arXiv subjects

Jairo K. Mengue

Publications and source records attributed to Jairo K. Mengue.

13 recordsLinked to original sources

arXiv2025

Optimal transportation and pressure at zero temperature

Given two compact metric spaces $X$ and $Y$, a Lipschitz continuous cost function $c$ on $X \times Y$ and two probabilities $μ\in\mathcal{P}(X),\,ν\in\mathcal{P}(Y)$, we propose to study the Monge-Kantorovich problem and its duality from a zero temperature limit of a convex pressure function. We consider the entropy defined by $H(π) = -D_{KL}(π|μ\times ν)$, where $D_{KL}$ is the Kullback-Leibler divergence, and then the pressure defined by the variational principle \[P(βA) = \sup_{π\in Π(μ,ν)} \left[ \smallint βA\,dπ+ H(π)\right],\]where $β>0$ and $A=-c$. We will show that it admits a dual formulation and when $β\to+\infty$ we recover the solution for the usual Monge-Kantorovich problem and its Kantorovich duality. Such approach is similar to one which is well known in Thermodynamic Formalism and Ergodic Optimization, where $β$ is interpreted as the inverse of the temperature ($β= \frac{1}{T}$) and $β\to+\infty$ is interpreted as a zero temperature limit.

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arXiv2024

Invariant measures for place dependent idempotent iterated function systems

We study the set of invariant idempotent probabilities for place dependent idempotent iterated function systems defined in compact metric spaces. Using well-known ideas from dynamical systems, such as the Mañé potential and the Aubry set, we provide a complete characterization of the densities of such idempotent probabilities. As an application, we provide an alternative formula for the attractor of a class of fuzzy iterated function systems.

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arXiv2021

On information gain, Kullback-Leibler divergence, entropy production and the involution kernel

It is well known that in Information Theory and Machine Learning the Kullback-Leibler divergence, which extends the concept of Shannon entropy, plays a fundamental role. Given an {\it a priori} probability kernel $\hatν$ and a probability $π$ on the measurable space $X\times Y$ we consider an appropriate definition of entropy of $π$ relative to $\hatν$, which is based on previous works. Using this concept of entropy we obtain a natural definition of information gain for general measurable spaces which coincides with the mutual information given from the K-L divergence in the case $\hatν$ is identified with a probability $ν$ on $X$. This will be used to extend the meaning of specific information gain and dynamical entropy production to the model of thermodynamic formalism for symbolic dynamics over a compact alphabet (TFCA model). In this case, we show that the involution kernel is a natural tool for better understanding some important properties of entropy production.

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arXiv2020

Thermodynamic Formalism for Haar systems in Noncommutative Integration: transverse functions and entropy of transverse measures

We consider here a class of groupoids obtained via an equivalence relation (the subgroupoids of pair groupoids). We generalize to Haar Systems in these groupoids some results related to entropy and pressure which are well known in Thermodynamic Formalism. We introduce a transfer operator, where the equivalence relation (which defines the groupoid) plays the role of the dynamics and the corresponding transverse function plays the role of the {\it a priori} probability. We also introduce the concept of invariant transverse probability and of entropy for an invariant transverse probability, as well as of pressure for transverse functions. Moreover, we explore the relation between quasi-invariant probabilities and transverse measures. Our results are on measurable category.

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arXiv2018

Quantum Spin probabilities at positive temperature are Hölder Gibbs probabilities

We consider the KMS state associated to the Hamiltonian $H= σ^x \otimes σ^x$ over the quantum spin lattice $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes ...$. For a fixed observable of the form $L \otimes L \otimes L \otimes ...$, where $L:\mathbb{C}^2 \to \mathbb{C}^2 $ is self adjoint, and for positive temperature $T$ one can get a naturally defined stationary probability $μ_T$ on the Bernoulli space $\{1,2\}^\mathbb{N}$. The Jacobian of $μ_T$ can be expressed via a certain continued fraction expansion. We will show that this probability is a Gibbs probability for a Hölder potential. Therefore, this probability is mixing for the shift map. For such probability $μ_T$ we will show the explicit deviation function for a certain class of functions. When decreasing temperature we will be able to exhibit the explicit transition value $T_c$ where the set of values of the Jacobian of the Gibbs probability $μ_T$ changes from being a Cantor set to being an interval. We also present some properties for quantum spin probabilities at zero temperature (for instance, the explicit value of the entropy).

