arXiv subjects

A. O. Lopes

Publications and source records attributed to A. O. Lopes.

18 recordsLinked to original sources

arXiv2025

A noncommutative Ruelle's Theorem for a normalized potential taking values on positivity-improving operators

Let $\mathcal{A}$ be a finite-dimensional real (or complex) C*-algebra, $Ω_{A}$ an aperiodic subshift of finite type, and $\mathcal{C}(Ω_{A}; \mathcal{A})$ the set of continuous functions from $Ω_{A}$ to $\mathcal{A}$. The shift $σ$ provides dynamics. Given a real Lipschitz potential $φ\in \mathcal{C}(Ω_{A}; \mathfrak{L}(\mathcal{A}))$, where $\mathfrak{L}(\mathcal{A})$ is the set of linear operators acting on a real $\mathcal{A}$, we introduce a noncommutative analogue of Ruelle's operator, which acts on $\mathcal{C}(Ω_{A}; \mathcal{A})$. Assuming the positivity-improving hypothesis, we prove a version of Ruelle's Theorem whenever the operator is normalized. An eigenstate (a linear functional) invariant for the action of the noncommutative Ruelle's operator will play the role of the Gibbs probability of Thermodynamic Formalism; to be called a Gibbs eigenstate. We introduce the concept of entropy for a Gibbs eigenstate (obtained from a certain family of potentials $φ$) - generalizing the classical one. In our setting, there is currently no direct relationship with cocycles and Lyapunov exponents. We present examples illustrating the novelty of the cases that can be considered, ranging from topics related to quantum channels to Pauli matrices. Interesting cases: $\mathcal{A}=M_{N \times N}(\mathbb{R})$ and $\mathcal{A}= \mathbb{R}^{N}$.

math.OA↗

arXiv2025

Grand-canonical Thermodynamic Formalism via IFS: volume, temperature, gas pressure and grand-canonical topological pressure

We consider here a dynamic model for a gas in which a variable number of particles $N \in \mathbb{N}_0 := \mathbb{N} \cup \{0\}$ can be located at a site. This point of view leads us to the grand-canonical framework and the need for a chemical potential. The dynamics is played by the shift acting on the set of sequences $Ω:= \mathcal{A}^\mathbb{N}$, where the alphabet is $\mathcal{A} := \{1,2,...,r\}$. Introducing new variables like the number of particles $N$ and the chemical potential $μ$, we adapt the concept of grand-canonical partition sum of thermodynamics of gases to a symbolic dynamical setting considering a Lipschitz family of potentials $% (A_N)_{N \in \mathbb{N}_0}$, $A_N:Ω\to \mathbb{R}$. Our main results will be obtained from adapting well-known properties of the Thermodynamic Formalism for IFS with weights to our setting. In this direction, we introduce the grand-canonical-Ruelle operator: $\mathcal{L}_{β, μ}(f)=g$, when, $β>0,μ<0,$ and \medskip $\,\,\,\,\,\,\,\,\,\,\,\,\,\,g(x)= \mathcal{L}_{β, μ}(f) (x) =\sum_{N \in \mathbb{N}_0} e^{β\, μ\, N }\, \sum_{j \in \mathcal{A}} e^{- \,β\, A_N(jx)} f(jx). $ \medskip We show the existence of the main eigenvalue, an associated eigenfunction, and an eigenprobability for $\mathcal{L}_{β, μ}^*$. We can show the analytic dependence of the eigenvalue on the grand-canonical potential. Considering the concept of entropy for holonomic probabilities on $Ω\times \mathcal{A}^{\mathbb{N}_0}$, we relate these items with the variational problem of maximizing grand-canonical pressure. In another direction, in the appendix, we briefly digress on a possible interpretation of the concept of topological pressure as related to the gas pressure of gas thermodynamics.

math.DS↗

arXiv2024

Idempotent approach to level-2 variational principles in Thermodynamical Formalism

