# From Lieb-Robinson Bounds to Automorphic Equivalence

Bruno Nachtergaele  Department of Mathematics and Center for Quantum
Mathematics and Physics
University of California, Davis
Davis, CA 95616, USA<EMAIL_ADDRESS>
###### Abstract.

I review the role of Lieb-Robinson bounds in characterizing and utilizing the
locality properties of the Heisenberg dynamics of quantum lattice systems. In
particular, I discuss two definitions of gapped ground state phases and show
that they are essentially equivalent.

Based upon work supported by the National Science Foundation under
DMS–2108390. MSC: 82B10 (Primary), 82B20, 82C10. Keywords: Lieb-Robinson
bounds, quantum phase transitions, gapped ground states.

Dedicated to Elliott Lieb on the occasion of his 90th birthday.

##  1\. Introduction

Lieb and Robinson proved their now famous and ubiquitous propagation estimate
in a 1972 article in Communications in Mathematical Physics [18]. The result
immediately attracted attention. It clearly established an important
fundamental property of the dynamics of quantum spin systems when the
interactions are sufficiently short-range. With this 7-page paper they planted
the seed of a revolution. That seed lay dormant for over thirty years. It took
that long before anyone understood that Lieb-Robinson bounds are the key to
proving a number of important results. It was Matthew Hastings who realized
that a propagation estimate of the Lieb-Robinson type should hold and that it
could be used, for example, to prove exponential clustering in a gapped ground
state. He applied Lieb-Robinson bounds without knowing, at first, that such an
estimate already existed in the literature and without knowing its proof. The
estimate of a finite correlation length in terms of the ground state gap
played an essential role in Hastings’ breakthrough paper on the multi-
dimensional Lieb-Schultz-Mattis theorem [13]. An exponential clustering
theorem had been proved for a while in the context of relativistic quantum
field theories [11], where the speed of light provides an absolute bound on
the speed of propagation of signals. Rigorous proofs of the non-relativistic
exponential clustering theorem and the Lieb-Robinson bounds used in them
appeared in [24, 14].

Interest in deriving Lieb-Robinson bounds for new situations continues
unabated. The increasing variety of physical and ‘artificial’ quantum many-
body systems available for the experimentalist is the primary motivation for
this, but there are also theoretical considerations that prompt researchers to
try to improve and extend existing bounds. For the mathematical physicist this
continuing interest is both a blessing and a curse. On the one hand, new
questions prompt new interesting research but, on the other hand, the ‘need
for speed’ in some communities leads to a sometimes premature rush to
publication. Some potentially interesting results remain unverified to my
knowledge. In part for this reason, I will not focus on the latest
improvements and extensions that make Lieb-Robinson type bounds available for
an ever wider variety of physical systems. Instead, I focus on the role Lieb-
Robinson bounds play in the definition and classification of gapped ground
state phases. That topic too continues to develop and we restrict ourselves
here to reviewing the importance of locality properties and the fundamental
role Lieb-Robinson bounds play in expressing the locality properties
quantitatively.

##  2\. Lieb-Robinson bounds

Lieb-Robinson bounds are a quantitative expression of the (quasi-)locality of
the dynamics of a quantum many-body system. This locality property stems from
the decay of the interaction strength as a function of distance. There are
different ways of imposing sufficient decay conditions on the interactions
that allow one to prove a Lieb-Robinson bound. The form the bounds take
depends, among other things, on the form of the assumptions. It is not my goal
here to review the many options that have been discussed in the literature,
but rather to emphasize how the concept of locality is playing a central role
in current research on quantum many-body physics.

###  2.1. Locality of observables

For simplicity of the presentation, we will restrict ourselves to quantum spin
systems. It is useful to leave the ‘lattice’ as general as possible. This
means considering spins associated with the points in a discrete metric space
$(\Gamma,d)$ that obeys a growth condition on the number of sites in balls: we
will assume that there are constants $c,\nu>0$ such that the number of points
in a ball of radius $r$ is bounded by $1+cr^{\nu}$. Some authors assume that
the set is Delone, meaning that $\Gamma$ is embedded in ${\mathbb{R}}^{d}$,
for some $d$, that the bound on the growth of balls follows from a minimum
distance between points and, in addition, that there is also a radius
$r_{0}>0$, such that $|B_{x}(r)|\geq 2$ for all $r>r_{0},x\in\Gamma$. Such an
assumption is not needed for the main results I discuss here. As a concrete
example of $(\Gamma,d)$, one can think of ${\mathbb{Z}}^{\nu}$, $\nu\geq 1$,
with the usual $\ell^{1}$ distance.

For each $x\in\Gamma$, the observable algebra is a finite-dimensional complex
matrix algebra $\mathcal{A}_{\\{x\\}}$, and for finite $\Lambda\subset\Gamma$,

$\mathcal{A}_{\Lambda}=\bigotimes_{x\in\Lambda}\mathcal{A}_{\\{x\\}}.$

For $\Lambda_{1}\subset\Lambda_{2}$, $\mathcal{A}_{\Lambda_{1}}$ is naturally
embedded into $\mathcal{A}_{\Lambda_{2}}$ and therefore we can define

${\mathcal{A}}_{\rm loc}=\bigcup_{\mbox{\small finite
}\Lambda\subset\Gamma}{\mathcal{A}}_{\Lambda},\quad{\mathcal{A}}_{\Gamma}=\overline{{\mathcal{A}}_{\rm
loc}}^{\|\cdot\|},$

where the completion is taken with respect to the standard operator norm. This
turns ${\mathcal{A}}_{\Gamma}$ into a $C^{*}$-algebra. The algebras
${\mathcal{A}}^{\rm loc}_{\Gamma}$ and ${\mathcal{A}}_{\Gamma}$ are referred
to as the algebras of local and quasi-local observables, respectively.
$A\in\mathcal{A}_{\Lambda}$ is said to be supported in $\Lambda$, and the
support of $A$, $\operatorname{supp}A$, is the smallest $\Lambda$ for which
$A\in\mathcal{A}_{\Lambda}$.

By construction, for all $A\in{\mathcal{A}}_{\Gamma}$ and any increasing
sequence of finite $\Lambda_{n}\uparrow\Gamma$, there exists a sequence of
local observables $A_{n}\in{\mathcal{A}}_{\Lambda_{n}}$, such that $A_{n}\to
A$. We will characterize the rate of convergence by positive non-increasing
functions $f$ with $\lim_{n}f(n)=0$, which we will refer to as decay
functions.

For a given sequence $(\Lambda_{n})$, increasing to $\Gamma$, and a decay
function $f$, we define

(2.1) $\|A\|_{f}=\|A\|+\sup_{n}f(n)^{-1}\inf\\{\|A-A_{n}\|\mid
A_{n}\in{\mathcal{A}}_{\Lambda_{n}}\\},$

and

${\mathcal{A}}^{f}_{\Gamma}=\\{A\in{\mathcal{A}}_{\Gamma}\mid\|A\|_{f}<\infty\\}.$

It is not hard to show that ${\mathcal{A}}^{f}_{\Gamma}$ is a Banach ∗-algebra
with norm $\|\cdot\|_{f}$ [22].

A concrete sequence of local approximations of any
$A\in{\mathcal{A}}_{\Gamma}$ is obtained by using the conditional expectations
$\Pi_{\Lambda}$ determined by the tracial state, $\rho$, on
${\mathcal{A}}_{\Gamma}$,

$\Pi_{\Lambda}={\rm
id}_{{\mathcal{A}}_{\Lambda}}\otimes\rho\restriction_{{\mathcal{A}}_{\Gamma\setminus\Lambda}}.$

The local approximations given by $\Pi_{\Lambda_{n}}(A)$ are not necessarily
optimal but the error is always bounded by twice the optimal one: for any
$A_{n}\in{\mathcal{A}}_{\Lambda_{n}}$ we have $\Pi_{\Lambda_{n}}(A_{n})=A_{n}$
and, hence

$\inf\\{\|A-B\|\mid
B\in{\mathcal{A}}_{\Lambda_{n}}\\}\leq\|A-\Pi_{\Lambda_{n}}(A)\|\leq\|A-A_{n}\|+\|\Pi_{\Lambda_{n}}(A_{n}-A)\|\leq
2\|A-A_{n}\|,$

where, for the last step, we used $\|\Pi_{\Lambda_{n}}\|=1$. By replacing the
$\inf$ in (2.1) by $\|A-\Pi_{\Lambda_{n}}(A)\|$ one obtains a more explicit
equivalent norm.

For lattice systems with an infinite-dimensional Hilbert space at each site,
such as oscillator lattices, a normalized trace does not exist, yet the same
type of estimates can be obtained [23].

Lieb-Robinson bounds express locality of observables by a commutator estimate,
which is again equivalent to the error of local approximations up to a factor
of $2$:

$\|A-\Pi_{\Lambda}(A)\|\leq\sup_{B\in{\mathcal{A}}_{\Gamma\setminus\Lambda},\|B\|=1}\|[A,B]\|\leq
2\|A-\Pi_{\Lambda}(A)\|.$

This relation also shows that ${\mathcal{A}}_{\Gamma}$ for any countably
infinite $\Gamma$, is asymptotically abelian in the sense that

$\lim_{n}\sup_{B\in{\mathcal{A}}_{\Gamma\setminus\Lambda_{n}},\|B\|=1}\|[A,B]\|=0,$

for all $A\in{\mathcal{A}}_{\Gamma}$ and increasing sequences
$\Lambda_{n}\uparrow\Gamma$.

A common choice for the finite volumes $\Lambda_{n}$ is balls of radius $n$
centered at some $x\in\Gamma$. For many families of decay functions that have
been used in the literature so far, the choice of $x$ is not important as the
norms defined with different $x$ are all equivalent. For some purposes, it is
better to consider a Fréchet space of observables with a family of
(semi-)norms [16].

###  2.2. Interactions with sufficient decay

A quantum spin model is typically defined in terms of an interaction: a map
$\Phi$ defined on the finite subsets of $\Gamma$ such that
$\Phi(X)=\Phi(X)^{*}\in\mathcal{A}_{X}$, for all finite $X\subset\Gamma$. For
each finite $\Lambda\subset\Gamma$, a local Hamiltonian is defined as follows:

$H_{\Lambda}=\sum_{X\subset\Lambda}\Phi(X),$

which in turn defines the finite-volume Heisenberg dynamics:

$\tau_{t}^{\Lambda}(A)=U_{\Lambda}(t)^{*}AU_{\Lambda}(t),\mbox{ with
}U_{\Lambda}=e^{-itH_{\Lambda}}.$

$\\{\tau_{t}^{\Lambda}\\}_{t\in{\mathbb{R}}}$ is a group of ∗-automorphisms of
${\mathcal{A}}_{\Gamma}$ that leaves ${\mathcal{A}}_{\Lambda}$ invariant.

If the interaction depends on time, $t\mapsto\Phi(X,t)\in\mathcal{A}_{X}$,
bounded and measurable, then

$\displaystyle\frac{d}{dt}U_{\Lambda}(t,s)$
$\displaystyle=-iH_{\Lambda}(t)U_{\Lambda}(t,s)$ $\displaystyle
U_{\Lambda}(s,s)$
$\displaystyle={\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}},$

defines co-cycles of unitaries $U_{\Lambda}(t,s)$ and automorphisms

$\tau_{t,s}^{\Lambda}(A)=U_{\Lambda}(t,s)^{*}AU_{\Lambda}(t,s).$

In applications we often make the further assumption that the $t$-dependence
is piecewise differentiable and $t$ may take values in a finite or infinite
interval $I\subset{\mathbb{R}}$.

If for all $x\in\Gamma$, $\sum_{X,x\in X}\|\Phi(X)\|<\infty$, we can define a
derivation $\delta:\mathcal{A}_{\rm loc}\to\mathcal{A}_{\Gamma}$, by

$\delta(A)=\sum_{X,X\cap Y\neq\emptyset}[\Phi(X),A],\quad
A\in\mathcal{A}_{Y},\mbox{ finite }Y\subset\Gamma.$

Under conditions of sufficient decay, $\delta$ can be exponentiated to define
$\tau^{\Phi}_{t}=e^{it\delta}$, a dynamics on $\mathcal{A}_{\Gamma}$. In
practice, one finds conditions on $\Phi$ such that $\tau_{t}^{\Lambda}(A)$
converges for all $A\in\mathcal{A}_{\rm loc}$ as $\Lambda\uparrow\Gamma$,
uniformly in $t$ in compact intervals.

