arXiv subjects

Artur O. Lopes

Publications and source records attributed to Artur O. Lopes.

At least 19 recordsLinked to original sources

arXiv2026

Ergodic Theory in Classical and Bayesian Inference

We begin by presenting the mathematical rationale underlying classical deductive inference. We then introduce the foundational ideas of the Bayesian inference framework. Results lying at the interface of Statistics and Ergodic Theory are outlined, providing a theoretical framework applicable to the prediction and analysis of real-world phenomena from random data. This text is expository in nature - no new results are presented; rather, recently published results are described in a didactic manner. Throughout, we work with Hölder equilibrium measures, which encompass a substantially more general class of processes than i.i.d. ones.

math.ST↗

arXiv2026

Ergodic Optimization and Ground States: a brief Introduction

Our goal in this short note is to briefly and succinctly describe some basic concepts and properties of Ergodic Optimization for readers unfamiliar with the subject. We avoid technical issues in order to provide a global overview of this topic. We will not attempt to cover all of the many contributions of various authors, who have greatly enriched the theory with invaluable results. The author has made a personal selection of the topics to be addressed, keeping in mind two main objectives: to motivate the reasons for studying the subject, and to describe schematically and pictorially its relationship with relevant concepts and properties of Statistical Mechanics, which is one of the sources of inspiration for the theory. We will not present new results or detailed proofs. Some examples will be provided. We describe some procedures that may help in obtaining explicit solutions. We present some references that by no means aim to exhaust the bibliography on the subject, where possible, minimizing the number of references.

math.DS↗

arXiv2026

A Dynamical Approach to Non-Extensive Thermodynamics

We develop a non-extensive thermodynamic formalism for the one-sided shift on a finite alphabet, inspired by Tsallis' generalization of Boltzmann entropy in statistical physics. We introduce notions of $q$-entropy, $q$-pressure, and $q$-transfer operators which extend the classical thermodynamic formalism when $q=1$. We prove a Bowen-type relation linking the $q$-pressure with a $(2-q)$-Ruelle transfer operator and show that $q$-equilibrium states correspond to classical equilibrium states for a related potential. We establish the existence and uniqueness of $q$-equilibrium states for Lipschitz potentials, prove the differentiability of the $q$-pressure, and obtain variational principles for both the $q$-pressure and a related asymptotic pressure. Finally, we study cohomological equations associated with $(2-q)$-transfer operators and prove the differentiable dependence of their solutions on the potential, yielding an alternative construction of eigenfunctions for classical Ruelle operators.

math.DS↗

arXiv2025

Thermodynamic formalism for continuous-time quantum Markov semigroups: the detailed balance condition, entropy, pressure and equilibrium quantum processes

$M_n(\mathbb{C})$ denotes the set of $n$ by $n$ complex matrices. Consider continuous time quantum semigroups $\mathcal{P}_t= e^{t\, \mathcal{L}}$, $t \geq 0$, where $\mathcal{L}:M_n(\mathbb{C}) \to M_n(\mathbb{C})$ is the infinitesimal generator. If we assume that $\mathcal{L}(I)=0$, we will call $e^{t\, \mathcal{L}}$, $t \geq 0$ a quantum Markov semigroup. Given a stationary density matrix $ρ= ρ_{\mathcal{L}}$, for the quantum Markov semigroup $\mathcal{P}_t$, $t \geq 0$, we can define a continuous time stationary quantum Markov process, denoted by $X_t$, $t \geq 0.$ Given an {\it a priori} Laplacian operator $\mathcal{L}_0:M_n(\mathbb{C}) \to M_n(\mathbb{C})$, we will present a natural concept of entropy for a class of density matrices on $M_n(\mathbb{C})$. Given an Hermitian operator $A:\mathbb{C}^n\to \mathbb{C}^n$ (which plays the role of an Hamiltonian), we will study a version of the variational principle of pressure for $A$. A density matrix $ρ_A$ maximizing pressure will be called an equilibrium density matrix. From $ρ_A$ we will derive a new infinitesimal generator $\mathcal{L}_A$. Finally, the continuous time quantum Markov process defined by the semigroup $\mathcal{P}_t= e^{t\, \mathcal{L}_A}$, $t \geq 0$, and an initial stationary density matrix, will be called the continuous time equilibrium quantum Markov process for the Hamiltonian $A$. It corresponds to the quantum thermodynamical equilibrium for the action of the Hamiltonian $A$.

