.001\pm 1\times 10^{-4}$, $1.003\pm 2\times 10^{-4}$, $1.003\pm 0.001$, $0.972\pm 0.002$, $1.028\pm 0.002$, and $1.007\pm 0.001$. $N_{\rm trk}^{\rm offline}$ |  $R_{1}$ (fm) |  $R_{2}$ (fm) | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$ |  $\beta$ (GeV-2) |  $\chi^{2}$/ndf
---|---|---|---|---|---|---|---
$0-4$ | $1.57\pm 0.15$ | $0.51\pm 0.01$ | $0.680\pm 0.034$ | $0.320\pm 0.034$ | $0.062\pm 0.006$ | $0.79\pm 0.18$ | 185.4/92
$10-12$ | $2.40\pm 0.07$ | $0.39\pm 0.02$ | $0.865\pm 0.007$ | $0.135\pm 0.007$ | $0.136\pm 0.009$ | $0.99\pm 0.07$ | 137.1/92
$31-33$ | $3.37\pm 0.07$ | $0.48\pm 0.02$ | $0.910\pm 0.004$ | $0.090\pm 0.004$ | $0.048\pm 0.004$ | $1.06\pm 0.12$ | 83.3/92
$80-84$ | $3.76\pm 0.03$ | $0.49\pm 0.01$ | $0.866\pm 0.007$ | $0.061\pm 0.001$ | $0.026\pm 0.001$ | $1.53\pm 0.06$ | 81.6/92
$105-109$ | $4.02\pm 0.06$ | $0.57\pm 0.01$ | $0.867\pm 0.149$ | $0.050\pm 0.002$ | $0.020\pm 0.001$ | $1.06\pm 0.07$ | 107.0/92
$130-250$ | $3.79\pm 0.22$ | $0.46\pm 0.09$ | $0.857\pm 0.051$ | $0.040\pm 0.011$ | $0.030\pm 0.014$ | $1.54\pm 0.51$ | 77.6/92
Note: When no constraint is applied for $\lambda_{1}$ and $\lambda_{2}$, we
obtain the following figures:
$0-4$ | $2.76\pm 0.30$ | $0.49\pm 0.02$ | $1.085\pm 0.083$ | $0.477\pm 0.052$ | $0.112\pm 0.051$ | $1.78\pm 0.57$ | 126.6/92
$10-12$ | $2.50\pm 0.09$ | $0.37\pm 0.02$ | $0.947\pm 0.033$ | $0.168\pm 0.025$ | $0.165\pm 0.028$ | $1.17\pm 0.13$ | 128.8/92
$31-33$ | $3.43\pm 0.11$ | $0.48\pm 0.02$ | $0.928\pm 0.029$ | $0.092\pm 0.004$ | $0.048\pm 0.004$ | $1.06\pm 0.12$ | 83.0/92
Figure 3: The long-range correlations (LRCs), see Eq. (4) for six intervals.

Among the six intervals shown in Fig. 3, the red (solid) line and green
(dashed) line appear to be exceptional. They are probably related to the
normalization factors ($0.980\pm 0.004$ and $1.031\pm 0.001$). In other words,
in those regions there is very small freedom or noise which cannot be
described by Eqs. (2)–(4).

##  3 Test of the two-component model for 3-pion BEC

Here, we investigate the 3-pion BEC using the two-component model. Since there
is currently no information from CMS on the multiplicity distribution $P(n)$
at 13 TeV, it is challenging to determine the ratio between the contributions
of the first and the second components. We use the diagrams in Fig. 4.

Figure 4: Diagrams for the third-order BEC. The matrix indicates the exchange
of identical pions.