A convenient way to state such a decay condition uses the notion of
$F$-function, defined as follows.

  1. i.

$F:[0,\infty)\to(0,\infty)$, non-decreasing;

  2. ii.

$\|F\|_{1}:=\sup_{y\in\Gamma}\sum_{x\in\Gamma}F(d(x,y))<\infty$;

  3. iii.

$\sum_{z\in\Gamma}F(d(x,z))F(d(z,y))\leq C_{F}F(d(x,y|))$ for all
$x,y\in\Gamma$ and a suitable constant $C_{F}$.

Then, for an interaction $\Phi$ as above, possibly time-dependent, we define

$\|\Phi\|_{F}(t)=\sup_{x,y\in\Gamma}\frac{1}{F(d(x,y))}\sum_{X,x,y\in
X}\|\Phi(X,t)\|,t\in I.$

Given $F$ and $I$ we can then define the Banach space of interactions
continuous on $I$:

${\mathcal{B}}_{F}(I)=\\{\Phi(\cdot,\cdot)\mid\Phi(X,\cdot)\mbox{ continuous
for all finite }X\subset\Gamma,\mbox{ and }\|\Phi(t)\|_{F}\mbox{ is locally
bounded}\\}.$

For $\Gamma={\mathbb{Z}}^{\nu}$, it is easy to check that

$F(r)=\frac{1}{(1+r)^{\nu+\epsilon}},\mbox{ for any }\epsilon>0$

is an $F$-function. Furthermore $F$ of the following form are $F$-functions:

$F(r)=e^{-ag(r)}F_{0}(r),$

where $F_{0}$ is an $F$-function and $g:[0,\infty)\to[0,\infty)$ is sub-
additive. Useful choices for $g$ are $g(r)=r^{\theta},\theta\in(0,1]$, and
$g(r)=r/(\log(1+r)^{2})$. The first characterizes interactions with
exponential and stretched exponential decay and the second applies to the
typical case of interactions defining the quasi-adiabatic evolution (see
(3.2)). The spaces of interactions defined by $F$-functions with stretched
exponential pre-factor will also be denoted by ${\mathcal{B}}_{a,\theta}$.

If for some $F$-function $F$, $\|\Phi\|_{F}(t)$ is locally bounded, it follows
that the limit $\tau^{\Phi}_{t}(A)$ defines a strongly continuous one-
parameter group of automorphisms of $\mathcal{A}_{\Gamma}$,
$\tau^{\Phi}_{t}=e^{it\overline{\delta}}$. In this situation,
$\overline{\delta}$ is the closure of $\delta$, i.e., ${\mathcal{A}}_{\rm
loc}$ is a core for $\overline{\delta}$.

In general, there are many interactions $\Phi$ that generate the same
dynamics, and this freedom is useful. For example, if $(\Gamma,d)$ is a Delone
subset of ${\mathbb{R}}^{\nu}$, one can assume that $\Phi$ is supported on
balls $b_{x}(n)\subset\Gamma$ and express decay by a condition of the form

$\|\Phi(b_{x}(n))\|\leq\|\Phi\|_{f}f(n),\mbox{ for all }x\in\Gamma.$

A procedure to do this is described in [28, Section 2.5]. The generator
$\delta$ can then be given on ${\mathcal{A}}_{g}$, for some $g$, as

$\delta(A)=\sum_{x}[\Phi_{x},A],\quad\Phi_{x}=\sum_{n\geq 1}\Phi(b_{x}(n)).$

For suitable $f$ and $g$, there is $h$ for which
$\delta(A)\in{\mathcal{A}}_{h}$. A concrete example of practical use is the
following. If $f(n)=g(n)=e^{-an^{\theta}}$, for $a>0$, and $\theta\in(0,1]$,
one can take $h(n)=e^{-a^{\prime}n^{\theta}}$, with $a^{\prime}<a$.

###  2.3. Lieb-Robinson bounds

For interactions $\Phi\in{\mathcal{B}}_{F}({\mathbb{R}})$, we have the
following theorem.

######  Theorem 2.1 (Lieb-Robinson Bound).

Let $\Phi\in{\mathcal{B}}_{F}(I)$, and $X,Y$ finite subsets of $\Gamma$ with
$X\cap Y=\emptyset$. Then, for all $\Lambda\subset\Gamma$,
$A\in{\mathcal{A}}_{X},B\in{\mathcal{A}}_{Y}$, and $t<s\in I$, we have

(2.2) $\left\|[\tau^{\Phi,\Lambda}_{t,s}(A),B]\right\|\leq
C_{F}^{-1}2\|A\|\|B\|\left(e^{2\int_{s}^{t}\|\Phi(r)\|_{F}dr}-1\right)\sum_{x\in
X,y\in Y}F(d(x,y)).$

To prove this theorem one proves it first for finite $\Lambda$. The finite-
volume result can then be used to prove the existence of the thermodynamic of
the dynamics, which then also satisfies the bound stated in the theorem. It is
worth noting that this result for the thermodynamic limit of the dynamics for
quantum spin systems goes beyond the traditional approach [3, 33] if one
considers interactions in which all many-body terms may be non-vanishing. This
is important for proving the locality properties of the quasi-adiabatic
evolution (also known as the spectral flow). See [27] for the details.

For time-independent interactions that decay exponentially, that is
$\Phi\in{\mathcal{B}}_{F}$, with $F(r)=e^{-ar}F_{0}$ for another $F$-function
$F_{0}$, one easily recovers the original form of the Lieb-Robinson bounds. It
suffices to estimate the sum in the right-hand side of (2.2) as follows:

$\sum_{x\in X,y\in Y}F(d(x,y))\leq e^{-ad(X,Y)}\sum_{x\in X,y\in
Y}F_{0}(d(x,y))\leq\|F_{0}\|_{1}\min(|X|,|Y|)e^{-ad(X,Y)}.$

Using the time-independence of $\|\Phi\|_{F}$, one then obtains immediately
the usual exponential form:

$\displaystyle\left\|[\tau^{\Phi}_{t}(A),B]\right\|$ $\displaystyle\leq
C_{F}^{-1}2\|A\|\|B\|\left(e^{2|t|\|\Phi\|_{F}}-1\right)\|F_{0}\|_{1}\min(|X|,|Y|)e^{-ad(X,Y)}$
$\displaystyle\leq
2\|A\|\|B\|C_{F}^{-1}\|F_{0}\|_{1}\min(|X|,|Y|)\left(e^{2|t|\|\Phi\|_{F}}-1\right)e^{-ad(X,Y)}$
(2.3) $\displaystyle\leq C\|A\|\|B\|\min(|X|,|Y|)e^{a(v_{LR}|t|-d(X,Y))},$

with

$v_{LR}=2a^{-1}\|\Phi\|_{F},\quad C=2C_{F}^{-1}\|F_{0}\|_{1}.$

I already mentioned the applications of Lieb-Robinson bounds to proving
exponential clustering of gapped ground states and generalizations of the
Lieb-Schultz-Mattis theorem to higher dimensions [14, 24, 13, 25]. In a few
years after Hastings’ rediscovery of Lieb-Robinson bounds, numerous
applications appeared at an accelerating rate, in particular in quantum
information theory and its applications to quantum many-body systems. In many
cases they allow properties valid for finite systems to be extended to the
thermodynamic limit by proving that they hold ‘uniformly in the volume’. A
basic example of this type is the following theorem, which provides a
continuity estimate for the dependence of the infinite-system dynamics on the
interaction $\Phi$.

######  Theorem 2.2 ([27]).

Let $\Phi,\Psi\in{\mathcal{B}}_{F}(I)$, $X\subset\Gamma$ finite, and
$A\in{\mathcal{A}}_{X}$. Then, for all $s\leq t\in I$,

$\|\tau_{t,s}^{\Phi}(A)-\tau_{t,s}^{\Psi}(A)\|\leq
2\|F\|_{1}C_{F}^{-1}|X|\|A\||t-s|e^{2|t-s|\min(\|\Phi\|_{F},\|\Psi\|_{F})}\|\Phi-\Psi\|_{F}.$

The Lieb-Robinson bound of Theorem 2.1 applies to interactions that decay as a
power law with sufficiently large exponent. This suffices for proving the
existence of the thermodynamic limit of the dynamics and some other
applications but in other situations one wants something better. For example,
for exponentially decaying interactions the bound (2.3) shows the existence of
a ‘light cone’ that contains the essential support of time-evolved local
observables. To obtain an analogue of such a light cone for systems with power
law interactions, a great deal of effort has been dedicated to improving the
estimates in Lieb-Robinson bounds for such cases [32, 19, 10, 17, 35]. Another
generalization with interesting applications concerns situations where $X$ or
$Y$, or both, are infinite [7, 30].

##  3\. Automorphic equivalence of gapped phases

Gapped ground state phases are represented by open regions in some space of
models (interactions). Much attention has been given in recent years to the
topological classification of these phases. Instead of the usual notion of
order parameters to characterize an ordered phase in statistical mechanics,
one uses topological invariants (Chern numbers, various indices) to classify
the topological structure of interest. Regardless of whether one is interested
specifically in topological order or more generally in gapped ground state
phases including those characterized by so called Landau order, one needs a
mathematical description of which models and their ground state(s) belong to
the same phase.

Concretely, we would like to define an equivalence relation and identify
phases with equivalence classes. Two seemingly different approaches have
emerged in the literature. In the first, the emphasis is on parameter
dependent model Hamiltonians. Two models are defined to represent the same
gapped ground state phase if there is an interpolating curve of model
Hamiltonians (interactions) along which the ground state gap satisfies a
common positive lower bound. In the second, the focus is on the ground states
themselves. Two states (which may or may not be assumed ground states of
gapped Hamiltonians) are defined to belong to the same phase if there is a
continuous family of sufficiently quasi-local automorphisms interpolating
between them. Before investigating the relation between these two points of
view, I give a precise formulation of each.

For concreteness, we will consider the class of interactions
$\Phi\in{\mathcal{B}}_{F}$, for an $F$-function $F$ on $(\Gamma,d)$, of
stretched exponential form. Denote by $\delta^{\Phi}$ the corresponding
generator of the Heisenberg dynamics on ${\mathcal{A}}_{\Gamma}$. Furthermore,
we restrict to the case of a finite number of mutually disjoint pure ground
states ${\mathcal{S}}^{\Phi}=\\{\omega_{1},\ldots,\omega_{n}\\}$. We say that
the model defined by $\Phi$ has gapped ground states with gap $\gamma$,
$\gamma>0$, if for all $i=1,\ldots,n$, we have

(3.1) $\omega_{i}(A^{*}\delta^{\Phi}(A))\geq\gamma\omega(A^{*}A),\mbox{ for
all }A\in{\mathcal{A}}^{\rm loc},\mbox{ with }\omega_{i}(A)=0.$

This is equivalent to saying that the GNS Hamiltonian for the system in the
ground state $\omega_{i}$ is a non-negative self-adjoint operator with a one-
dimensional kernel and a spectral gap above zero of size $\geq\gamma$. Such
ground states have been called locally unique gapped ground states by Tasaki
[34].

###  3.1. Equivalence of interactions

Suppose $\Phi_{0}$ and $\Phi_{1}$ are two interactions in the class
${\mathcal{B}}_{a,\theta}$, with ground state sets ${\mathcal{S}}^{\Phi_{0}}$
and ${\mathcal{S}}^{\Phi_{1}}$, respectively.

######  Definition 3.1.

[8, 27] The interactions $\Phi_{0}$ and $\Phi_{1}$ belong to the same phase if
there exists a differentiable curve of interactions $[0,1]\ni s\mapsto\Phi(s)$
such that the following hold:

  1. i.

$\Phi(0)=\Phi_{0},\Phi(1)=\Phi_{1}$;

  2. ii.

There exists a constant $\gamma^{\prime}>0$, such that for all $s\in[0,1]$
$\Phi(s)$ has gapped ground states with gap $\gamma^{\prime}>0$.