In this work we introduce an idempotent pressure to level-2 functions and its associated density entropy. All this is related to idempotent pressure functions which is the natural concept that corresponds to the meaning of probability in the level-2 max-plus context. In this general framework the equilibrium states, maximizing the variational principle, are not unique. We investigate the connections with the general convex pressure introduced recently to level-1 functions by Biś, Carvalho, Mendes and Varandas. Our general setting contemplates the dynamical and not dynamical framework. We also study a characterization of the density entropy in order to get an idempotent pressure invariant by dynamical systems acting on probabilities; this is therefore a level-2 result. We are able to produce idempotent pressure functions at level-2 which are invariant by the dynamics of the pushforward map via a form of Ruelle operator.

math.DS↗

arXiv2024

Thermodynamic Formalism on the Skorokhod space: the continuous time Ruelle operator, entropy, pressure, entropy production and expansiveness

Consider the semi-flow given by the continuous time shift $Θ_t:\mathcal{D} \to \mathcal{D} $, $t \geq 0$, acting on the $\mathcal{D} $ of \textit{càdlàg} paths $w: [0,\infty) \to S^1$, where $S^1$ is the unitary circle. We equip the space $\mathcal{D} $ with the Skorokhod metric, and we show that the semi-flow is expanding. We also introduce a stochastic semi-group $e^{t\, L}$, $t \geq 0,$ where $L$ acts linearly on continuous functions $f:S^1\to\mathbb{R}$. This stochastic semigroup and an initial vector of probability $π$ define an associated stationary shift-invariant probability $\mathbb{P}$ on the Polish space $\mathcal{D} $. Given such $\mathbb{P}$ and an Hölder potential $V:S^1 \to \mathbb{R}$, we define a continuous time Ruelle operator, which is described by a family of linear operators $ \mathbb{L}^t_V$, $t\geq 0,$ acting on continuous functions $φ: S^1 \to \mathbb{R}$. More precisely, given any Hölder $V$ and $t\geq 0$, the operator $ \mathbb{L}^t_V$, is defined by $φ\to ψ(y) = \mathbb{L}^t_V(φ)(y)= \int_{w(t)=y} e^{ \int_0^t V(w(s)) ds} φ(w(0)) d \mathbb{P}(w).$ For some specific parameters we show the existence of an eigenvalue $λ_V$ and an associated Hölder eigenfunction $φ_V>0$.After a coboundary procedure we obtain another stochastic semigroup, with infinitesimal generator $L_V$, and this will define a new probability $\mathbb{P}_V$ on $\mathcal{D}$, which we call the Gibbs (or, equilibrium) probability for the potential $V$. In this case, we define entropy for some shift-invariant probabilities on $\mathcal{D}$, and we consider a variational problem of pressure. Finally, we define entropy production and present our main result: we analyze its relation with time-reversal and symmetry of $L$. We also show that the continuous-time shift $Θ_t$, acting on the Skorohod space $D$, is expanding.

math.DS↗

arXiv2024

Diffusion Processes: entropy, Gibbs states and the continuous time Ruelle operator

We consider a Riemmaniann compact manifold $M$, the associated Laplacian $Δ$ and the corresponding Brownian motion $X_t$, $t\geq 0.$ Given a Lipschitz function $V:M\to\mathbb R$ we consider the operator $\frac{1}{2}Δ+V$, which acts on differentiable functions $f: M\to\mathbb R$ via the operator $$\frac{1}{2} Δf(x)+\,V(x)f(x) ,$$ for all $x\in M$. Denote by $P_t^V$, $t \geq 0,$ the semigroup acting on functions $f: M\to\mathbb R$ given by $$P_{t}^V (f)(x)\,:=\, \mathbb E_{x} \big[e^{\int_0^{t} V(X_r)\,dr} f(X_t)\big].\,$$ We will show that this semigroup is a continuous-time version of the discrete-time Ruelle operator. Consider the positive differentiable eigenfunction $F: M \to \mathbb{R}$ associated to the main eigenvalue $λ$ for the semigroup $P_t^V$, $t \geq 0$. From the function $F$, in a procedure similar to the one used in the case of discrete-time Thermodynamic Formalism, we can associate via a coboundary procedure a certain stationary Markov semigroup. The probability on the Skhorohod space obtained from this new stationary Markov semigroup can be seen as a stationary Gibbs state associated with the potential $V$. We define entropy, pressure, the continuous-time Ruelle operator and we present a variational principle of pressure for such a setting.