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arXiv2018

Selection of calibrated subaction when temperature goes to zero in the discounted problem

Consider $T(x)= d \, x$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1 \to \mathbb{R}$, $0<λ<1$ and the unique function $b_λ:S^1 \to \mathbb{R}$ satisfying $ b_λ(x) = \max_{T(y)=x} \{ λ\, b_λ(y) + A(y)\}.$ We will show that, when $λ\to 1$, the function $b_λ- \frac{m(A)}{1-λ}$ converges uniformly to the calibrated subaction $V(x) = \max_{μ\in \mathcal{ M}} \int S(y,x) \, d μ(y)$, where $S$ is the Mañe potential, $\mathcal{ M}$ is the set of invariant probabilities with support on the Aubry set and $m(A)= \sup_{μ\in \mathcal{M}} \int A\,dμ$. For $β>0$ and $λ\in (0,1)$, there exists a unique fixed point $u_{λ,β} :S^1\to \mathbb{R}$ for the equation $e^{u_{λ,β}(x)} = \sum_{T(y)=x}e^{βA(y) +λu_{λ,β}(y)}$. It is known that as $λ\to 1$ the family $e^{[u_{λ,β}- \sup u_{λ,β}]}$ converges uniformly to the main eigenfuntion $ϕ_β$ for the Ruelle operator associated to $βA$. We consider $λ=λ(β)$, $β(1-λ(β))\to+\infty$ and $λ(β) \to 1$, as $β\to\infty$. Under these hypothesis we will show that $\frac{1}β(u_{λ,β}-\frac{P(βA)}{1-λ})$ converges uniformly to the above $V$, as $β\to \infty$. The parameter $β$ represents the inverse of temperature in Statistical Mechanics and $β\to \infty$ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when $β\to \infty$.

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arXiv2017

Large Deviations for Quantum Spin probabilities at temperature zero

We consider certain self-adjoint observables for the KMS state associated to the Hamiltonian $H= σ^x \otimes σ^x$ over the quantum spin lattice $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes ...$. For a fixed observable of the form $L \otimes L \otimes L \otimes ...$, where $L:\mathbb{C}^2 \to \mathbb{C}^2 $, and for the zero temperature limit one can get a naturally defined stationary probability $μ$ on the Bernoulli space $\{1,2\}^\mathbb{N}$. This probability is ergodic but it is not mixing for the shift map. It is not a Gibbs state for a continuous normalized potential but its Jacobian assume only two values almost everywhere. Anyway, for such probability $μ$ we can show that a Large Deviation Principle is true for a certain class of functions. The result is derived by showing the explicit form of the free energy which is differentiable.

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arXiv2017

Large deviations for equilibrium measures and selection of subaction

Given a Lipschitz function $f:\{1,...,d\}^\mathbb{N} \to \mathbb{R}$, for each $β>0$ we denote by $μ_β$ the equilibrium measure of $βf$ and by $h_β$ the main eigenfunction of the Ruelle Operator $L_{βf}$. Assuming that $\{μ_β\}_{β>0}$ satisfy a large deviation principle, we prove the existence of the uniform limit $V= \lim_{β\to\infty}\frac{1}β\log(h_β)$. Furthermore, the expression of the deviation function is determined by its values at the points of the union of the supports of maximizing measures. We study a class of potentials having two ergodic maximizing measures and prove that a L.D.P. is satisfied. The deviation function is explicitly exhibited and does not coincide with the one that appears in the paper by Baraviera-Lopes-Thieullen which considers the case of potentials having a unique maximizing measure.

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arXiv2016

A Large Deviation Principle for Gibbs States on Markov Shifts at Zero Temperature

Let $Σ_{A}(\mathbb{N})$ be a topologically mixing countable Markov shift with the BIP property over the alphabet $\mathbb{N}$ and $f: Σ_{A}(\mathbb{N}) \rightarrow \mathbb{R}$ a potential satisfying the Walters condition with finite Gurevich pressure. Under suitable hypotheses, we prove the existence of a Large Deviation Principle for the family $(μ_β)_{β> 0}$ where each $μ_β$ is the Gibbs measure associated to the potential $βf$. Our main theorem generalizes from finite to countable alphabets and also to a larger class of potentials a previous result of A. Baraviera, A. O. Lopes and P. Thieullen.