  3. iii.

There exist $a^{\prime}>0,\theta^{\prime}\in(0,1]$, such that
$\Phi(\cdot)\in{\mathcal{B}}^{1}_{a^{\prime},\theta^{\prime}}([0,1])$, defined
as the Banach space of interactions for which, with
$F(r)=e^{-a^{\prime}r^{\theta^{\prime}}}F_{0}(r)$,

$\sup_{x,y\in\Gamma}\frac{1}{F(d(x,y))}\sum_{{\rm finite}X:x,y\in
X}\left(\|\Phi(X,s)\|+|X|\|\Phi^{\prime}(X,s)\|\right)$

is bounded by a bounded measurable function of $s$.

It is easy to show that ‘belonging to the same phase’ defines an equivalence
relation on a set of interactions.

###  3.2. Equivalence of states

Suppose ${\mathcal{S}}_{0}$ and ${\mathcal{S}}_{1}$ are two finite sets of
pure states of ${\mathcal{A}}_{\Gamma}$.

######  Definition 3.2.

[8, 1] The sets of states ${\mathcal{S}}_{0}$ and ${\mathcal{S}}_{1}$ are
automorphically equivalent (in the stretched exponential locality class) if
there exists a continuous curve of interactions $[0,1]\ni s\mapsto\Psi(s)$
such that the following hold:

  1. i.

There exist $a^{\prime}>0,\theta^{\prime}\in(0,1]$, such that for all
$s\in[0,1]$, $\Psi(s)\in{\mathcal{B}}_{a^{\prime},\theta^{\prime}}$;

  2. ii.

$[0,1]\ni s\mapsto\Psi(s)$ is continuous in the norm of
${\mathcal{B}}_{a^{\prime},\theta^{\prime}}([0,1])$;

  3. iii.

The family of automorphisms $\alpha_{s,0}$ generated by $\Psi(s)$ satisfies

${\mathcal{S}}_{1}=\\{\omega\circ\alpha_{1,0}\mid\omega\in{\mathcal{S}}_{0}\\}.$

It is also straightforward to show that ‘automorphic equivalence’ defines an
equivalence relation on sets of states.

###  3.3. Equivalent equivalences

We now show that the equivalence relations defined Definition 3.1 on sets of
interactions on the one hand and Definition 3.2 on sets of states on the other
hand are compatible in both directions.

######  Theorem 3.3 (From equivalence of states to equivalent interactions).

Let ${\mathcal{S}}_{0}$ be a set of mutually disjoint pure ground states with
a gap bounded below by $\gamma>0$ for the dynamics with generator $\delta_{0}$
defined by an interaction $\Phi_{0}\in{\mathcal{B}}_{a,\theta}$, for some
$a>0,\theta\in(0,1]$. If a set of states ${\mathcal{S}}_{1}$ is automorphic
equivalent to ${\mathcal{S}}_{0}$ in the stretched exponential locality class,
then there exists a differentiable curve of interactions of class
${\mathcal{B}}^{1}_{a^{\prime},\theta^{\prime}}([0,1])$, for some
$a^{\prime}>0,\theta^{\prime}\in(0,1]$, $\Phi(s),s\in[0,1]$, with
$\Phi(0)=\Phi_{0}$, and such that ${\mathcal{S}}_{1}$ are gapped ground states
with gap bounded below by $\gamma$ for the dynamics generated by $\Phi(1)$.

###### Proof.

Since each $\omega_{0}\in{\mathcal{S}}_{0}$ is a gapped ground state with gap
$\geq\gamma$ for the dynamics generated by $\delta_{0}$ defined in terms of an
interaction $\Phi(0)$, we have

$\delta_{0}(A)=\sum_{X\subset\Gamma,X\cap\operatorname{supp}(A)\neq\emptyset}[\Phi_{0}(X),A],\mbox{
for all }A\in{\mathcal{A}}^{\rm loc},$

and

$\omega_{i}(A^{*}\delta_{0}(A))\geq\gamma\omega_{i}(A^{*}A),\mbox{ for all
}A\in{\mathcal{A}}^{\rm loc},\mbox{ with }\omega_{i}(A)=0.$

Let $\alpha_{s},s\in[0,1]$, be the curve of automorphisms implied by the
automorphic equivalence of ${\mathcal{S}}_{1}$ with ${\mathcal{S}}_{0}$, and
define

$\delta_{s}=\alpha_{s}^{-1}\circ\delta_{0}\circ\alpha_{s},s\in[0,1].$

It is straightforward to check that the action of the generator $\delta_{s}$
on local observables $A$ can be given as

$\delta_{s}(A)=\sum_{X\subset\Gamma,X\cap\operatorname{supp}(A)\neq\emptyset}[\alpha_{s}^{-1}(\Phi_{0}(X)),A].$

The map $X\mapsto\tilde{\Phi}(X,s):=\alpha_{s}^{-1}(\Phi_{0}(X))$ is not an
interaction in the usual sense because $\tilde{\Phi}(X,s)$ are generally not
local. It is shown in [27, Section VI.E.2] how to construct a proper
interaction of the stretched exponential class, $\Phi(s)$, that generates the
correct infinite system dynamics, meaning
$\tau^{\Phi(s)}_{t}=\alpha_{s}^{-1}\circ\tau^{\Phi_{0}}_{t}\circ\alpha_{s},s\in[0,1],t\in{\mathbb{R}}$.
Of course, $\Phi(s)$ also leads to the same generator $\delta_{s}$.

Then, using ${\mathcal{A}}_{\rm loc}\subset{\mathcal{A}}_{f}\subset\mathop{\rm
dom}\delta_{s}$, and $\alpha_{s}^{\pm
1}({\mathcal{A}}_{f})\subset{\mathcal{A}}_{f}$, for suitable $f$, we can
verify that the gap remains bounded below by $\gamma$ along the curve (in
fact, the gaps for each pure state are constant as a function of $s$). To see
this, take $A\in{\mathcal{A}}^{\rm loc}$, such that $\omega_{s}(A)=0$. Then,
by the assumptions, $\alpha_{s}(A)\in\mathop{\rm dom}\delta_{0}$ and
$\omega_{0}(\alpha_{s}(A))=0$, we have

$\displaystyle\omega_{s}(A^{*}\delta_{s}(A))$
$\displaystyle=\omega_{0}(\alpha_{s}(A^{*})\alpha_{s}(\delta_{s}(A)))$
$\displaystyle=\omega_{0}(\alpha_{s}(A^{*})\delta_{0}(\alpha_{s}(A)))$
$\displaystyle\geq\gamma\omega_{0}(\alpha_{s}(A^{*})\alpha_{s}(A))$
$\displaystyle=\gamma\omega_{s}(A^{*}A).$

This proves that $\omega_{1}$ is a ground state for $\delta_{1}$ and with the
same gap as $\omega_{0}$ for $\delta_{0}$, and $\omega_{s},s\in[0,1]$ are
gapped ground states of the interpolating family $\Phi^{(s)}$. Since
$\omega_{0}\in{\mathcal{S}}_{0}$ was arbitrary, this conclude the proof. ∎

######  Theorem 3.4 (From equivalent interactions to automorphic equivalence).

Suppose $s\mapsto\Phi(s)$ is a differentiable curve of interactions of class
${\mathcal{B}}^{1}_{a,\theta}([0,1])$, such that there exists $\gamma>0$ and
sets of mutually disjoint pure gapped ground states ${\mathcal{S}}_{s}$,
$s\in[0,1]$, with gap bounded below by $\gamma$. Then, there exists a strongly
continuous curve of automorphisms $\alpha_{s}$ of class
${\mathcal{B}}_{a^{\prime},\theta^{\prime}}([0,1])$, such that

${\mathcal{S}}_{s}=\\{\omega\circ\alpha_{s,0}\mid\omega\in{\mathcal{S}}_{0}\\}.$

###### Proof.

For any differentiable curve of interactions one can define the Hastings
generator of what is often referred to as the quasi-adiabatic evolution[13,
15, 1, 27]. It is defined by transforming the following ‘non-local
interaction’ into a standard interaction:

$\tilde{\Psi}(X,s)=\int_{-\infty}^{\infty}w_{\xi}(t)\int_{0}^{t}\tau_{u}^{(s)}\left(\frac{d}{ds}\Phi(X,s)\right)du\,dt.$

Here, there are in principle many possible choices for the function
$w_{\xi}(t)$. A specific choice is the $L^{1}$-normalized positive function
discussed in detail in [27, Section VI.B]. It satisfies

(3.2) $w_{\xi}(t)\leq c\xi|t|e^{-\eta\frac{\xi|t|}{(\ln(\xi|t|))^{2}}}$

for some numerical constants $c>0$ and $\eta>2/7$, and $\xi$ is a parameter
that needs to satisfy $\xi\in(0,\gamma)$. A larger value of $\xi$ generally
leads to better locality properties for the automorphisms $\alpha_{s}$
generated by the $\Psi(s)$. See [27, Section VI] for an in depth discussion.
The upshot is that the curve $\alpha_{s}$ is generated by a ‘time-dependent’
interaction $\Psi(s)$ of a stretched exponential decay class:

$\frac{d}{ds}\alpha_{s}(A)=i\alpha_{s}(\delta^{\Psi(s)}(A)),A\in{\mathcal{A}}^{\rm
loc}.$

The crucial property of Hastings’ quasi-adiabatic evolution $\alpha_{s}$ is
then that for any $\omega_{0}\in{\mathcal{S}}_{0}$,
$\omega_{s}:=\omega_{0}\circ\alpha_{s}\in{\mathcal{S}}_{s}$, which proves the
automorphic equivalence of the sets of states ${\mathcal{S}}_{0}$ and
${\mathcal{S}}_{1}$. ∎

##  4\. Remarks

There are a number of topics closely related to the notion of gapped ground
state phase discussed in this paper that I have not discussed. The first
obvious question one would raise in connection to the gapped phases is their
stability. Are the interactions that lead to gapped ground states part of an
open set of interactions for which the gap persists? This type of stability
property has been the goal of much research over the past several decades.
Some recent approaches include [36, 9, 12]. A very successful approach that
uses Lieb-Robinson bounds at its core is the Bravyi-Hastings-Michalakis
strategy of [4, 5, 20] further developed in [28, 29].

For concreteness and brevity, I focused the discussion on quantum spin
systems, but note that Lieb-Robinson bounds and automorphic equivalence can be
proved and applied in the same way for lattice fermion systems [14, 6, 26, 21,
2, 31].

## Acknowledgements

As so many others, I continue to draw inspiration from Elliott Lieb’s
pathbreaking work on quantum lattice systems. This acknowledgement is too
short to do justice to the profound influence Elliott has had on my way of
doing mathematical physics. Of the many collaborators with whom I had the
pleasure of working on the topics discussed in this paper, I want to
acknowledge two in particular: Robert Sims (U Arizona) and Amanda Young (TU
Munich & MCQST).

## Funding

This paper is based upon work supported by the National Science Foundation
under grant DMS–2108390.