math-ph↗

arXiv2025

An introduction to Coupling

In this review paper, we describe the use of couplings in several different mathematical problems. We consider the total variation norm, maximal coupling, and the $\bar{d}$-distance. We present a detailed proof of a result recently proved: the dual of the Ruelle operator is a contraction with respect to $1$-Wasserstein distance. We also show exponential convergence to equilibrium in the state space for finite-state Markov chains when the transition matrix $\mathcal{P}$ has all entries positive.} In this new version, we describe in more detail the line of reasoning followed in the work previously published as a chapter in ``Modeling, Dynamics, Optimization and Bioeconomics II'', Springer Verlag (2017).

math.PR↗

arXiv2025

Nonlinear Thermodynamic Formalism: Mean-field Phase Transitions, Large Deviations and Bogoliubov's Variational Principle

Let $Ω=\{1,2,\ldots ,d\}^{\mathbb{N}}$, $T$ be the shift acting on $Ω$, $\mathcal{P}(T)$ the set of $T$-invariant probabilities. Given a Hölder potential $A$ and a continuous function $F$, we investigate the probabilities $ρ_{F,A}$ that are maximizers of the nonlinear pressure $\mathfrak{P}_{F,A}:=\sup_{ρ\in \mathcal{P}(T)}\{ F(\int A(x)ρ(\mathrm{d}x))+h(ρ)\} .$ $ρ_{F,A}$} is called a nonlinear equilibrium; a nonlinear phase transition occurs when there is more than one. In the case $F$\ is convex or concave, we combine Varadhan's lemma and Bogoliubov's variational principle to characterize them via the linear pressure problem and self-consistency conditions. Let $μ\in \mathcal{P}(T)$ be the maximal entropy measure, $φ_{n}(x)=n^{-1}(φ(x)+φ(T(x))+\cdots +φ(T^{n-1}(x)))$ and $β>0$.}\newline (I) We also consider the limit measure $\mathfrak{m}$ on $ Ω$, so that $\forall ψ\in C(Ω)$, $\int ψ(x)\,\mathfrak{m}\,( \mathrm{d}x)\,\,=\lim_{n\rightarrow \infty }\frac{\,\int \,ψ(x)\,\,\,e^{ \frac{βn}{2}\,\,A_{n}((x)^{2}}\,\,μ\,(\mathrm{d}x)\,}{\int e^{\frac{ βn}{2}\,\,A_{n}((x)^{2}}μ\,(\mathrm{d}x)\,\,}.$ We call $\mathfrak{m}$ a \textit{quadratic mean-field Gibbs probability (II) Via subsequences $n_{k}$, $k\in \mathbb{N}$, we study the limit measure $\mathfrak{M}$ on $Ω$, so that $\forall ψ\in C(Ω)$, $\int ψ(x)\mathfrak{M}(\mathrm{d} x)=\lim_{k\rightarrow \infty }\frac{\,\int ψ_{n_{k}}(x)e^{\frac{βn_{k}}{2}A_{n_{k}}(x)^{2}}μ(\mathrm{d}x)}{\int e^{\frac{βn_{k}}{2} A_{n_{k}}(x)^{2}}μ(\mathrm{d}x)}.$ We call $\mathfrak{M}$ a quadratic mean-field equilibrium probability; it is shift-invariant. Explicit examples are given.