The formula that corresponds to the diagrams in Fig. 4 [16, 17, 18] is
expressed as

$\displaystyle F_{i}^{(3)}=1.0+3\lambda_{i}E_{\rm BE_{i}}+2(\lambda_{i}E_{\rm
BE_{i}})^{3/2}$ (5)

By assuming an equal weight for the first and the second components,
$F_{1}^{(3)}$ and $F_{2}^{(3)}$, we obtain the following normalized expression

$\displaystyle F^{(3+:3-)}=1.0+\frac{1}{2}\left(3\lambda_{1}E_{\rm
BE_{1}}+2(\lambda_{1}E_{\rm
BE_{1}})^{3/2}\right)+\frac{1}{2}\left(3\lambda_{2}E_{\rm
BE_{2}}+2(\lambda_{2}E_{\rm BE_{2}})^{3/2}\right),$ (6)

where $\lambda_{1}$, $\lambda_{2}$, $R_{1}$, and $R_{2}$ are fixed by using
the numerical values in Table3. Typical figures are presented in Fig. 5. We
could calculate the ratio if the CMS Collaboration reports the multiplicity
distributions $P(n)$ [2], as this would allow us to understand the ensemble
property of the BEC through the multiplicity distribution. It is worth noting
that the ATLAS Collaboration has already observed the multiplicity
distributions $P(n)$ [19] and BEC [20] to be considered in [21].

Figure 5: Prediction of upper limit of the 3$\pi$ BEC in pp collisions at 13
TeV by means of Eq. (6) with Eqs. (2)-(4). $N^{\rm(BG)}$ means $N_{\rm
MC}^{(3+:3-)}$, because of no-BEC in the Monte Carlo events.

In the near future, we may be able to further test the two-component model
when the CMS Collaboration analyzes the 3-$\pi$ BEC. If we observe the same
extensions as in Fig. 2, we could conclude that the two-component model is a
viable approach.

##  4 Concluding remarks

C1)

Our analysis of CMS BEC at 13 TeV using the $\tau$-model with Eq. (1) confirms
the applicability of this model. This is evidenced by the values of $\chi^{2}$
in Table 1.

C2)

As portrayed in Table 1, the interaction ranges $r$ in the Lévy-type
expression $e^{-(Qr)^{\alpha_{\tau}}}$ increase as the range of the interval
$N_{\rm trk}^{\rm offline}$ increases. However, it appears that the
interaction ranges from 30 to 50 fm are large in $pp$ collisions at 13 TeV.

C3)

To gain a better understanding of the results obtained from the $\tau$-model,
we have analyzed the BEC using the $\tau$-model with Eq. (4). This has led to
improved estimations, as shown in Appendix A.

C4)

We look forward to future analyses by the CMS Collaboration of the
multiplicity distributions and the third-order BEC at 13 TeV

Hereafter, we summarize the results of the two-component model using Eqs.
(2)–(4).

C5)

In Table 2, we mentioned how to propose LRC(Gauss), i.e., Eq. (4). To
investigate the remarks mentioned in C2) above using the two-component model,
we utilized Eqs. (2)–(4). Our results are presented in Table 3. The large
extensions are approximately 4 fm, and they appear to be reasonable.

C6)

Furthermore, to test the availability of the two-component model, we
calculated the 3-pion BEC by making use of the estimated values and diagrams
presented in Fig. 4. Interestingly, as $N_{\rm trk}^{\rm offline}$ increases,
the 3-pion BEC rapidly decreases, due to the changes in the extension $R_{1}$
(1 fm to 4 fm). Moreover, the intercepts at $Q=0.0$ GeV are about 3.0,
providing the equal weight.

C7)

To investigate the role of the LRC(Gauss), i.e., Eq. (4), we reanalyzed the
BEC at 0.9 and 7 TeV, with the results presented in Appendix B. The estimated
$\chi^{2}$ values became smaller than that of LRC(linear) [7]. We also
analyzed the data with the constraint, $0.1\leq k_{T}\leq 0.3$ GeV and $2\leq
N_{\rm ch}\leq 9$. We observe that $R_{1}$’s increase and $R_{2}$’s are almost
constant.

C8)

As portrayed in Table 3, the BEC in the intervals $0\leq N_{\rm trk}^{\rm
offline}\leq 4$ and $10\leq N_{\rm trk}^{\rm offline}\leq 12$ cannot be
analyzed with better $\chi^{2}$ values. A more complicated model may be
necessary.