## References

  * [1] S. Bachmann, S. Michalakis, B. Nachtergaele, and R. Sims, Automorphic equivalence within gapped phases of quantum lattice systems. _Comm. Math. Phys._ 309 (2012), 835–871
  * [2] C. Bourne and Y. Ogata, The classification of symmetry protected topological phases of one-dimensional fermion systems. _Forum of Mathematics, Sigma_ 9 (2021), E25
  * [3] O. Bratteli and D. W. Robinson, _Operator algebras and quantum statistical mechanics_. 2 edn., 2, Springer Verlag, 1997
  * [4] S. Bravyi, M. Hastings, and S. Michalakis, Topological quantum order: stability under local perturbations. _J. Math. Phys._ 51 (2010), 093512
  * [5] S. Bravyi and M. B. Hastings, A short proof of stability of topological order under local perturbations. _Commun. Math. Phys._ 307 (2011), 609
  * [6] J.-B. Bru and W. de Siquiera Pedra, _Lieb-Robinson bounds for multi-commutators and applications to response theory_. Springer Briefs in Mathematical Physics 13, Springer Nature, 2017
  * [7] M. Cha, P. Naaijkens, and B. Nachtergaele, On the stability of charges in infinite quantum spin systems. _Commun. Math. Phys._ 373 (2020), 219–264
  * [8] X. Chen, Z.-C. Gu, and X.-G. Wen, Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. _Phys. Rev. B_ 82 (2010), no. 15, 155138
  * [9] W. De Roeck and M. Salmhofer, Persistence of exponential decay and spectral gaps for interacting fermions. _Commun. Math. Phys._ 365 (2019), 773–796
  * [10] D. V. Else, F. Machado, C. Nayak, and N. Y. Yao, An improved Lieb-Robinson bound for many-body Hamiltonians with power-law interactions. _Phys. Rev. A_ 101 (2020), 022333
  * [11] K. Fredenhagen, A remark on the cluster theorem. _Commun. Math. Phys._ 97 (1985), 461–463
  * [12] J. Fröhlich and A. Pizzo, Lie-Schwinger block-diagonalization and gapped quantum chains. _Commun. Math. Phys._ 375 (2020), 2039–2069
  * [13] M. B. Hastings, Lieb-Schultz-Mattis in higher dimensions. _Phys. Rev. B_ 69 (2004), 104431
  * [14] M. B. Hastings and T. Koma, Spectral gap and exponential decay of correlations. _Commun. Math. Phys._ 265 (2006), 781–804
  * [15] M. B. Hastings and X. G. Wen, Quasi-adiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance. _Phys. Rev. B_ 72 (2005), 045141
  * [16] A. Kapustin and N. Sopenko, Local Noether theorem for quantum lattice systems and topological invariants of gapped states. 2022, arXiv:2201.01327
  * [17] T. Kuwahara and K. Saito, Strictly linear light cones in long-range interacting systems of arbitrary dimensions. _Phys. Rev. X_ 10 (2020), 031010
  * [18] E. Lieb and D. Robinson, The finite group velocity of quantum spin systems. _Commun. Math. Phys._ 28 (1972), 251–257
  * [19] T. Matsuta, T. Koma, and S. Nakamura, Improving the Lieb–Robinson bound for long-range interactions. _Ann. H. Poincaré_ 18 (2017), 519–528
  * [20] S. Michalakis and J. Zwolak, Stability of frustration-free Hamiltonians. _Commun. Math. Phys._ 322 (2013), 277–302
  * [21] A. Moon and B. Nachtergaele, Stability of gapped ground state phases of spins and fermions in one dimension. _J. Math. Phys._ 59 (2018), 091415
  * [22] A. Moon and Y. Ogata, Automorphic equivalence within gapped phases in the bulk. _J. Funct. Analysis_ 278 (2020), 108422
  * [23] B. Nachtergaele, V. Scholz, and R. F. Werner, Local approximation of observables and commutator bounds. In _Operator methods in mathematical physics_ , edited by J. Janas, P. Kurasov, A. Laptev, and S. Naboko, pp. 143–149, Operator Theory: Advances and Applications 227, Birkäuser Verlag, 2013
  * [24] B. Nachtergaele and R. Sims, Lieb-Robinson bounds and the exponential clustering theorem. _Commun. Math. Phys._ 265 (2006), 119–130
  * [25] B. Nachtergaele and R. Sims, A multi-dimensional Lieb-Schultz-Mattis theorem. _Commun. Math. Phys._ 276 (2007), 437–472
  * [26] B. Nachtergaele, R. Sims, and A. Young, Lieb-Robinson bounds, the spectral flow, and stability for lattice fermion systems. In _Mathematical results in quantum physics_ , edited by F. Bonetto, D. Borthwick, E. Harrell, and M. Loss, pp. 93–115, Contemporary Mathematics 717, Amer. Math. Soc., 2018
  * [27] B. Nachtergaele, R. Sims, and A. Young, Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms. _J. Math. Phys._ 60 (2019), 061101
  * [28] B. Nachtergaele, R. Sims, and A. Young, Quasi-locality bounds for quantum lattice systems and perturbations of gapped ground states II. Perturbations of frustration-free spin models with gapped ground states. _Ann. H. Poincaré_ (2021)
  * [29] B. Nachtergaele, R. Sims, and A. Young, Stability of the bulk gap for frustration-free topologically ordered quantum lattice systems. 2021, arXiv:2102.07209
  * [30] Y. Ogata, A derivation of braided $C^{*}$-tensor categories from gapped ground states satisfying the approximate Haag duality. 2021, arXiv:2106.15741
  * [31] Y. Ogata, An invariant of symmetry protected topological phases with on-site finite group symmetry for two-dimensional Fermion systems. 2021, arXiv:2110.04672
  * [32] P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A. Gorshkov, and C. Monroe, Non-local propagation of correlations in quantum systems with long-range interactions. _Nature_ 511 (2014), 198
  * [33] B. Simon, _The statistical mechanics of lattice gases_. 1, Princeton University Press, 1993
  * [34] H. Tasaki, The Lieb-Schultz-Mattis theorem: A topological point of view. 2022, arXiv:2202.06243
  * [35] M. Tran, A. Guo, C. Baldwin, A. Ehrenberg, A. Gorshkov, and A. Lucas, The Lieb-Robinson light cone for power-law interactions. 2021, arXiv:2103.15828
  * [36] D. A. Yarotsky, Ground states in relatively bounded quantum perturbations of classical lattice systems. _Commun. Math. Phys._ 261 (2006), 799–819


# ML framework for global river flood predictions based on the Caravan dataset

Ioanna Bouri111University of Helsinki, Finland., Manu Lahariya222University of
Ghent, Belgium., Omer Nivron333University of Cambridge, UK., Enrique Portales
Julia444University of Valencia, Spain.,
Dietmar Backes555University of Luxembourg, Luxembourg., Piotr
Bilinski666University of Oxford, UK., Guy Schumann777RSS-Hydro/University of
Bristol, Luxembourg/UK..

###### Abstract

Reliable prediction of river floods in the first 72 hours can reduce harm
because emergency agencies have sufficient time to prepare and deploy for help
at the scene. Such river flood prediction models already exist and perform
relatively well in most high-income countries. But, due to the limited
availability of data, these models are lacking in low-income countries. Here,
we offer the first global river flood prediction framework based on the newly
published Caravan dataset. Our framework aims to serve as a benchmark for
future global river flood prediction research. To support generlizability
claims we include custom data evaluation splits. Further, we propose and
evaluate a novel two-path LSTM architecture (2P-LSTM) against three baseline
models. Finally, we evaluate the generated models on different locations in
Africa and Asia that were not part of the Caravan dataset.

##  1 Introduction

Floods can be devastating for humans, economies and nature. Recent
catastrophic flooding events in Pakistan and Australia have highlighted the
increasing severity of catastrophic flooding events. Between 1995 and 2015,
over 2.6 billion people were affected by floods, accounting for 56% of all
people exposed to weather-related disasters [3, 2]. The impact of floods is
worse in low and middle-income countries, which are home to 89% of the world’s
flood-exposed population [12].

Informed rescue operations, crisis management and planning can reduce the harm
to the effected areas by floods. For example, people who receive flood
warnings before the flood event are twice as likely to evacuate [11]. For such
warning systems to have the desired impact, the UN highlights the importance
of the first 72 hours after an event occurs [4]. Flood forecasting systems
with 72-hour lead-time currently only exist at a local level, mostly in high-
income countries. This may be due to lack of globally available data and the
complexity associated with hydrological models.

In this paper we offer the first global data-driven river flood prediction
framework. This is made possible due to the newly published Caravan dataset,
which is a global dataset that includes meteorological forcing data, catchment
attributes and discharge data for basins around the world [7].

Our framework provides an end-to-end solution to train and test machine
learning models for river flood predictions on a global scale. This is the
first generalizable global flood prediction framework based on the Caravan
dataset, to the best of our knowledge. The main contributions of this paper
are: (i) a framework to train and evaluate a global flood prediction model.
The framework evaluates the generalizability of the models based on a test
dataset defined using evaluation splits, i.e., if the predictions are
generalizable to basins within the Caravans dataset, as well as to worldwide
gauge locations that are not included in the Caravans dataset. (ii) we propose
and evaluate a novel model architecture (2P-LSTM) for river flood prediction,
and, (iii) Design experiments to evaluate the framework, train models using
real-world Caravans data, and compare the 2P-LSTM model against three
baselines which it outperforms for 72-hour ahead predictions. Furthermore, we
extrapolate our predictions in basin areas located in low-income, not well-
documented countries that were not part of the Caravans dataset. Code for the
framework and trained models are provided in GitHub
(https://github.com/mlahariya/floodcastai). Python was used for the models
developed in this article.

###  1.1 Application context

Why do humanitarian aid and disaster response agencies need a 72-hour notice
of river-floods over the globe? The first 72 hours after a natural disaster
occurs are crucial, according to the UN Office for the Coordination of
Humanitarian Affairs (OCHA). Response must begin during that time to save
lives. The first few hours are dedicated to assess the impact of the disaster.
Then, OCHA deploys skilled staff to the scene, releases inter-agency appeal
within 24-72 hours, and mobilizes initial funding within 72 hours [4]. In
short, if we are able to predict 72 hours before an event, disaster response
agencies can initiate their 72-hour plans before the negative impact has
already started.

Our research provides the first framework for global river flood predictions
with up to 72-hours notice before the start of a flooding event.

##  2 Data

In this work we use the newly published Caravan dataset, which is a global
dataset of 2532 gauge stations in river basins and their attributes (Figure 4
shows the gauge locations). These attributes can be put in two categories:
dynamic attributes and static attributes888For the full list of attributes
refer to Kratzert et al.[7]. Dynamic attributes are measurements that change
over time such as the daily temperatures, precipitation, etc. Static
attributes are measurements that are considered to be constant such as the
elevation of the river basin.

The framework we developed aims to identify when the water level tops the
river bank given previous dynamic and static attributes. The proxy for water
level in the Caravan dataset is the streamflow variable (mm/day) — which
describes the volume of water flowing through a river section divided by the
area of the basin.

The Caravan dataset $\cal D$ is structured in the form of a daily time-series,
one for each river basin. $t$ is on a daily granularity and $t\in[1980,2022]$.
For example, $t=1998.0026$ describes January 2nd 1998. The target variable for
daily mean streamflow (mm/day) at time $t$ will be denoted by $y_{t}$, where
$y_{t}\in\mathbb{R}$. Furthermore, we denote the dynamic attributes (e.g.
precipitation) at time $t$ as $x_{t}^{d}$, where $x_{t}^{d}\in\mathbb{R}^{40}$
and the static attributes (e.g. % vegetation cover) of the specific basin
which we denote by $x^{s}$, where $x^{s}\in\mathbb{R}^{209}$.

To ensure the model predictions can generalize to worldwide gauge locations
that are not included in the Caravans dataset, we have curated a new
independent dataset $\cal R$ of 195 river basins located in low and middle
income countries are not represented that are not represented on Caravan. The
data sources for our new dataset are: Global Runoff-Data Center [1] for our
target streamflow measurements, the ERA5 project [10] from the European Centre
for Medium-Range Weather Forecasts (ECMWF) for our dynamic attributes, and
HydroATLAS [9] for the static attributes. These are the same sources that have
been used to create the Caravan dataset.

##  3 Methodology

In this section, we describe the framework used to train and evaluate a global
flood prediction model. The machine learning pipeline is shown in Figure 1.

Figure 1: Overview of the framework to train model _f_.

###  3.1 Models

Our machine learning pipeline focuses on the daily mean streamflow prediction
at each gauge station. We develop a framework that predicts the streamflow at
each time step $t,t+1,t+2,t+3$, as to achieve 72-hour lead-time predictions.
Following the mathematical notation introduced in section 2, the learning
setting is:

$\hat{y}_{t:t+3}=f(x_{t}^{d},\dots,x_{t-k}^{d},x^{s},y_{t-1}),$ (1)

where $f$ denotes each model written as a function, mapping the input
variables to the predicted streamflow sequence $\hat{y}_{t:t+3}$ and $k$ is
the length of the look-back sequence. Since the temporal resolution of the
data is daily, each time step $t+m$ represents $m$ days-ahead quantities. The
estimated sequence $\hat{y}_{t:t+3}$ depends on the last $k$ dynamic
variables, meaning $\\{x^{d}_{t},...x^{d}_{t-k}\\}$, the static variables
$x^{s}$ and streamflow value at the previous time step $y_{t-1}$. To extend
the prediction task to a 72-hour lead-time, we estimate
$\hat{y}_{t},\hat{y}_{t+1},\hat{y}_{t+2},\hat{y}_{t+3}$.