math.DS↗

arXiv2025

The Dirac operator for the pair of Ruelle and Koopman operators, and a generalized Boson formalism

Denote by $\mathbfμ$ the maximal entropy measure for the shift map $σ$ acting on $Ω= \{0, 1\}^\mathbb{N}$, by $L$ the associated Ruelle operator and by $K = L^{\dagger}$ the Koopman operator, both acting on $\mathscr{L}^2(\mathbfμ)$. The Ruelle-Koopman pair can determine a generalized boson system in the sense of \cite{Kuo}. Here $2^{-\frac{1}{2}} K$ plays the role of the creation operator and $ 2^{-\frac{1}{2}} L$ is the annihilation operator. We show that $[L,K]$ is the projection on the kernel of $L.$ In $C^*$-algebras the Dirac operator $\mathcal{D}$ represents derivative. Akin to this point of view we introduce a dynamically defined Dirac operator $\mathcal{D}$ associated with the Ruelle-Koopman pair and a representation $π$. Given a continuous function $f$, denote by $M_f$ the operator $ g \to M_f(g)=f\, g.$ Among other dynamical relations we get $$\|\left[ \mathcal{D} , π(M_f) \right]\| = \sup_{x \in Ω} \sqrt{\frac{|f(x) - f(0x)|^{2}}{2} + \frac{|f(x) - f(1x)|^2}{2}} = \left|\sqrt{L |K f - f|^{2}}\right|_{\infty}$$ which concerns a form of discrete-time mean backward derivative. We also derive an inequality for the discrete-time forward derivative $f \circ σ-f$: $$ |f \circ σ-f |_{\infty} = |K f - f|_{\infty} \geq \|\left[ \mathcal{D} , π(M_f) \right]\| \geq |f - L f|_{\infty}.$$ Moreover, we get $\|\, \left[\mathcal{D} ,π(K L)\right] \,\|=1$. The Number operator is $\frac{1}{\sqrt{2}}K \frac{1}{\sqrt{2}} L.$ The Connes distance requires to ask when an operator $A$ satisfies the inequality $\|\, \left[\mathcal{D} ,π(A)\right] \,\|\leq 1$; the Lipschtiz constant of $A$ smaller than $1$.

math.FA↗

arXiv2025

The Dirac operator for the Ruelle-Koopman pair on L^p-spaces: an interplay between Connes distance and symbolic dynamics

Denote by $\bmμ$ the maximal entropy measure for the shift $σ$ acting on $Ω= \{0, 1\}^\mathbb{N}$, by $\ruelle$ the associated Ruelle operator and by $\koopman = \ruelle^{\dagger}$ the Koopman operator, both acting on $\lp{2}(\bmμ)$. Using a diagonal representation $π$, the Ruelle-Koopman pair can be used for defining a dynamical Dirac operator $\mathcal{D},$ as in \cite{BL}. $\mathcal{D}$ plays the role of a derivative. In \cite{lpspec}, the notion of a spectral triple was generalized to $\lp{p}$-operator algebras; in consonance, here, we generalize results for $\mathcal{D}$ to results for a Dirac operator $\mathcal{D}_p$ , and the associated Connes distance $d_p$, to this new $\lp{p}$ context, $p \geq 1$. Given the states $η, ξ$: $d_{p}(η, ξ) \defn \sup \{ \,|η(a) - ξ(a) | where a \in \mathcal{A} and \norm{\left[\mathcal{D}_p,π(a)\right]} \leq 1\}$. The operator $M_f$ acts on $L^p (μ).$ We explore the relationship of $\mathcal{D}_p$ with dynamics, in particular with $f \circ σ- f$, the discrete-time derivative of a continuous $f:Ω\to \mathbb{R}$. Take $p,p^{\prime}>0$ satisfying $\frac{1}{p} + \frac{1}{ p^{\prime}}=1$. We show for any continuous function $f$: $\norm{\left[ \dirac_p, π(\mult_f) \right]} = | \sqrt[λ]{\ruelle \abs{f \circ σ- f}^λ} |_{\infty}$, where $λ= \max\{p, p^\prime\}$. Furthermore, we show $\norm{\left[ \mathcal{D}_p, π(\koopman^{n} \mathcal{L}^{n})]\right]}=1$ for all $n \geq 1$. We also prove a formula analogous to the Kantorovich duality formula for minimizing the cost of tensor products.