math.PR↗

arXiv2024

Level-2 IFS Thermodynamic Formalism: Gibbs probabilities in the space of probabilities and the push-forward map

We will denote by $\mathcal{M}$ the space of Borel probabilities on the symbolic space $Ω=\{1,2...,m\}^\mathbb{N}$. $\mathcal{M}$ is equipped Monge-Kantorovich metric. We consider here the push-forward map $\mathfrak{T}:\mathcal{M} \to \mathcal{M}$ as a dynamical system. The space of Borel probabilities on $\mathcal{M}$ is denoted by $\mathfrak{M}$. Given a continuous function $A: \mathcal{M}\to \mathbb{R}$, an {\it a priori} probability $Π_0$ on $\mathcal{M}$, and a certain convolution operation acting on pairs of probabilities on $\mathcal{M}$, we define an associated Level-2 IFS Ruelle operator. We show the existence of an eigenfunction and an eigenprobability $\hatΠ\in\mathfrak{M}$ for such an operator. Under a normalization condition for $A$, we show the existence of some $\mathfrak{T}$-invariant probabilities $\hatΠ\in\mathfrak{M}.$ We are able to define the variational entropy of such $\hatΠ$ and a related maximization pressure problem associated to $A$. In some particular examples, we show how to get eigenprobabilities solutions on $\mathfrak{M}$ for the Level-2 Thermodynamic Formalism problem from eigenprobabilities on $\mathcal{M}$ for the classical (Level-1) Thermodynamic Formalism. These examples highlight the fact that our approach is a natural generalization of the classic case.

math.DS↗

arXiv2022

Dynamical hypothesis tests and Decision Theory for Gibbs distributions

We consider the problem of testing for two Gibbs probabilities $μ_0$ and $μ_1$ defined for a dynamical system $(Ω,T)$. Due to the fact that in general full orbits are not observable or computable, one needs to restrict to subclasses of tests defined by a finite time series $h(x_0), h(x_1)=h(T(x_0)),..., h(x_n)=h(T^n(x_0))$, $x_0\in Ω$, $n\ge 0$, where $h:Ω\to\mathbb R$ denotes a suitable measurable function. We determine in each class the Neyman-Pearson tests, the minimax tests, and the Bayes solutions, and show the asymptotic decay of their risk functions, as $n\to\infty$. In the case of $Ω$ being a symbolic space, for each $n\in \mathbb{N}$, these optimal tests rely on the information of the measures for cylinder sets of size $n$.

math.ST↗

arXiv2021

Explicit Bivariate Rate Functions for Large Deviations in AR(1) and MA(1) Processes with Gaussian Innovations

We investigate large deviations properties for centered stationary AR(1) and MA(1) processes with independent Gaussian innovations, by giving the explicit bivariate rate functions for the sequence of random vectors $(\boldsymbol{S}_n)_{n \in \N} = \left(n^{-1}(\sum_{k=1}^n X_k, \sum_{k=1}^n X_k^2)\right)_{n \in \N}$. In the AR(1) case, we also give the explicit rate function for the bivariate random sequence $(\W_n)_{n \geq 2} = \left(n^{-1}(\sum_{k=1}^n X_k^2, \sum_{k=2}^n X_k X_{k+1})\right)_{n \geq 2}$. Via Contraction Principle, we provide explicit rate functions for the sequences $(n^{-1} \sum_{k=1}^n X_k)_{n \in \N}$, $(n^{-1} \sum_{k=1}^n X_k^2)_{n \geq 2}$ and $(n^{-1} \sum_{k=2}^n X_k X_{k+1})_{n \geq 2}$, as well. In the AR(1) case, we present a new proof for an already known result on the explicit deviation function for the Yule-Walker estimator.

math.PR↗

arXiv2019

The KMS Condition for the homoclinic equivalence relation and Gibbs probabilities