C9)

From Table 3, we can observe behaviors of $R_{1}$ and $R_{2}$’s in Fig. 6. The
larger extension $R_{1}$’s seem to be saturated at larger $N_{\rm trk}^{\rm
offline}$. To confirm that, of course, more data are needed. Compare Fig. 11
and Table 6 in Appendix B, where data at 7 TeV with the constraints $0.1\leq
k_{T}\leq 0.3$ GeV and and $2\leq N_{\rm ch}\leq 9$ are analyzed. See also
discussions on two kinds of extensions mentioned in Ref. [13].

Figure 6: Behaviors of $R_{1}$ and $R_{2}$’s estimated in Table 3. The smaller
extension $R_{2}$’s are almost the constant.

Acknowledgments. One of the authors (M.B.) would like to thank his colleagues
at the Department of Physics, Shinshu University.

##  Appendix A Analysis of BEC at 13 TeV using the $\tau$-model with Eq. (4)

We are interested in the influence of Eq. (4) on the $\tau$-model. To
investigate this, we reanalyzed the BEC using the following formula

$\displaystyle F_{\tau\mathchar 45\relax{\rm
Gauss}}=[1+\lambda\cos((r_{0}Q)^{2}+\tan(\alpha_{\tau}\pi/4)(Qr)^{\alpha_{\tau}})e^{-(Qr)^{\alpha_{\tau}}}]\cdot{\rm
LRC_{(Gauss)}}$ (7)

Our findings are presented in Fig. 7 and Table 4. It can be seen that the
interaction range $r$ values are smaller than 10 fm.

Figure 7: Fit to the BEC measurements by CMS in pp collisions at 13 TeV by Eq. (7) with Eq. (4). Table 4: Fit parameters of the CMS measurements of BEC in pp collisions at 13 TeV ($0.0\leq k_{T}\leq 1.0$ GeV) using the $\tau$-model with Eq. (4). $N_{\rm trk}^{\rm offline}$ |  $r_{0}$ (fm) |  $r$ (fm) | $\lambda$ | $\alpha_{\tau}$ | $\alpha$ | $\beta$ |  $\chi^{2}$/ndf
---|---|---|---|---|---|---|---
$0-4$ | $0.22\pm 0.02$ | $2.12\pm 0.52$ | $1.02\pm 0.12$ | $0.595\pm 0.047$ | $\\!\\!\\!\\!-0.169\pm 0.017$ | $5.60\pm 0.49$ | 169.6/92
$10-12$ | $0.25\pm 0.01$ | $9.89\pm 1.69$ | $2.49\pm 0.25$ | $0.417\pm 0.014$ | $0.099\pm 0.060$ | $0.17\pm 0.13$ | 138.7/92
$31-33$ | $0.16\pm 0.01$ | $3.27\pm 0.45$ | $1.76\pm 0.10$ | $0.566\pm 0.022$ | $0.177\pm 0.024$ | $19.86\pm 1.02$ | 85.9/92
$80-84$ | $0.00$ | $3.76\pm 0.06$ | $1.65\pm 0.02$ | $0.566\pm 0.002$ | $0.159\pm 0.003$ | $23.83\pm 0.31$ | 166.2/92
$105-109$ | $0.13\pm 0.01$ | $5.61\pm 0.54$ | $1.84\pm 0.08$ | $0.517\pm 0.012$ | $0.119\pm 0.010$ | $25.48\pm 0.73$ | 103.1/92
$130-250$ | $0.18\pm 0.02$ | $9.02\pm 3.43$ | $2.18\pm 0.42$ | $0.468\pm 0.037$ | $0.076\pm 0.024$ | $21.63\pm 3.39$ | 78.5/92

Figure 8: $\lambda_{\rm eff}$’s and LRCs of BEC measurements by CMS in pp
collisions at 13 TeV by Eq. (7). The vertical line at $Q=2.0$ GeV represents
the effective range of the LRC ($0\leq Q\leq 2$ GeV).