math-ph↗

arXiv2024

Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities

We consider here the discrete time dynamics described by a transformation $T:M \to M$, where $T$ is either the action of shift $T=σ$ on the symbolic space $M=\{1,2,...,d\}^\mathbb{N}$, or, $T$ describes the action of a $d$ to $1$ expanding transformation $T:S^1 \to S^1$ of class $C^{1+α}$ (\,for example $x \to T(x) =d\, x $ (mod $1) $\,), where $M=S^1$ is the unit circle. It is known that the infinite-dimensional manifold $\mathcal{N}$ of equilibrium probabilities for Hölder potentials $A:M \to \mathbb{R}$ is an analytical manifold and carries a natural Riemannian metric associated with the asymptotic variance. We show here that under the assumption of the existence of a Fourier-like Hilbert basis for the kernel of the Ruelle operator there exists geodesics paths. When $T=σ$ and $M=\{0,1\}^\mathbb{N}$ such basis exists. In a different direction, we also consider the KL-divergence $D_{KL}(μ_1,μ_2)$ for a pair of equilibrium probabilities. If $D_{KL}(μ_1,μ_2)=0$, then $μ_1=μ_2$. Although $D_{KL}$ is not a metric in $\mathcal{N}$, it describes the proximity between $μ_1$ and $μ_2$. A natural problem is: for a fixed probability $μ_1\in \mathcal{N}$ consider the probability $μ_2$ in a convex set of probabilities in $\mathcal{N}$ which minimizes $D_{KL}(μ_1,μ_2)$. This minimization problem is a dynamical version of the main issues considered in information projections. We consider this problem in $\mathcal{N}$, a case where all probabilities are dynamically invariant, getting explicit equations for the solution sought. Triangle and Pythagorean inequalities will be investigated.

math.DS↗

arXiv2024

Directional derivatives and the central limit theorem on compact general one-dimensional lattices

We will show the central limit theorem for the general one-dimensional lattice where the space of symbols is a compact metric space. We consider the CLT for Lipschitz-Gibbs probabilities and in the proof we use several properties of the Ruelle operator defined on our setting; this will require fixing an {\em a priori probability}. An important issue in the proof of the CLT is the existence of a certain second-order derivative, and this will follow from the analytic properties that will be described in detail throughout the paper. As additional results of independent interest, we will also describe some explicit estimates of the first and second directional derivatives of some dynamical entities like entropy and pressure. For example: given a fixed potential $f$, and a variable observable $η$ on the Kernel of the Ruelle operator $\mathcal{L}_f$, we consider the equilibrium probability $μ_{f + t \,η}$ for $f + t \,η$. We estimate the values $ \frac{d}{dt} h (μ_{f + t \,η})|_{t=0}$ and $ \frac{d^2}{dt^2} h (μ_{f + t \,η})|_{t=0}$, where $h (μ_{f + t \,η})$ is the entropy of $ μ_{C + t \,η}$. For fixed $f$ we can find conditions that can indicate the $η$ attaining the maximal possible value of $ \frac{d}{dt} h (μ_{f + t \,η})|_{t=0}$ (up to a natural normalization of $η)$, entirely in terms of elements on the kernel of $\mathcal{L}_f$. We also consider directional derivatives of the eigenfunction.