#### Two-path architecture (2P-LSTM)

The 2P-LSTM architecture focuses on modelling separately the dynamic and
static attributes. Such an approach can be effective when modelling
hydrological data that often consists of dynamic variables (e.g. hydrological
signatures) and static attributes (e.g. catchment properties). Separate
modelling of static and dynamic attributes has been previously studied in
recent literature and has provided promising results in the hydrological
domain [8, 5, 6].

As it is illustrated in figure 2, the 2P-LSTM architecture separates the input
data into two modelling paths. The dynamic variables follow a path through a
sequence-to-sequence LSTM network (Seq2Seq LSTM), and the static attributes
follow a path through a feedforward network [13]. The outputs of the two paths
are then concatenated and serve as input to a feedforward network. Finally,
the output of the network consists of a sequence of predicted streamflow
values over the time steps $t,t+1,t+2,t+3$.

Figure 2: An overview of the 2P-LSTM architecture.

The Seq2Seq LSTM that models the dynamic attributes consists of two networks:
an encoder and a decoder. The encoder learns to predict a streamflow value
using the time-dependent features in the look-back sequence $k$. The outputs
of the encoder are discarded and its cell state and hidden state information
are used to initialize the decoder network. The decoder then generates the
output sequence predicting the streamflow values in the next time step. In
training mode, the decoder takes as input a streamflow sequence of
$y_{t-1},y_{t},y_{t+1},y_{t+2}$, and predicts an estimated streamflow sequence
$\hat{y}_{t},\hat{y}_{t+1},\hat{y}_{t+2},\hat{y}_{t+3}$. However, in inference
mode, the input streamflow sequences are not available. In this case, only the
streamflow value $y_{t-1}$ is used as input, and, in a recursive manner, the
resulting estimate for $\hat{y}_{t}$ is used as input to predict the
streamflow at the next time step.

We compare the proposed 2P-LSTM with three baseline models: linear regression
(LR), random forest (RF) ensembles, and a feedforward neural network (NN).

###  3.2 Evaluation

A generalizable model should be able to make predictions in ungauged basins
and in locations where there is no training data. Additionally, the model
should generalize to unrepresented climates zones.Traditional model validation
methods (such as random k-fold validation) are wanting in appropriate estimate
of models performance on ungauged basins and unrepresented climates – to
ensure its generalizability globally. To ensure models global
generalizability, we define custom _Evaluation splits_ on split Caravan
dataset into train, validation and test datasets. These test datasets are
prepared based on two types of splits, namely, (i) holdout split, and (ii)
stratified split.

In _Holdout split_ , the test set contains the data from ungauged basin in
unrepresented climates. Thus, historical information of the gauges and climate
zones in the test set is not used to train the model. In _Stratified split_ ,
the test set contains the data from ungauged basins in seen climates. For each
climate zone, we keep 10% of the gauges in the test set, 10% in the validation
set, and 80% in the training set. Thus, we do stratified sampling of gauges
based on climate zones based on the climate zone in the basin area.

##  4 Experimental Results

Tables 2 and 2 report the RMSE, obtained from the stratified and holdout test
sets respectively. First, notice that the errors for the stratified split are
$\sim$ 1 mm/day for $y_{t}$. The predictions for $y_{t+1},y_{t+2},y_{t+3}$
show bigger errors, as expected due to cumulative time-series error. Second,
the errors for the holdout split are slightly higher. The prediction task for
the holdout split is harder because the test set contains river basins from
unrepresented climate zones.

Table 1: RMSE for models on _stratified split_ test dataset | Time-step |
---|---|---
_(mm/day)_ | yt | yt+1 | yt+2 | yt+3 |
LR | 1.81 | 2.10 | 2.45 | 2.55 |
RF | 1.42 | 1.76 | 2.07 | 2.23 |
NN | 1.86 | 2.46 | 2.70 | 2.84 |
2P-LSTM | 1.43 | 1.86 | 1.97 | 2.13 |

Table 2: RMSE for models on _holdout split_ test dataset  | Time-step |
---|---|---
_(mm/day)_ | yt | yt+1 | yt+2 | yt+3 |
LR | 2.44 | 3.18 | 3.55 | 3.55 |
RF | 1.64 | 1.92 | 2.18 | 2.31 |
NN | 2.10 | 2.69 | 2.89 | 3.00 |
2P-LSTM | 1.58 | 1.94 | 2.25 | 2.49 |

Figure 3: 72-hour ahead streamflow predictions produced by the 2P-LSTM model
trained using the stratified split. On the right, predictions are extrapolated
to a gauge in Africa, outside the Caravan data.

With respect to the models, RF and 2P-LSTM are the best performing ones, with
similar results for both stratified and holdout test splits. However, 2P-LSTM
yields the lowest error in predicting $y_{t+3}$. Since our aim is to have the
best model for 72-hours ahead predictions, we chose 2P-LSTM to illustrate some
predictions. Figure 3 shows a comparison between observed and predicted
streamflow for two cases: one interpolated case where the test gauge belongs
to Caravan data (left) and an extrapolated case, where the test gauge belongs
to the non-Caravan dataset from Africa (right). In most cases, the model
accurately captures the location and magnitude of the high flow peaks, which
are possible cases of river overflow. In this non-Caravan test set, the
2P-LSTM evaluation error was 2.59 mm/day, which is an example of accurate
predictions and generalization capabilities considering that these basins are
unrepresented in training.

##  5 Conclusions

In this paper, we propose the first global river flood prediction framework
based on the newly published Caravan dataset. Our framework aims to serve as a
benchmark for future global river flood prediction research. We show the high
potential of our framework by learning and evaluating three baseline models
and a novel 2P-LSTM architecture. Based on the evaluation, the 2P-LSTM model
outperforms the baseline models, and provides 2.13 mm/day RMSE in the gauges
in Caravan dataset and 2.59 mm/day RMSE when extrapolating to worldwide gauges
outside of the Caravans dataset. This is an important step towards river flood
predictions in low-income, poorly-documented areas around the world. Future
work includes, (i) flood extent derivation using the predicted streamflow in
combination with Digital Elevation Models (DEMs), and, (ii) evaluating the
potential of other sources of data in model inference, such as forecasted data
and satellite data.

## References

  * [1] Global Runoff Data Base (GRDC).  https://www.bafg.de/GRDC/EN/01_GRDC/13_dtbse/database_node.html.  Accessed: 2022-07-31.
  * [2] Making development sustainable: the future of disaster risk management.
  * [3] The human cost of weather related disasters.  https://www.cred.be/HCWRD.  Accessed: 2022-09-29.
  * [4] UN Office for the Coordination of Humanitarian Affairs.  https://www.unocha.org/story/five-essentials-first-72-hours-disaster-response.  Accessed: 2022-09-29.
  * [5] Aatish Anshuman and TI Eldho.  Entity aware sequence to sequence learning using lstms for estimation of groundwater contamination release history and transport parameters.  Journal of Hydrology, 608:127662, 2022.
  * [6] Frederik Kratzert, Daniel Klotz, Guy Shalev, Günter Klambauer, Sepp Hochreiter, and Grey Nearing.  Towards learning universal, regional, and local hydrological behaviors via machine learning applied to large-sample datasets.  Hydrology and Earth System Sciences, 23(12):5089–5110, 2019.
  * [7] Frederik Kratzert, Grey Nearing, Nans Addor, Tyler Erickson, Martin Gauch, Oren Gilon, Lukas Gudmundsson, Avinatan Hassidim, Daniel Klotz, Sella Nevo, Guy Shalev, and Yossi Matias.  Caravan - A global community dataset for large- sample hydrology.  May 2022.
  * [8] Anna Leontjeva and Ilya Kuzovkin.  Combining static and dynamic features for multivariate sequence classification.  In 2016 IEEE International Conference on Data Science and Advanced Analytics (DSAA), pages 21–30. IEEE, 2016.
  * [9] Simon Linke, Bernhard Lehner, Camille Ouellet Dallaire, Joseph Ariwi, Günther Grill, Mira Anand, Penny Beames, Vicente Burchard-Levine, Sally Maxwell, Hana Moidu, et al.  Global hydro-environmental sub-basin and river reach characteristics at high spatial resolution.  Scientific data, 6(1):1–15, 2019.
  * [10] Joaquín Muñoz-Sabater, Emanuel Dutra, Anna Agustí-Panareda, Clément Albergel, Gabriele Arduini, Gianpaolo Balsamo, Souhail Boussetta, Margarita Choulga, Shaun Harrigan, Hans Hersbach, et al.  Era5-land: A state-of-the-art global reanalysis dataset for land applications.  Earth System Science Data, 13(9):4349–4383, 2021.
  * [11] Sella Nevo, Gal Elidan, Avinatan Hassidim, Guy Shalev, Oren Gilon, Grey Nearing, and Yossi Matias.  Ml-based flood forecasting: Advances in scale, accuracy and reach.  2020\.
  * [12] Jun Erik Rentschler, Melda Salhab, and Bramka Arga Jafino.  Flood exposure and poverty in 188 countries.  Nature Communications, 13, 2022.
  * [13] Ilya Sutskever, Oriol Vinyals, and Quoc V Le.  Sequence to sequence learning with neural networks.  Advances in neural information processing systems, 27, 2014.

##  Appendix A Appendix

Figure 4: Locations of the basins contained in Caravans dataset. Figure 5:
72-hour ahead streamflow predictions produced by the 2P-LSTM model trained
using the holdout split.


# Monotonicity of the set of zeros of the Lyapunov exponent with respect to
shift embeddings

Oleg Safronov<EMAIL_ADDRESS>Department of Mathematics and Statistics,
UNCC, Charlotte, NC

###### Abstract.

We consider the discrete Schrödinger operators with potentials whose values
are read along the orbits of a shift of finite type. We study a certain subset
of the collection of energies at which the Lyapunov exponent is zero and prove
monotonicity of this set with respect to the shift embeddings. Then we
introduce a certain function ${\mathcal{J}}(A,\mu)$ determined by the position
of these zeros and prove monotonicity of ${\mathcal{J}}(A,\mu)$ with respect
to embeddings.

In this short paper, we study the discrete Schrödinger operators $H_{\omega}$
defined on $\ell^{2}({\mathbb{Z}})$ by

$\bigl{[}H_{\omega}u\bigr{]}(n)=u(n+1)+u(n-1)+V(T^{n}\omega)u(n),\qquad\omega\in\Omega.$

Here $\Omega$ is a campact metric space of infinite sequences
$\\{\omega_{n}\\}_{n\in{\mathbb{Z}}}$ such that
$\omega_{n}\in\\{1,\dots,\ell\\}={\mathcal{A}}$ for each $n$. There are
sequences in ${\mathcal{A}}^{\mathbb{Z}}$ that are not allowed to be in
$\Omega$ and one of the assumptions we make is that forbidden words are of
length $2$. The metric $d(\cdot,\cdot)$ on $\Omega$ is defined by

$d(\omega,\omega^{\prime})=e^{-N(\omega,\omega^{\prime})},$

where $N(\omega,\omega^{\prime})$ is the largest nonnegative integer such that
$\omega_{n}=\omega_{n}^{\prime}$ for all $|n|<N(\omega,\omega^{\prime})$. The
mapping $T:\,\Omega\to\Omega$ is assumed to be a subshift of finite type
defined by

$\bigl{(}T\omega\bigr{)}_{n}=\omega_{n+1},\qquad\forall n\in{\mathbb{Z}}.$

Finally, the function $V$ is assumed to be locally constant on $\Omega$ in the
sense of the following definition.