As illustrated in Fig. 8, three LRC’s appear to be various. Therein the
behavior of LRC for $0\leq N_{\rm trk}^{\rm offline}\leq 4$ is related to the
negative $\alpha$. For the sake of reference, we demonstrate the effective
degree of coherence in the $\tau$-model,

$\lambda_{\rm
eff}=\lambda\cos((r_{0}Q)^{2}+\tan(\alpha_{\tau}\pi/4)(Qr)^{\alpha_{\tau}})$

in Fig. 8. By making use of $\lambda_{\rm eff}$’s and LRCs, we can estimate
the intercepts at $Q=0.0$ GeV, which are shown in Fig. 7.

##  Appendix B Reanalysis of CMS BEC at 0.9 and 7 TeV [2] by LRC, expressed by
Eq. (4)

We examined the changes in the values of $\chi^{2}$ when LRC(linear) was
replaced with Eq. (4) in the reanalysis of BEC at 0.9 and 7 TeV [2]. Our new
results obtained using Eq. (4) are presented in Fig. 9 and Table 5 and
compared with those obtained elsewhere [7], where the linear form for the
${\rm LRC}=C(1+\delta Q)$ was used. These results are also shown in Table 5.
We show the LRCs in Fig. 10.

Figure 9: Fit to the CMS BEC measurements in pp collisions at 0.9 and 7.0 TeV by Eqs. (2)–(4). Table 5: Fit parameters of the CMS BEC measurements in pp collisions at 0.9 and 7.0 TeV by Eqs. (2)–(4). |  $R_{1}$ (fm) |  $R_{2}$ (fm) | $\lambda_{1}$ | $\lambda_{2}$ |  $\\!\\!\\!\delta\,({\rm GeV}^{-1})$ or $(\alpha,\ \beta\,({\rm GeV}^{-2}))\\!\\!\\!$ |  $\chi^{2}$/ndf
---|---|---|---|---|---|---
$\sqrt{s}=0.9$ TeV |  |  |  |  |  |
LRC(linear) [7] | $1.42\pm 0.03$ | $3.37\pm 0.42$ | $0.53\pm 0.02$ | $0.23\pm 0.04$ | $0.030\pm 0.001$ | 417/192
Eq. (4) | $0.82\pm 0.10$ | $2.27\pm 0.13$ | $0.41\pm 0.01$ | $0.30\pm 0.02$ | $(0.10\pm 0.02,\ 1.38\pm 0.10)$ | 234/191
$\sqrt{s}=7$ TeV |  |  |  |  |  |
LRC(linear) [7] | $3.88\pm 0.18$ | $0.71\pm 0.01$ | $0.84\pm 0.03$ | $0.12\pm 0.01$ | $0.023\pm 0.001$ | 540/192
Eq. (4) | $3.13\pm 0.13$ | $0.51\pm 0.02$ | $0.80\pm 0.03$ | $0.10\pm 0.01$ | $(0.06\pm 0.01,\ 1.46\pm 0.11)$ | 217/191
Figure 10: LRCs by LRC(linear) [7] and LRC(Gauss) (Eq. (4)) of CMS BEC
measurements in pp collisions at 0.9 and 7.0 TeV are presented. The vertical
line at $Q=2.0$ GeV represents the effective range of the LRC ($0\leq Q\leq 2$
GeV).

It can be said that the Gaussian distribution of the LRC in the two-component
model is better than that of the linear form, because the LRC(Gauss) converges
to 1.0 in the region $Q\geq 2.0$ GeV. The reason is as follows: The emitting
source functions and/or the LRC’s in the Euclidean space
($Q^{\prime}=\sqrt{({\bf p}_{1}-{\bf p}_{2})^{2}+(E_{1}-E_{2})^{2}}$ and
$\xi^{\prime}=\sqrt{({\bf r}_{1}-{\bf r}_{2})^{2}+(t_{1}-t_{2})^{2}}$) are
calculated as