math.DS↗

arXiv2024

Parametrized Families of Gibbs Measures and their Statistical Inference

For Hölder continuous functions $f_i$, $i=0,\ldots ,d$, on a subshift of finite type and $Θ\subset \mathbb \R^d$ we consider a parametrized family of potentials $\{F_θ= f_0+\sum_{i=1}^d θ_i f_i : θ\in Θ\}$. We show that the maximum likelihood estimator of $θ$ for a family of Gibbs measures with potentials $F_θ$ is consistent and determine its asymptotic distribution under the associated shift-invariant distribution. A second part discusses applications; from confidence intervals through testing problems to connections to Bernoulli distributions and stationary Markov chains.

math.DS↗

arXiv2024

Thermodynamic Formalism for a family of cellular automata and duality with the shift

We will consider a family of cellular automata $Φ: \{1,2,...,r\}^\mathbb{N}\circlearrowright$ that are not of algebraic type. Our first goal is to determine conditions that result in the identification of probabilities that are at the same time $σ$-invariant and $Φ$-invariant, where $σ$ is the full shift. Via the use of versions of the Ruelle operator $\mathcal{L}_{A,σ}$ and $\mathcal{L}_{B,Φ}$ we will show that there is an abundant set of measures with this property; they will be equilibrium probabilities for different Lispchitz potentials $A,B$ and for the corresponding dynamics $σ$ and $Φ$. Via the use of a version of the involution kernel $W$ for a $(σ,Φ)$-mixed skew product $\hatΦ: \{1,2,...,r\}^\mathbb{Z}\circlearrowright$, given $A$ one can determine $B$, in such way that the integral kernel $e^W$ produce a duality between eigenprobabilities $ρ_A$ for $(\mathcal{L}_{A,σ})^*$ and eigenfunctions $ψ_B$ for $\mathcal{L}_{B,Φ}$. In another direction, considering the non-mixed extension $\hatΦ_n : \{1,2,...,r\}^\mathbb{Z}\circlearrowright$ of $Φ$, given a Lispchitz potential $\hat{A} : \{1,2,...,r\}^\mathbb{Z}\to \mathbb{R}$, we can identify a Lipschitz potential $A:\{1,2,...,r\}^\mathbb{N} \to \mathbb{R} $, in such away that relates the variational problem of $\hatΦ_n$-Topological Pressure for $\hat{A}$ with the $Φ$-Topological Pressure for $A$. We also present a version of Livsic's Theorem. Whether or not $Φ$ (or $\hatΦ)$ can eventually be conjugated with another shift of finite type is irrelevant in our context.

math.DS↗

arXiv2024

Spectral Triples on Thermodynamic Formalism and Dixmier Trace Representations of Gibbs: theory and examples

In this paper we study spectral triples and non-commutative expectations associated to expanding and weakly expanding maps. In order to do so, we generalize the Perron-Frobenius-Ruelle theorem and obtain a polynomial decay of the operator, which allows to prove differentiability of a dynamically defined $ζ$-function at its critical parameter. We then generalize Sharp's construction of spectral triples to this setting and provide criteria when the associated spectral metric is non-degenerate and when the non-commutative expectation of the spectral triple is colinear to the integration with respect to the associated equilibrium state from thermodynamic formalism. Due to our general setting, we are able to simultaneously analyse expanding maps on manifolds or connected fractals, subshifts of finite type as well as the Dyson model from statistical physics, which underlines the unifying character of noncommutative geometry. Furthermore, we derive an explicit representation of the $ζ$-function associated to a particular class of pathological continuous potentials, giving rise to examples where the representation as a non-commutative expectation via the associated zeta function holds, and others where it does not hold.