D. Ruelle considered a general setting where he is able to characterize equilibrium states for Hölder potentials based on properties of conjugating homeomorphism in the so called Smale spaces. On this setting he also shows a relation of KMS states of $C^*$-algebras and equilibrium probabilities of Thermodynamic Formalism. A later paper by N. Haydn and D. Ruelle presents a shorter proof of this equivalence. Here we consider similar problems but now on the symbolic space $Ω= \{1,2,...,d\}^{\mathbb{Z} - \{ 0 \} }$ and the dynamics will be given by the shift $τ$. In the case of potentials depending on a finite coordinates we will present a simplified proof of the equivalence mentioned above which is the main issue of the papers by D. Ruelle and N. Haydn. The class of conjugating homeomorphism is explicit and reduced to a minimal set of conditions. We also present with details (following D. Ruelle) the relation of these probabilities with the KMS dynamical $C^*$-state on the $C^*$-Algebra associated to the groupoid defined by the homoclinic equivalence relation. The topics presented here are not new but we believe the main ideas of the proof of the results by Ruelle and Haydn will be quite transparent in our exposition.

math.DS↗

arXiv2017

Spectral Properties of the Ruelle Operator for Product Type Potentials on Shift Spaces

We study a class of potentials $f$ on one sided full shift spaces over finite or countable alphabets, called potentials of product type. We obtain explicit formulae for the leading eigenvalue, the eigenfunction (which may be discontinuous) and the eigenmeasure of the Ruelle operator. The uniqueness property of these quantities is also discussed and it is shown that there always exists a Bernoulli equilibrium state even if $f$ does not satisfy Bowen's condition. We apply these results to potentials $f:\{-1,1\}^\mathbb{N} \to \mathbb{R}$ of the form $$ f(x_1,x_2,\ldots) = x_1 + 2^{-γ} \, x_2 + 3^{-γ} \, x_3 + ...+n^{-γ} \, x_n + \ldots $$ with $γ>1$. For $3/2 < γ\leq 2$, we obtain the existence of two different eigenfunctions. Both functions are (locally) unbounded and exist a.s. (but not everywhere) with respect to the eigenmeasure and the measure of maximal entropy, respectively.

math.DS↗

arXiv2014

Entropy, Pressure and Duality for Gibbs plans in Ergodic Transport

Let $X$ be a finite set and $Ω=\{1,...,d\}^{\mathbb{N}}$ be the Bernoulli space. Denote by $σ$ the shift map acting on $Ω$. For a fixed probability $μ$ on $X$ with supp($μ$)$=X$, define $Π(μ,σ)$ as the set of all Borel probabilities $π\in P(X\times Ω)$ such that the $x$-marginal of $π$ is $μ$ and the $y$-marginal of $π$ is $σ$-invariant. We consider a fixed Lipschitz cost function $c: X \times Ω\to \mathbb{R}$ and an associated Ruelle operator. We introduce the concept of Gibbs plan, which is a probability on $X \times Ω$. Moreover, we define entropy, pressure and equilibrium plans. The study of equilibrium plans can be seen as a generalization of the optimal cost problem where the concept of entropy is introduced. We show that an equilibrium plan is a Gibbs plan. Our main result is a Kantorovich duality Theorem on this setting. The pressure plays an important role in the establishment of the notion of admissible pair. Finally, given a parameter $β$, which plays the role of the inverse of temperature, we consider equilibrium plans for $βc$ and its limit $π_\infty$, when $β\to \infty$, which is also known as ground state. We compare this with other previous results on Ergodic Transport in temperature zero.

math.DS↗

arXiv2013

Ergodic optimization, zero temperature limits and the max-plus algebra

Lecture notes of a course at the Brazilian Mathematical Colloquium. We review some basic notions in ergodic theory and thermodynamic formalism, as well as introductory results in the context of max-plus algebra, in order to exhibit some properties of equilibrium measures when temperature goes to zero.

math.DS↗

arXiv2013

Ergodic Transport Theory, periodic maximizing probabilities and the twist condition