Definition. A function $V:\,\Omega\to{\mathbb{R}}$ is said to be locally
constant, if there is an $\epsilon>0$ such that

$V(\omega^{\prime})=V(\omega)\qquad\text{whenever}\qquad
d(\omega^{\prime},\omega)<\epsilon.$

Spectral properties of $H_{\omega}$ are related to the behavior of solutions
to the equation

(0.1) $u(n+1)+u(n-1)+V(T^{n}\omega)u(n)=Eu(n),\qquad n\in{\mathbb{Z}},$

for $E\in{\mathbb{R}}$.

On the other hand, all solutions to (0.1) can be described in terms of the
Schrödinger cocycles $(T,A^{E})$ with $A=A^{E}:\,\Omega\to{\rm
SL}(2,{\mathbb{R}})$ defined by

$A^{E}(\omega)=\begin{pmatrix}E-V(\omega)&-1\\\ 1&0\end{pmatrix}$

Namely, $u$ is a solution of (0.1) if and only if

$\begin{pmatrix}u(n)\\\
u(n-1)\end{pmatrix}=A_{n}(\omega)\cdot\begin{pmatrix}u(0)\\\
u(-1)\end{pmatrix},\qquad\forall n\in{\mathbb{Z}},$

where

$A_{n}(\omega)=\begin{cases}A(T^{n-1}\omega)\cdots
A(\omega)\quad\text{if}\quad n\geq 1;\\\
[A_{-n}(T^{n}\omega)]^{-1}\quad\text{if}\quad n\leq-1;\\\ {\rm
Id}\quad\text{if}\quad n=0.\end{cases}$

Since $\Omega$ is a metric space, we can talk about the Borel $\sigma$-algebra
of subsets of $\Omega$ and consider probability measures on $\Omega$. Let
$\mu$ be a $T$-ergodic probability measure on $\Omega$. The Lyapunov exponent
for $A$ and $\mu$ is defined by

$L(A,\mu)=\lim_{n\to\infty}\frac{1}{n}\int\ln(\|A_{n}(\omega)\|)d\mu(\omega).$

By Kingman’s subaddive ergodic theorem,

$\frac{1}{n}\ln(\|A_{n}(\omega)\|)\qquad\text{converges to}\quad
L(A,\mu)\qquad\text{as}\quad n\to\infty,$

for $\mu$-almost every $\omega\in\Omega$. For simplisity, we write
$L(E)=L(A,\mu).$

One of the main theorems of the paper [1] gives sufficient conditions
guaranteeing that the set

(0.2)
${\mathfrak{L}}(A,\mu)=\bigl{\\{}E\in{\mathbb{R}}:\,\,L(A,\mu)=0\bigr{\\}}$

is finite. One of these conditions is that $\mu$ has a local product
structure.

Let us now give a formal definition of a measure having such a property. We
first define the spaces of semi-infinite sequences

$\Omega_{+}=\\{\\{\omega_{n}\\}_{n\geq 0}:\,\,\omega\in\Omega\\}\quad\text{
and }\quad\Omega_{-}=\\{\\{\omega_{n}\\}_{n\leq 0}:\,\,\omega\in\Omega\\}.$

Then using the natural projection $\pi_{\pm}$ from $\Omega$ onto
$\Omega_{\pm}$, we define $\mu_{\pm}=(\pi_{\pm})_{*}\mu$ on $\Omega_{\pm}$ to
be the pushforward measure of $\mu$. After that, for each $1\leq j\leq\ell$,
we introduce the cylinder sets

$[0;j]=\\{\omega\in\Omega:\,\,\omega_{0}=j\\}\quad\text{and}\quad[0;j]_{\pm}=\\{\omega\in\Omega_{\pm}:\,\,\omega_{0}=j\\}.$

A local product structure is a relation between the measures
$\mu_{j}=\mu\bigl{|}_{[0;j]}$ and the measures
$\mu_{j}^{\pm}=\mu_{\pm}\bigl{|}_{[0;j]}$. To describe this relation, we need
to consider the natural homeomorphisms

$P_{j}:[0;j]\to[0;j]_{-}\times[0;j]_{+}$

defined by

$P_{j}(\omega)=\bigl{(}\pi_{-}\omega,\pi_{+}\omega\bigr{)},\qquad\forall\omega\in\Omega.$

Definition. We say that $\mu$ has a local product structure if there is a
positive $\psi:\Omega\to(0,\infty)$ such that for each $1\leq j\leq\ell$, the
function $\psi\circ P_{j}^{-1}$ belongs to
$L^{1}\bigl{(}[0;j]_{-}\times[0;j]_{+},\mu^{-}_{j}\times\mu^{+}_{j}\bigr{)}$
and

$\bigl{(}P_{j}\bigr{)}_{*}d\mu_{j}=\psi\circ
P_{j}^{-1}\,d(\mu^{-}_{j}\times\mu^{+}_{j}).$

We will shortly divide points of the set (0.2) into two groups: removable and
unremovable points. We will show that unremovable points in
${\mathfrak{L}}(A,\mu)$ do not disappear in the process of passing from
$\Omega$ to a subshift $\tilde{\Omega}\subset\Omega$ with an ergodic measure
$\tilde{\mu}$ on it.

A point $p\in\Omega$ is said to be periodic for $T$ provided there is a
positive integer $n_{p}$ for which $T^{n_{p}}\,p=p$. If $p\in\Omega$ is
periodic, then $V(T^{n}p)$ is a periodic function of $n$, because
$V(T^{n_{p}+n}p)=V(T^{n}p)$ for every $n\in{\mathbb{Z}}$. For a periodic point
$p$ of period $n_{p}$, define $\Delta_{p}(E)$ to be the trace of the monodromy
matrix $A_{n_{p}}(p)$

$\Delta_{p}(E)={\rm Tr}(A_{n_{p}}(p)).$

By ${\rm Per}(T)$, we denoted the collection of all periodic points of $T$.

Definition. A point $E\in{\mathfrak{L}}(A,\mu)$ is said to be unremovable from
${\mathfrak{L}}(A,\mu)$ provided

either 1) there exists a $T$\- periodic point $p\in\Omega$ for which
$0<|\Delta_{p}(E)|<2$,

or 2) $|\Delta_{p}(E)|\in\\{0,2\\}$ for all $p\in{\rm Per}(T)$.

The collection of unremovable from ${\mathfrak{L}}(A,\mu)$ points will be
denoted by ${\mathfrak{U}}(A,\mu)$.

######  Theorem 1.

Let $T:\,\Omega\to\Omega$ be a subshift of finite type. Assume that $\mu$ is a
$T$-ergodic measure on $\Omega$ that has a local product structure and the
property ${\rm supp}(\mu)=\Omega$. Let $V$ be a real-valued locally constant
function on $\Omega$. Then for any subshift
$\tilde{T}:\,\tilde{\Omega}\to\tilde{\Omega}$ of $T$ and any
$\tilde{T}$-ergodic measure $\tilde{\mu}$ on $\tilde{\Omega}\subset\Omega$,

(0.3) ${\mathfrak{U}}(A,\mu)\,\subseteq{\mathfrak{U}}(\tilde{A},\tilde{\mu}),$

where $\tilde{A}$ is the restriction of $A$ to $\tilde{\Omega}$.

######  Corollary 2.

Let $T:\,\Omega\to\Omega$ be a subshift of finite type. Assume that $\mu$ and
$\tilde{\mu}$ are $T$-ergodic measures on $\Omega$ that have a local product
structure and have the property ${\rm supp}(\mu)={\rm
supp}(\tilde{\mu})=\Omega$. Let $V$ be a real-valued locally constant function
on $\Omega$. Then

(0.4) ${\mathfrak{U}}(A,\mu)\,={\mathfrak{U}}(A,\tilde{\mu}).$

Several definitions below ivolve the set

${\mathfrak{S}}(T,\mu)=\bigcup_{p\in{\rm
Per}(T)}\bigl{\\{}E\in{\mathbb{R}}:\,\,\Delta_{p}(E)\in(-2,0)\cup(0,2)\bigr{\\}}.$

This set may only become smaller when one passes from $T$ to $\tilde{T}$,

${\mathfrak{S}}(\tilde{T},\tilde{\mu})\subseteq{\mathfrak{S}}(T,\mu),$

while ${\mathfrak{U}}(A,\mu)$ may only increase due to the property (0.3).
This observation allows one to construct a real-valued function
${\mathcal{J}}(A,\mu)$ that decreases when either ${\mathfrak{S}}(T,\mu)$
becomes smaller, or ${\mathfrak{U}}(A,\mu)$ becomes larger. For this purpose,
we recall that if $T$ has a fixed point, then there are at most finitely many
points in the set ${\mathfrak{U}}(A,\mu)$ (see Theorem 1.2 in [1]). Thus
${\mathfrak{U}}(A,\mu)=\\{E_{1},E_{2},\dots,E_{l}\\}$ where
$E_{1}<E_{2}<\dots<E_{l}$. We first enlarge the collection
${\mathfrak{U}}(A,\mu)$ by adding the two points $E_{0}=-5/2-\|V\|_{\infty}$
and $E_{l+1}=5/2+\|V\|_{\infty}$. Then, for each interval $(E_{j},E_{j+1})$
whose intersection with ${\mathfrak{S}}(T,\mu)$ is not empty, we define
$N_{j}$ by

$N_{j}=\text{ the integer part of
}\Bigl{[}\frac{2|E_{l+1}-E_{0}|}{|(E_{j},E_{j+1})\cap{\mathfrak{S}}(T,\mu)\bigr{|}}\Bigr{]}.$

Here, $|X|$ in the denominator denotes the Lebesgue measure of a Borel set
$X\subset{\mathbb{R}}$. If
$(E_{j},E_{j+1})\cap{\mathfrak{S}}(T,\mu)=\emptyset$, then we define $N_{j}$
to be equal to $2$. Finally, after setting

$N(A,\mu)=\max_{0\leq j\leq l}N_{j},$

we define the function

${\mathcal{J}}(A,\mu)=\sum_{j=0}^{l}{\mathcal{E}}_{j}(A,\mu),$

where

$\begin{split}{\mathcal{E}}_{j}(A,\mu)=\frac{\bigl{|}(E_{j},E_{j+1})\cap{\mathfrak{S}}(T,\mu)\bigr{|}}{\lambda}\,\,\ln\Bigl{(}\frac{\bigl{|}(E_{j},E_{j+1})\cap{\mathfrak{S}}(T,\mu)\bigr{|}}{\lambda}\Bigr{)}+\\\
+\frac{\bigl{|}(E_{0},E_{l+1})\setminus{\mathfrak{S}}(T,\mu)\bigr{|}}{\lambda\cdot(l+1)}\,\,\ln\Bigl{(}\frac{\bigl{|}(E_{0},E_{l+1})\setminus{\mathfrak{S}}(T,\mu)\bigr{|}}{N(A,\mu)\cdot\lambda}\Bigr{)}\end{split}$

and $\lambda=|E_{l+1}-E_{0}|$. The next result establishes monotonicity of the
function ${\mathcal{J}}(A,\mu)$ with respect to embeddings of the subshifts.