math.DS↗

arXiv2024

On the quantum Guerra-Morato Action Functional

Given a smooth potential $W:\mathrm{T}^{n} \to \mathbb{R}$ on the torus, the Quantum Guerra-Morato action functional is given by \smallskip $ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, I(ψ) = \int\,(\, \, \,\frac{D v\, D v^*}{2}(x) - W(x) \,) \,\,a(x)^2 dx,$ \smallskip \noindent where $ψ$ is described by $ψ= a\, e^{i\,\frac{ u }{h}} $, $ u =\, \frac{v + v^*}{2},$ $a=e^{\,\frac{v^*\,-\,v}{2\, \hbar} }$, $v,v ^*$ are real functions, $\int a^2 (x) d x =1$, and $D$ is derivative on $x \in \mathrm{T}^{n}$. It is natural to consider the constraint $ \mathrm{d}\mathrm{i}\mathrm{v}(a^{2}Du)=0$, which means flux zero. The $a$ and $u$ obtained from a critical solution (under variations $τ$) for such action functional, fulfilling such constraints, satisfy the Hamilton-Jacobi equation with a quantum potential. Denote $'=\frac{d}{dτ}$. We show that the expression for the second variation of a critical solution is given by \smallskip $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int a^{2}\,D[ v' ]\, D [(v ^*)']\, dx.$ \smallskip Introducing the constraint $\int a^2 \,D u \,dx =V$, we also consider later an associated dual eigenvalue problem. From this follows a transport and a kind of eikonal equation.

math-ph↗

arXiv2023

The generalized IFS Bayesian method and an associated variational principle covering the classical and dynamical cases

We introduce a general IFS Bayesian method for getting posterior probabilities from prior probabilities, and also a generalized Bayes' rule, which will contemplate a dynamical, as well as a non-dynamical setting. Given a loss function ${l}$, we detail the prior and posterior items, their consequences and exhibit several examples. Taking $Θ$ as the set of parameters and $Y$ as the set of data (which usually provides {random samples}), a general IFS is a measurable map $τ:Θ\times Y \to Y$, which can be interpreted as a family of maps $τ_θ:Y\to Y,\,θ\inΘ$. The main inspiration for the results we will get here comes from a paper by Zellner (with no dynamics), where Bayes' rule is related to a principle of minimization of {information.} We will show that our IFS Bayesian method which produces posterior probabilities (which are associated to holonomic probabilities) is related to the optimal solution of a variational principle, somehow corresponding to the pressure in Thermodynamic Formalism, and also to the principle of minimization of information in Information Theory. Among other results, we present the prior dynamical elements and we derive the corresponding posterior elements via the Ruelle operator of Thermodynamic Formalism; getting in this way a form of dynamical Bayes' rule.

math.DS↗

arXiv2023

The sectional curvature of the infinite dimensional manifold of Hölder equilibrium prababilities

Here we consider the discrete time dynamics described by a transformation $T:M \to M$, where $T$ is the shift and $M=\{1,2,...,d\}^\mathbb{N}$. It is known that the infinite-dimensional manifold $\mathcal{N}$ of Hölder equilibrium probabilities is an analytical manifold and carries a natural Riemannian metric. Given a normalized Hölder potential $A$ denote by $μ_A \in \mathcal{N}$ the associated equilibrium probability. The set of tangent vectors $X$ to the manifold $\mathcal{N}$ at the point $μ_A$ coincides with the kernel of the Ruelle operator for $A$. The Riemannian norm $|X|=|X|_A$ of the vector $X$, which is tangent to $\mathcal{N}$ at the point $μ_A$, is described via the asymptotic variance, that is, satisfies $|X|^2\,\,= \langle X, X \rangle =\lim_{n \to \infty} \frac{1}{n} \int (\sum_{i=0}^{n-1} X\circ T^i )^2 \,d μ_A$. Consider an orthonormal basis $X_i$, $i \in \mathbb{N}$, for the tangent space at $μ_A$. Given two unit tangent vectors $X$ and $Y$ the curvature $K(X,Y)$ satisfies $\,\,\,\,K(X,Y) = \frac{1}{4}[\, \sum_{i=1}^\infty ( \int X \,Y\, X_i \,d μ_A)^2 - \sum_{i=1}^\infty \int X^2 X_i \,d μ_A\, \,\int Y^2 X_i \,d μ_A \,].$ When the equilibrium probabilities $μ_A$ is the set of invariant Markov probabilities on $\{0,1\}^\mathbb{N}\subset \mathcal{N}$, introducing an orthonormal basis $\hat{a}_y$, indexed by finite words $y$, we show explicit expressions for $K(\hat{a}_x,\hat{a}_z)$, which is a finite sum. These values can be positive or negative depending on $A$ and the words $x$ and $z$. Words $x,z$ with large length can eventually produce large negative curvature $K(\hat{a}_x,\hat{a}_z)$. If $x, z$ do not begin with the same letter, then $K(\hat{a}_x,\hat{a}_z)=0$.