The present paper is a follow up of another one by A. O. Lopes, E. Oliveira and P. Thieullen which analyze ergodic transport problems. Our main focus will a more precise analysis of case where the maximizing probability is unique and is also a periodic orbit. Consider the shift T acting on the Bernoulli space Σ={1, 2, 3,.., d}^\mathbb{N} $ and $A:Σ\to \mathbb{R} a Holder potential. Denote m(A)=max_{νis an invariant probability for T} \int A(x) \; dν(x) and, μ_{\infty,A}, any probability which attains the maximum value. We assume this probability is unique (a generic property). We denote \T the bilateral shift. For a given potential Holder A:Σ\to \mathbb{R}, we say that a Holder continuous function W: \hatΣ \to \mathbb{R} is a involution kernel for A, if there is a Holder function A^*:Σ\to \mathbb{R}, such that, A^*(w)= A\circ \T^{-1}(w,x)+ W \circ \T^{-1}(w,x) - W(w,x). We say that A^* is a dual potential of A. It is true that m(A)=m(A^*). We denote by V the calibrated subaction for A, and, V^* the one for A^*. We denote by I^* the deviation function for the family of Gibbs states for βA, when β\to \infty. For each x we get one (more than one) w_x such attains the supremum above. That is, solutions of V(x) = W(w_x,x) - V^* (w_x)- I^*(w_x). A pair of the form (x,w_x) is called an optimal pair. If \T is the shift acting on (x,w) \in {1, 2, 3,.., d}^\mathbb{Z}, then, the image by \T^{-1} of an optimal pair is also an optimal pair. Theorem - Generically, in the set of Holder potentials A that satisfy (i) the twist condition, (ii) uniqueness of maximizing probability which is supported in a periodic orbit, the set of possible optimal w_x, when x covers the all range of possible elements x in \in Σ, is finite.

math.DS↗

arXiv2011

On the general one-dimensional XY Model: positive and zero temperature, selection and non-selection

We consider $(M,d)$ a connected and compact manifold and we denote by $\mathcal{B}_i$ the Bernoulli space $M^{\Z}$ of sequences represented by $$x=(... x_{-3},x_{-2},x_{-1},x_0,x_1,x_2,x_3,...),$$ where $x_i$ belongs to the space (alphabet) $M$. The case where $M=\mathbb{S}^1$, the unit circle, is of particular interest here. The analogous problem in the one-dimensional lattice $\mathbb{N}$ is also considered. %In this case we consider the potential $A: {\cal B}=M^\mathbb{N} \to \mathbb{R}.$ Let $A: \mathcal{B}_i \rar \R$ be an {\it observable} or {\it potential} defined in the Bernoulli space $\mathcal{B}_i$. The potential $A$ describes an interaction between sites in the one-dimensional lattice $M^\mathbb{Z}$. Given a temperature $T$, we analyze the main properties of the Gibbs state $\hatμ_{\frac{1}{T} A}$ which is a certain probability measure over ${\cal B}_i$. We denote this setting "the general XY model". In order to do our analysis we consider the Ruelle operator associated to $\frac{1}{T} A$, and, we get in this procedure the main eigenfunction $ψ_{\frac{1}{T} A}$. Later, we analyze selection problems when temperature goes to zero: a) existence, or not, of the limit (on the uniform convergence) $$V:=\lim_{T\to 0} T\, \log(ψ_{\frac{1}{T} A}),\,\,\,\,\text{a question about selection of subaction},$$ and, b) existence, or not, of the limit (on the weak$^*$ sense) $$\tildeμ:=\lim_{T\to 0} \hatμ_{\frac{1}{T}\, A},\,\,\,\,\text{a question about selection of measure}.$$ The existence of subactions and other properties of Ergodic Optimization are also considered.

math.DS↗

arXiv2011

On the selection of subaction and measure for a subclass of potentials defined by P. Walters

Suppose $σ$ is the shift acting on Bernoulli space $X=\{0,1\}^\mathbb{N}$, and, consider a fixed function $f:X \to \mathbb{R}$, under the Waters's conditions (defined in a paper in ETDS 2007). For each real value $t\geq 0$ we consider the Ruelle Operator $L_{tf}$. We are interested in the main eigenfunction $h_t$ of $L_{tf}$, and, the main eigenmeasure $ν_t$, for the dual operator $L_{tf}^*$, which we consider normalized in such way $h_t(0^\infty)=1$, and, $\int h_t \,d\,ν_t=1, \forall t>0$. We denote $μ_t= h_t ν_t$ the Gibbs state for the potential $t\, f$. By selection of a subaction $V$, when the temperature goes to zero (or, $t\to \infty$), we mean the existence of the limit $$V:=\lim_{t\to\infty}\frac{1}{t}\log(h_{t}).$$ By selection of a measure $μ$, when the temperature goes to zero (or, $t\to \infty$), we mean the existence of the limit (in the weak$^*$ sense) $$μ:=\lim_{t\to\infty} μ_t.$$ We present a large family of non-trivial examples of $f$ where the selection of measure exists. These $f$ belong to a sub-class of potentials introduced by P. Walters. In this case, explicit expressions for the selected $V$ can be obtained for a certain large family of potentials.