######  Theorem 3.

Let $T:\,\Omega\to\Omega$ be a subshift of finite type. Assume that $\mu$ is a
$T$-ergodic measure on $\Omega$ that has a local product structure and the
property ${\rm supp}(\mu)=\Omega$. Let $V$ be a real-valued locally constant
function on $\Omega$. Suppose $\tilde{T}:\,\tilde{\Omega}\to\tilde{\Omega}$ is
a further subshift of $T$ and $\tilde{\mu}$ is a $\tilde{T}$-ergodic measure
on $\tilde{\Omega}\subset\Omega$ for which the set
${\mathfrak{U}}(\tilde{A},\tilde{\mu})$ is finite (by $\tilde{A}$, we denote
the restriction of $A$ to $\tilde{\Omega}$). Then

${\mathcal{J}}(A,\mu)\,\geq{\mathcal{J}}(\tilde{A},\tilde{\mu}).$

##  1\. Begining of the proof of Theorem 1. Main ingredients

Note that a Schrödinger cocycle $A=A^{E}$ with a locally constant potential
$V:\Omega\to{\mathbb{R}}$ is also locally constant. Put differently, there is
an $\epsilon>0$ such that

$A(\omega^{\prime})=A(\omega)\qquad\text{whenever}\qquad
d(\omega^{\prime},\omega)<\epsilon.$

Definition . Let $T:\Omega\to\Omega$ be a subshift of finite type. The local
stable set of a point $\omega\in\Omega$ is defined by

$W^{s}(\omega)=\\{\omega^{\prime}\in\Omega:\,\,\omega^{\prime}_{n}=\omega_{n}\quad\text{for}\quad
n\geq 0\\}$

and the local unstable set of $\omega$ is defined by

$W^{u}(\omega)=\\{\omega^{\prime}\in\Omega:\,\,\omega^{\prime}_{n}=\omega_{n}\quad\text{for}\quad
n\leq 0\\}.$

For $\omega^{\prime}\in W^{s}(\omega)$, define
$H_{\omega^{\prime},\omega}^{s,n}$ to be

$H_{\omega,\omega^{\prime}}^{s,n}=\bigl{[}A_{n}(\omega^{\prime})\bigr{]}^{-1}A_{n}(\omega).$

Since $d(T^{j}\omega^{\prime},T^{j}\omega)\leq e^{-j}$ tends to $0$ as
$j\to\infty$, there is an index $n_{0}$ for which

$H_{\omega,\omega^{\prime}}^{s,n}=H_{\omega,\omega^{\prime}}^{s,n_{0}}\qquad\text{for
}\quad n\geq n_{0}.$

In this case, we define the stable holonomy $H_{\omega,\omega^{\prime}}^{s}$
by

$H_{\omega,\omega^{\prime}}^{s}=H_{\omega,\omega^{\prime}}^{s,n_{0}}.$

The unstable holonomy $H_{\omega,\omega^{\prime}}^{u}$ for $\omega^{\prime}\in
W^{u}(\omega)$ is defined similarly by

$H_{\omega,\omega^{\prime}}^{u}=\bigl{[}A_{n}(\omega^{\prime})\bigr{]}^{-1}A_{n}(\omega)\qquad\text{for
all}\quad n\leq-n_{0}.$

The general theory of dynamical systems tells us that the cocycle

$(T,A):\Omega\times{\mathbb{R}\mathbb{P}}^{1}\to{\mathbb{R}\mathbb{P}}^{1}$

defined by

$(T,A)(\omega,\xi)=(T\omega,A(\omega)\xi)$

has an invariant probability measure $m$ on
$\Omega\times{\mathbb{R}\mathbb{P}}^{1}.$ We say that such a measure $m$
projects to $\mu$ if $m(\Delta\times{\mathbb{R}\mathbb{P}}^{1})=\mu(\Delta)$
for all Borel subsets $\Delta$ of $\Omega$. Given any $T$-invariant measure
$\mu$ on $\Omega$, one can find a $(T,A)$-ivariant measure $m$ that projects
to $\mu$ by applying the Krylov-Bogolyubov trick.

Definition. Suppose $m$ is a $(T,A)$-invariant probability measure on
$\Omega\times{\mathbb{R}\mathbb{P}}^{1}$ that projects to $\mu$. A
disintegration of $m$ is a measurable family
$\\{m_{\omega}:\quad\omega\in\Omega\\}$ of probability measures on
${\mathbb{R}\mathbb{P}}^{1}$ having the property

$m(D)=\int_{\Omega}m_{\omega}(\\{\xi\in{\mathbb{R}\mathbb{P}}^{1}:\,\,(\omega,\xi)\in
D\\})d\mu(\omega)$

for each measurable set $D\subset\Omega\times{\mathbb{R}\mathbb{P}}^{1}.$

Existence of such a disintegration is guaranteed by Rokhlin’s theorem.
Moreover, $\\{\tilde{m}_{\omega}:\quad\omega\in\Omega\\}$ is another
disintegration of $m$ then $m_{\omega}=\tilde{m}_{\omega}$ for $\mu$-almost
every $\omega\in\Omega$. It is easy to see that $m$ is $(T,A)$-invariant if
and only if $A(\omega)_{*}m_{\omega}=m_{T\omega}$ for $\mu$-almost every
$\omega\in\Omega$.

Definition. A $(T,A)$-invariant measure $m$ on
$\Omega\times{\mathbb{R}\mathbb{P}}^{1}$ that projects to $\mu$ is said to be
an su-state for $A$ provided it has a disintegration
$\\{m_{\omega}:\quad\omega\in\Omega\\}$ such that for $\mu$-almost every
$\omega\in\Omega$,

1)

$\quad A(\omega)_{*}m_{\omega}=m_{T\omega},$

2)

$\bigl{(}H^{s}_{\omega,\omega^{\prime}}\bigr{)}_{*}m_{\omega}=m_{\omega^{\prime}}\qquad\text{
for every}\quad\omega^{\prime}\in W^{s}(\omega).$

3)

$\bigl{(}H^{u}_{\omega,\omega^{\prime}}\bigr{)}_{*}m_{\omega}=m_{\omega^{\prime}}\qquad\text{
for every}\quad\omega^{\prime}\in W^{u}(\omega)$

The following statement was proved in [1] (Proposition 4.7) for a
significantly larger class of functions $A$.

######  Proposition 4.

Let $A$ be locally constant. Suppose $\mu$ has a local product structure and
$L(A,\mu)=0$. If the support of the measure $\mu$ coincides with all of
$\Omega$, then there exists an su-state for $A$.

We apply the following method to extend $m_{\omega}$ to a continuous function
of $\omega$ on all of $\Omega$. For each $1\leq j\leq\ell$, we select a point
$\omega^{(j)}$ in $[0;j]\cap\Omega_{0}$ for which the measure
$m_{\omega^{(j)}}$ is well defined. Then we set

(1.5)
$m_{\omega}=\Bigl{(}H^{u}_{\omega\wedge\omega^{(\omega_{0})},\omega}H^{s}_{\omega^{(\omega_{0})},\omega\wedge\omega^{(\omega_{0})}}\Bigr{)}_{*}m_{\omega^{(\omega_{0})}}.$

Obviously $m_{\omega}$ depends continuously on $\omega$.

Observe that ${\mathbb{R}\mathbb{P}}^{1}$ may be aslo viewed as
${\mathbb{R}}\cup\\{\infty\\}$, because any vector of the form
$(\xi,1)\in{\mathbb{R}\mathbb{P}}^{1}$ is uniquily characterized by
$\xi\in{\mathbb{R}}\cup\\{\infty\\}$. Aslo, ${\mathbb{C}\mathbb{P}}^{1}$ may
be aslo viewed as ${\mathbb{C}}\cup\\{\infty\\}$ because there is a 1:1
mapping of one set onto another. The part of ${\mathbb{C}\mathbb{P}}^{1}$ that
is mapped onto the extended upper half-plane
${\mathbb{C}}_{+}\cup\\{\infty\\}$ will be denoted by
${\mathbb{C}}_{+}{\mathbb{P}}^{1}$.

Now we will state Proposition 4.9 from [1] in the following more convenient
form:

######  Proposition 5.

For each probability measure $\nu$ on ${\mathbb{R}\mathbb{P}}^{1}$ containing
no atom of mass $\geq 1/2$, there is an unique point
$B(\nu)\in{\mathbb{C}_{+}\mathbb{P}}$, called the conformal barycenter of
$\nu$, such that

$B(P_{*}\nu)=P\cdot B(\nu)$

for each $P\in{\rm SL}(2,{\mathbb{R}})$.

Let $m$ be an su-state with a continuous disintegration $m_{\omega}$. If
$m_{\omega}$ does not have an atom of mass $\geq 1/2$, then we set
$Z(\omega)\subset{\mathbb{C}_{+}\mathbb{P}}$ to be $\\{B(m_{\omega})\\}$.
Otherwise $Z(\omega)$ is defined to be the collection of points $\xi$ at which
$m_{\omega}(\\{\xi\\})\geq 1/2$. Since $m_{\omega}$ is a probability measure,
the set $Z(\omega)$ can contain at most two points. The following theorem is a
consequence of Proposition 5.

######  Theorem 6.

Let $A$ be locally constant. Suppose $\mu$ has a local product structure and
$L(A,\mu)=0$. Then

$A(\omega)Z(\omega)=Z(T\omega)\qquad\text{for each}\quad\omega\in\Omega.$

If $\omega^{\prime},\omega$ are two points in $\Omega$ such that
$\omega_{0}^{\prime}=\omega_{0}$, then

(1.6)
$Z(\omega)=\Bigl{(}H^{u}_{\omega\wedge\omega^{\prime},\omega}H^{s}_{\omega^{\prime},\omega\wedge\omega^{\prime}}\Bigr{)}Z(\omega^{\prime}).$

In particular, the number of the points in $Z(\omega)$ does not depend on
$\omega$. Moreover, if $Z(\omega)$ is real for one $\omega$, then it is real
for all $\omega\in\Omega$.

The last two lines of the theorem follow from the fact that for any two points
$\omega$ and $\omega^{\prime}$ in $\Omega$, there is a real matrix $P\in{\rm
SL}(2,{\mathbb{R}})$ for which $Z(\omega)=P\cdot Z(\omega^{\prime})$. Indeed,
if $\omega^{\prime}_{0}=\omega_{0}$, then this property is guaranteed by
(1.6). On the other hand, since $T$ is transitive, for any two points
$\omega^{\prime}$ and $\omega$, there is an index $n$ and a point
$\tilde{\omega}$ such that
$(T^{n}\tilde{\omega}\bigr{)}_{0}=\omega^{\prime}_{0}$ while
$\tilde{\omega}_{0}=\omega_{0}$. Therefore

$Z(T^{n}\tilde{\omega})=A_{n}(\tilde{\omega})Z(\tilde{\omega})=\Bigl{(}H^{u}_{T^{n}\tilde{\omega}\wedge\omega^{\prime},T^{n}\tilde{\omega}}H^{s}_{\omega^{\prime},T^{n}\tilde{\omega}\wedge\omega^{\prime}}\Bigr{)}Z(\omega^{\prime}),$

which implies that

$Z(\tilde{\omega})=[A_{n}(\tilde{\omega})]^{-1}\Bigl{(}H^{u}_{T^{n}\tilde{\omega}\wedge\omega^{\prime},T^{n}\tilde{\omega}}H^{s}_{\omega^{\prime},T^{n}\tilde{\omega}\wedge\omega^{\prime}}\Bigr{)}Z(\omega^{\prime}).$

It remains to note that

$Z(\omega)=\Bigl{(}H^{u}_{\omega\wedge\tilde{\omega},\omega}H^{s}_{\tilde{\omega},\omega\wedge\tilde{\omega}}\Bigr{)}Z(\tilde{\omega}).$

##  2\. End of the proof of Theorem 1

Let $E\in{\mathfrak{U}}(A,\mu)$. We must show that
$E\in{\mathfrak{U}}(\tilde{A},\tilde{\mu})$.

Assume first that $0<|\Delta_{p}(E)|<2$ for some $T$\- periodic point
$p\in\Omega$. By the symbol $n_{p}$, we denote the period of $p$. We also set

$L(A,p)=\lim_{n\to\infty}\frac{1}{n}\ln\bigl{(}\|A_{n}(p)\|\bigr{)}.$

It is easy to see that $Z(p)$, viewed as a set of complex numbers, is not
real. In fact, $Z(p)$ consists of one point
$(a+i\sqrt{4-(\Delta_{p}(E))^{2}})/b$, where $a$ and $b\neq 0$ are the two
elements of the first row of the matrix $A_{n_{p}}(p)$. Since, for any
periodic point $q\in\Omega$, the set $Z(q)$ is the image of $Z(p)$ under an
${\rm SL}(2,{\mathbb{R}})$ transformation, $Z(q)$ is not real and consists of
one point. Therefore, for any periodic point $q\in\Omega$, the matrix
$A_{n_{q}}(q)$ has two complex eigenvalues that belong to the unit circle. The
latter observation leads to the conclusion that $E\in\sigma(H_{q})$ and

(2.7) $L(A,q)=0\qquad\text{for all periodic points}\quad q\in\Omega.$

In particular, $L(A,q)=L(\tilde{A},q)=0$ for all periodic points that belong
to $\tilde{\Omega}$.

Now we use the following result proved in a much more general setting by
Kalinin (see Theorem 1.4 in [12]).