math.DS↗

arXiv2023

Noncommutative integration, quantum mechanics, Tannaka's theorem for compact groupoids and examples

We consider topological groupoids in finite and also in a compact settings. In the initial sections, we introduce definitions of typical observables and we studied them in the context of statistical mechanics and quantum mechanics. We exhibit explicit examples and one of them will be the so-called quantum ratchet. This is related to Schwinger's algebra of selective measurements. Here we consider $\mathcal{G}$-kernels, transverse functions, modular functions, and quasi-invariant measures for Haar systems. Later we present our main result which is a version of Tannaka's theorem for Hausdorff compact groupoids - extending the original proof of T. Tannaka.

math-ph↗

arXiv2022

The involution kernel and the dual potential for functions in the Walters family

Our notation: Points in $\{0,1\}^{\mathbb{Z}-\{0\}} =\{0,1\}^\mathbb{N}\times \{0,1\}^\mathbb{N}=Ω^{-} \times Ω^{+}$, are denoted by $( y|x) =(...,y_2,y_1|x_1,x_2,...)$, where $(x_1,x_2,...) \in \{0,1\}^\mathbb{N}$, and $(y_1,y_2,...) \in \{0,1\}^\mathbb{N}$. The bijective map $\hatσ(...,y_2,y_1|x_1,x_2,...)= (...,y_2,y_1,x_1|x_2,...)$ is called the bilateral shift and acts on $\{0,1\}^{\mathbb{Z}-\{0\}}$. Given $A: \{0,1\}^\mathbb{N}=Ω^+\to \mathbb{R}$ we express $A$ in the variable $x$, like $A(x)$. In a similar way, given $B: \{0,1\}^\mathbb{N}=Ω^{-}\to \mathbb{R}$ we express $B$ in the variable $y$, like $B(y)$. Finally, given $W: Ω^{-} \times Ω^{+}\to \mathbb{R}$, we express $W$ in the variable $(y|x)$, like $W(y|x)$. By abuse of notation we write $A(y|x)=A(x)$ and $B(y|x)=B(y).$ The probability $μ_A$ denotes the equilibrium probability for $A: \{0,1\}^\mathbb{N}\to \mathbb{R}$. Given a continuous potential $A: Ω^+\to \mathbb{R}$, we say that the continuous potential $A^*: Ω^{-}\to \mathbb{R}$ is the dual potential of $A$, if there exists a continuous $W: Ω^{-} \times Ω^{+}\to \mathbb{R}$, such that, for all $(y|x) \in \{0,1\}^{\mathbb{Z}-\{0\}}$ $$ A^* (y) = \left[ A \circ \hatσ^{-1} + W \circ \hatσ^{-1} - W \right] (y|x). $$ We say that $W$ is an involution kernel for $A$. The function $W$ allows you to define an spectral projection in the linear space of the main eigenfunction of the Ruelle operator for $A$. Given $A$, we describe explicit expressions for $W$ and the dual potential $A^*$, for $A$ in a family of functions introduced by P. Walters. We present conditions for $A$ to be symmetric and to be of twist type.

math.DS↗