$\displaystyle F_{\rm
source}(\xi^{\prime},\,R)=\frac{1}{(2\pi)^{2}\xi^{\prime}}\int_{0}^{\infty}Q^{\prime
2}\,E_{\rm BE}(Q^{\prime},\,R)\,J_{1}(Q^{\prime}\xi^{\prime})dQ^{\prime}$ (8)

where $J_{1}(Q\xi)$ is the Bessel function. For the LRC, we should replace
$E_{\rm BE}$ with (LRC$-1.0$) and $R$ with $\beta$ in Eq. (8), respectively.
In other words, the $({\rm LRC_{(Gauss)}-1.0})=\sum_{k=1}^{\infty}(-\alpha
e^{-\beta Q^{2}})^{k}$ is preferable to the $({\rm LRC_{(linear)}-1.0)}=\delta
Q$, because the former converges, as $Q$ is large. Finally, we should adopt
the inverse Wick rotation for $\xi^{\prime}$ [12, 15]; $\xi=\sqrt{({\bf
r}_{1}-{\bf r}_{2})^{2}-(t_{1}-t_{2})^{2}}$.

Finally we analyzed data on BEC at 7 TeV with $0.1\leq k_{T}\leq 0.3$ GeV and
$2\leq N_{\rm ch}\leq 9$ in Ref. [2] by means of Eqs. (2)-(4). The smaller
extensions are almost constant. This fact is similar to Fig. 6. From estimated
parameters with the constraint $2\leq N_{\rm ch}\leq 9$ (fixed), we know that
both $R_{1}$ and $R_{2}$’s are approximately constant.

Figure 11: Fit to the BEC measurements by CMS in pp collisions at 7 TeV with $0.1\leq k_{T}\leq 0.3$ GeV by Eqs. (2)–(4). Table 6: Fit parameters of the CMS BEC measurements in pp collisions at 7.0 TeV with $0.1\leq k_{T}\leq 0.3$ GeV and $2\leq N_{\rm ch}\leq 9$ by Eqs. (2)–(4) with $0\leq\lambda_{1}\leq 1$ and $\lambda_{2}=1-\lambda_{1}$. $N_{\rm ch}$ means the charged particle multiplicity. $C$ (top to bottom): $1.037\pm 0.011$, $1.013\pm 0.001$, $1.008\pm 0.001$, $1.010\pm 0.008$, and $1.037\pm 0.011$. |  $R_{1}$(fm) |  $R_{2}$(fm) | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$ |  $\beta$ (GeV-2) |  $\chi^{2}$/ndf
---|---|---|---|---|---|---|---
$0.1\leq k_{T}\leq 0.3$ (fixed) |  |  |  |  |  |
$2\leq N_{\rm ch}\leq 9$ | $1.16\pm 0.31$ | $0.36\pm 0.19$ | $0.95\pm 0.10$ | $0.05\pm 0.10$ | $0.14\pm 0.04$ | $0.80\pm 0.28$ | 212/192
$10\leq N_{\rm ch}\leq 24$ | $2.05\pm 0.25$ | $0.46\pm 0.04$ | $0.95\pm 0.10$ | $0.05\pm 0.10$ | $0.14\pm 0.03$ | $2.42\pm 0.24$ | 188/192
$25\leq N_{\rm ch}\leq 80$ | $2.77\pm 0.14$ | $0.47\pm 0.04$ | $0.91\pm 0.02$ | $0.09\pm 0.02$ | $0.08\pm 0.02$ | $2.47\pm 0.26$ | 176/192
$2\leq N_{\rm ch}\leq 9$ (fixed) |  |  |  |  |  |
$0.1\leq k_{T}\leq 0.3$ | $1.16\pm 0.31$ | $0.36\pm 0.19$ | $0.95\pm 0.10$ | $0.05\pm 0.10$ | $0.14\pm 0.04$ | $0.80\pm 0.28$ | 212/192
$0.3\leq k_{T}\leq 0.5$ | $1.47\pm 0.20$ | $0.29\pm 0.05$ | $0.86\pm 0.03$ | $0.14\pm 0.03$ | $0.15\pm 0.04$ | $0.99\pm 0.23$ | 198/192
$0.5\leq k_{T}\leq 1.0$ | $0.97\pm 0.41$ | $0.25\pm 0.07$ | $0.71\pm 0.10$ | $0.29\pm 0.10$ | $0.43\pm 0.08$ | $1.16\pm 0.32$ | 177/192

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