######  Proposition 7.

Let $A$ be locally constant on $\tilde{\Omega}$. Then for each $\delta>0$
there is a periodic point $q\in\tilde{\Omega}$ such that
$|L(\tilde{A},q)-L(\tilde{A},\tilde{\mu})|<\delta$.

Combining Proposition 7 with the equality (2.7), we obtain that

$L(\tilde{A},\tilde{\mu})=0.$

Thus, $E\in{\mathfrak{U}}(\tilde{A},\tilde{\mu})$.

Now assume that $|\Delta_{p}(E)|\in\\{0,2\\}$ for all $p\in{\rm Per}(T)$. Then
$|\Delta_{p}(E)|\in\\{0,2\\}$ for all $p\in{\rm Per}(\tilde{T})$. In
particular, this implies that all eigenvalues of $A_{n_{p}}$ belong to the
unit circle and, hence, $L(\tilde{A},p)=0$ for any periodic
$p\in\tilde{\Omega}$. Thus, we infer from Proposition 7 that
$L(\tilde{A},\tilde{\mu})=0.$

The proof is complete. $\,\,\Box$

######  Corollary 8.

A point $E\in{\mathfrak{L}}(A,\mu)$ is unremovable from
${\mathfrak{L}}(A,\mu)$ if and only if the point $E$ belongs to the spectrum
of $H_{p}$ for each $p\in{\rm Per}(T)$.

The zeros of the Lyapunov exponent that belong to the set

${\mathfrak{S}}(T,\mu)=\bigcup_{p\in{\rm
Per}(T)}\bigl{\\{}E\in{\mathbb{R}}:\,\,\Delta_{p}(E)\in(-2,0)\cup(0,2)\bigr{\\}}$

have simple and interesting properties described in the following statement.

######  Corollary 9.

Let $V$ be a real-valued locally constant function on $\Omega$. Let
$T:\,\Omega\to\Omega$ be a transitive subshift of finite type. Assume that
$\mu$ is a $T$-ergodic measure on $\Omega$ that has a local product structure.
Then for any subshift $\tilde{T}:\,\tilde{\Omega}\to\tilde{\Omega}$ of $T$ and
any $\tilde{T}$-ergodic measure $\tilde{\mu}$ on
$\tilde{\Omega}\subset\Omega$,

${\mathfrak{S}}(T,\mu)\cap{\mathfrak{L}}(A,\mu)\,\subseteq\,\,{\mathfrak{S}}(T,\mu)\cap{\mathfrak{L}}(\tilde{A},\tilde{\mu}),$

where $\tilde{A}$ is the restriction of $A$ to $\tilde{\Omega}$.

##  3\. Proof of Theorem 3

Observe that under the assumptions of Theorem 3,

$N(\tilde{A},\tilde{\mu})\geq N(A,\mu).$

First consider the case where
${\mathfrak{U}}(A,\mu)={\mathfrak{U}}(\tilde{A},\tilde{\mu})$ while
${\mathfrak{S}}(\tilde{T},\tilde{\mu})\subset{\mathfrak{S}}(T,\mu)$ is a
proper inclusion. Then the inequality

(3.8) ${\mathcal{J}}(\tilde{A},\tilde{\mu})\leq{\mathcal{J}}(A,\mu)$

may be established by the means of Calculus. Indeed, since
$N(\tilde{A},\tilde{\mu})\geq N(A,\mu)$, we only need to show that the
derivative of ${\mathcal{J}}(A,\mu)$ with respect to
$x=|(E_{j_{0}},E_{j_{0}+1})\cap{\mathfrak{S}}(T,\mu)|$ is positive, provided
$|(E_{0},E_{l+1})\cap{\mathfrak{S}}(T,\mu)|$ is viewed as the linear function
$\tau-x$, where $\tau=E_{j_{0}+1}-E_{j_{0}}+\sum_{j\neq
j_{0}}|(E_{j},E_{j+1})\cap{\mathfrak{S}}(T,\mu)|$. Put differently, we must
show that the derivative of

$\psi(x)=\frac{x}{\lambda}\ln\Bigl{(}\frac{x}{\lambda}\Bigr{)}+\frac{\tau-x}{\lambda}\ln\Bigl{(}\frac{\tau-x}{N(A,\mu)\cdot\lambda}\Bigr{)}$

is positive.

The direct computation shows that

$\psi^{\prime}(x)=\frac{1}{\lambda}\ln\Bigl{(}\frac{N(A,\mu)\cdot
x}{\tau-x}\Bigr{)}.$

Thus, we infer that $\psi^{\prime}(x)>0$ from the finequality
$(N(A,\mu)+1)x>\tau$.

It remains to prove (3.8) in the case where
${\mathfrak{U}}(A,\mu)\subset{\mathfrak{U}}(\tilde{A},\tilde{\mu})$ is a
proper inclusion. In the case
${\mathfrak{U}}(A,\mu)={\mathfrak{U}}(\tilde{A},\tilde{\mu})$, the quantity
${\mathcal{J}}(A,\mu)$ was an increasing function of ${\mathfrak{S}}(T,\mu)$.
Therefore, it is enough to consider the case where
${\mathfrak{S}}(T,\mu)={\mathfrak{S}}(\tilde{T},\tilde{\mu})$.

Suppose ${\mathfrak{U}}(\tilde{A},\tilde{\mu})$ consists of the points
$\tilde{E}_{1}<\tilde{E}_{2}<\dots<\tilde{E}_{r}$. Define
$\tilde{E}_{0}=E_{0}$ and $\tilde{E}_{r+1}=E_{l+1}$. Then each interval
$[E_{j},E_{j+1})$ is the union of a finite collection of intervals
$[\tilde{E}_{k},\tilde{E}_{k+1})$:

$[E_{j},E_{j+1})=\bigcup_{k=k_{0}(j)}^{\tilde{k}(j)}[\tilde{E}_{k},\tilde{E}_{k+1}).$

Therefore,

$\begin{split}|[E_{j},E_{j+1})\cap{\mathfrak{S}}(T,\mu)|\ln\Bigl{(}|[E_{j},E_{j+1})\cap{\mathfrak{S}}(T,\mu)|\Bigr{)}\geq\\\
\sum_{k=k_{0}(j)}^{\tilde{k}(j)}|[E_{k},E_{k+1})\cap{\mathfrak{S}}(T,\mu)|\ln\Bigl{(}|[E_{k},E_{k+1})\cap{\mathfrak{S}}(T,\mu)|\Bigr{)},\end{split}$

which implies (3.8).

##  4\. A special case

Assume that $V$ depends only on the zero-coordinate $\omega_{0}$ of $\omega$.
Then all stable and unstable holonomies are identity operators. Consequently,
if $m_{\omega}$ is a continuous disintegration of an su-state, then

$m_{\omega}=m_{\omega^{\prime}}\quad\text{whenever}\quad\omega_{0}=\omega^{\prime}_{0}.$

But then the equality

$A(\omega)m_{\omega}=m_{T\omega}$

implies that $m_{T\omega^{\prime}}=m_{T\omega}$ whenever
$\omega_{0}=\omega^{\prime}_{0}$.

Let us now give a condition that makes the latter equality impossible. For
each $j_{0}\in{\mathcal{A}}$, define the set

(4.9) $D_{j_{0}}=\\{j\in{\mathcal{A}}:\,\,\exists\omega\in\Omega\,\text{such
that}\,\,\omega_{0}=j_{0},\,\omega_{1}=j\\}.$

Suppose that for each pair of symbols $j$ and $j^{\prime}$ in ${\mathcal{A}}$,
there is an ordered collection of letters $j_{1},j_{2},\dots,j_{k}$, such that
$j\in D_{j_{1}}$ and $j^{\prime}\in D_{j_{k}}$, while $D_{j_{n}}\cap
D_{j_{n+1}}\neq\emptyset$ for all $1\leq n\leq k-1$. Then $m_{\omega}$ is
constant on $\Omega$. This would imply that $Z(\omega)$ is constant on
$\Omega$, which would turn the relation $A^{E}(\omega)\cdot
Z(\omega)=Z(T\omega)$ into the equality

$A^{E}(\omega)\cdot Z(\omega)=Z(\omega).$

This cannot be true in the case where $V(\omega)$ takes at least two different
values. The obtained contradiction shows that there is no su-state for the
cocycle $A^{E}$, which implies that the Lyapunov exponent is positive for
every $E\in{\mathbb{R}}$.

######  Theorem 10.

Assume that $V$ depends only on the zero-coordinate $\omega_{0}$ of $\omega$
and takes at least two different values on $\Omega$. Let the sets $D_{j_{0}}$
be defined by (4.9). Suppose that for each pair of symbols $j$ and
$j^{\prime}$ in ${\mathcal{A}}$, there is an ordered collection of letters
$j_{1},j_{2},\dots,j_{k}$, such that $j\in D_{j_{1}}$ and $j^{\prime}\in
D_{j_{k}}$, while $D_{j_{n}}\cap D_{j_{n+1}}\neq\emptyset$ for all $1\leq
n\leq k-1$. Finally, assume that $\mu$ is a $T$-ergodic measure on $\Omega$
that has a local product structure and the property ${\rm
supp}(\mu)=\Omega$.Then the Lyapunov exponent is positive for each
$E\in{\mathbb{R}}$:

$L(A^{E},\mu)>0.$

## References

  * [1] A. Avila, D. Damanik, and Z. Zhang: Schrödinger operators with potentials generated by hyperbolic transformations: I–positivity of the Lyapunov exponent, Invent. Math. 231, (2023) 851-927.
  * [2] A. Avila, M. Viana: Extremal Lyapunov exponents: an invariance principle and applications, Invent. Math. 181 (2010), 115-189.
  * [3] L. Backes, A. Brown, C. Butler: Continuity of Lyapunov exponents for cocycles with invariant holonomies, J. Mod. Dyn. 12 (2018), 223-260.
  * [4] K. Bjerklöv: Positive Lyapunov exponent for some Schrödinger cocycles over strongly expanding circle endomorphisms, Commun. Math. Phys. 379 (2020), 353-360.
  * [5] C. Bonatti, M. Viana: Lyapunov exponents with multiplicity 1 for deterministic products of matrices, Ergodic Theory Dynam. Systems 24 (2004), 1295-1330.
  * [6] J. Bourgain, E. Bourgain-Chang: A note on Lyapunov exponents of deterministic strongly mixing potentials, J. Spectr. Theory 5 (2015), 1-15.
  * [7] J. Bourgain, M. Goldstein, W. Schlag: Anderson localization for Schrödinger operators on Z with potentials given by the skew-shift, Commun. Math. Phys. 220 (2001), 583-621.
  * [8] J. Bourgain, W. Schlag: Anderson localization for Schrödinger operators on Z with strongly mixing potentials, Commun. Math. Phys. 215 (2000), 143-175.
  * [9] D. Damanik: Schrödinger operators with dynamically defined potentials, Ergodic Theory Dynam. Systems 37 (2017), 1681-1764.
  * [10] D. Damanik, R. Killip: Almost everywhere positivity of the Lyapunov exponent for the doubling map, Commun. Math. Phys. 257 (2005), 287-290.
  * [11] D. Damanik, S. Tcheremchantsev: Power-law bounds on transfer matrices and quantum dynamics in one dimension, Commun. Math. Phys. 236 (2003), 513-534.
  * [12] B. Kalinin: Livs̃ic theorem for matrix cocycles, Ann. of Math. 173 (2011), 1025-1042.
  * [13] R. Leplaideur: Local product structure for equilibrium states, Trans. Amer. Math. Soc. 352 (2000), 1889-1912.
  * [14] C. Sadel, H. Schulz-Baldes: Positive Lyapunov exponents and localization bounds for strongly mixing potentials, Adv. Theo. Math. Phys. 12 (2008), 1377-1400.
  * [15] M. Viana: Almost all cocycles over any hyperbolic system have non-vanishing Lyapunov exponents, Ann. of Math. 167 (2008), 643-680.
  * [16] M. Viana, J. Yang: Continuity of Lyapunov exponents in the $C^{0}$ topology, Israel J. Math. 229 (2019), 461-485.
  * [17] Z. Zhang: Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger operators, J. Spectr. Theory 10 (2020) no. 4, 1471-1517.