math.DS↗

arXiv2011

A dynamical point of view of Quantum Information: entropy, pressure and Wigner measures

Quantum Information is a new area of research which has been growing rapidly since the last decade. This topic is very close to potential applications to the so called Quantum Computer. In our point of view it makes sense to develop a more "dynamical point of view" of this theory. We want to consider the concepts of entropy and pressure for "stationary systems" acting on density matrices which generalize the usual ones in Ergodic Theory (in the sense of the Thermodynamic Formalism of R. Bowen, Y. Sinai and D. Ruelle). We consider the operator $\mathcal{L}$ acting on density matrices $ρ\in \mathcal{M}_N$ over a finite $N$-dimensional complex Hilbert space $\mathcal{L}(ρ):=\sum_{i=1}^k tr(W_iρW_i^*)V_iρV_i^*,$ where $W_i$ and $V_i$, $i=1,2,... k$ are operators in this Hilbert space. $\mathcal{L}$ is not a linear operator. In some sense this operator is a version of an Iterated Function System (IFS). Namely, the $V_i (.) V_i^*=:F_i(.)$, $i=1,2,...,k$, play the role of the inverse branches (acting on the configuration space of density matrices $ρ$) and the $W_i$ play the role of the weights one can consider on the IFS. We also analyze the discrete Wigner function. We suppose that for all $ρ$ we have that $\sum_{i=1}^k tr(W_iρW_i^*)=1$. A family $W:=\{W_i\}_{i=1,..., k}$ determines a Quantum Iterated Function System (QIFS) $\mathcal{F}_{W}$, $\mathcal{F}_W=\{\mathcal{M}_N,F_i,W_i\}_{i=1,..., k}.$

math.DS↗

arXiv2011

A dynamical point of view of Quantum Information: entropy and pressure

Quantum Information is a new area of research which has been growing rapidly since last decade. This topic is very close to potential applications to the so called Quantum Computer. In our point of view it makes sense to develop a more "dynamical point of view" of this theory. We want to consider the concepts of entropy and pressure for "stationary systems" acting on density matrices which generalize the usual ones in Ergodic Theory (in the sense of the Thermodynamic Formalism of R. Bowen, Y. Sinai and D. Ruelle). We consider the operator $\mathcal{L}$ acting on density matrices $ρ\in \mathcal{M}_N$ over a finite $N$-dimensional complex Hilbert space $\mathcal{L}(ρ):=\sum_{i=1}^k tr(W_iρW_i^*)V_iρV_i^*,$ where $W_i$ and $V_i$, $i=1,2,...k$ are operators in this Hilbert space. $\mathcal{L}$ is not a linear operator. In some sense this operator is a version of an Iterated Function System (IFS). Namely, the $V_i\,(.)\,V_i^*=:F_i(.)$, $i=1,2,...,k$, play the role of the inverse branches (acting on the configuration space of density matrices $ρ$) and the $W_i$ play the role of the weights one can consider on the IFS. We suppose that for all $ρ$ we have that $\sum_{i=1}^k tr(W_iρW_i^*)=1$. A family $W:=\{W_i\}_{i=1,..., k}$ determines a Quantum Iterated Function System (QIFS) $\mathcal{F}_{W}$, $\mathcal{F}_W=\{\mathcal{M}_N,F_i,W_i\}_{i=1,..., k}.$

quant-ph↗

arXiv2011

A dynamical point of view of Quantum Information: Wigner measures

We analyze a known version of the discrete Wigner function and some connections with Quantum Iterated Funcion Systems. This paper is a follow up of "A dynamical point of view of Quantum Information: entropy and pressure" by the same authors.

quant-ph↗