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arXiv2015

Duality results for Iterated Function Systems with a general family of branches

For $X$, $Y$, $Z$ and $W$ compact metric spaces, consider two uniformly contractive IFS $\{τ_x: Z\to Z,\, x\in x\}$ and $\{τ_y:W\to W,\, y\in Y\}$. For a fixed $α\in \mathcal{P}(X)$ with $supp(α)=X$ we define the entropy of a holonomic measure $π\in \mathcal{P}(X\times Z)$ relative to $α$, the pressure of a continuous cost function $c(x,z)$ and show that for $c$ Lipschitz this pressure coincides with the spectral radius of the associated transfer operator. The same approach can be applied to the pair $Y,W$. For fixed probabilities $α\in \mathcal{P}(X)$ and $β\in \mathcal{P}(Y)$ with $supp(α)=X,\,supp(β)=Y$ we denote by $H_α(π), π\in Π(\cdot,\cdot,τ)$, the entropy of the $(X,Z)-$marginal of $π$ relative to $α$ and denote by $H_β(π)$, the entropy of the $(Y,W)-$marginal of $π$ relative to $β$. The marginal pressure of a continuous cost function $c \in C(X\times Y \times Z \times W)$ relative to $(α,β)$ will be defined by $P^{m}(c) = \sup_{π\inΠ(\cdot,\cdot,τ)} \int c\, dπ+ H_α(π) +H_β(π)$ and we will show the following duality result: \[\inf_{P^{m}(c -φ(x) -ψ(y))=0} \int φ(x)\,dμ+\int ψ(y)\,dν= \sup_{π\inΠ(μ,ν,τ)} \int c\, dπ+ H_α(π) +H_β(π).\] When $Z$ and $W$ have only one point and the entropy is unconsidered this equality can be rewritten as the Kantorovich Duality for compact spaces $X,Y$ and continuous cost $-c$: \[\inf_{c -φ(x) -ψ(y)\leq 0} \int φ(x)\,dμ+\int ψ(y)\,dν= \sup_{π\inΠ(μ,ν)} \int c\, dπ.\]

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arXiv2013

Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature

We generalize several results of the classical theory of Thermodynamic Formalism by considering a compact metric space $M$ as the state space. We analyze the shift acting on $M^\mathbb{N}$ and consider a general a-priori probability for defining the Transfer (Ruelle) operator. We study potentials $A$ which can depend on the infinite set of coordinates in $M^\mathbb{N}.$ We define entropy and by its very nature it is always a nonpositive number. The concepts of entropy and transfer operator are linked. If M is not a finite set there exist Gibbs states with arbitrary negative value of entropy. Invariant probabilities with support in a fixed point will have entropy equal to minus infinity. In the case $M=S^1$, and the a-priori measure is Lebesgue $dx$, the infinite product of $dx$ on $(S^1)^\mathbb{N}$ will have zero entropy. We analyze the Pressure problem for a Hölder potential $A$ and its relation with eigenfunctions and eigenprobabilities of the Ruelle operator. Among other things we analyze the case where temperature goes to zero and we show some selection results. Our general setting can be adapted in order to analyze the Thermodynamic Formalism for the Bernoulli space with countable infinite symbols. Moreover, the so called XY model also fits under our setting. In this last case M is the unitary circle $S^1$. We explore the differentiable structure of $(S^1)^\mathbb{N}$ by considering potentials which are of class $C^1$ and we show some properties of the corresponding main eigenfunctions.

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arXiv2012

Duality Theorems in Ergodic Transport

We analyze several problems of Optimal Transport Theory in the setting of Ergodic Theory. In a certain class of problems we consider questions in Ergodic Transport which are generalizations of the ones in Ergodic Optimization. Another class of problems is the following: suppose $σ$ is the shift acting on Bernoulli space $X=\{0,1\}^\mathbb{N}$, and, consider a fixed continuous cost function $c:X \times X\to \mathbb{R}$. Denote by $Π$ the set of all Borel probabilities $π$ on $X\times X$, such that, both its $x$ and $y$ marginal are $σ$-invariant probabilities. We are interested in the optimal plan $π$ which minimizes $\int c d π$ among the probabilities on $Π$. We show, among other things, the analogous Kantorovich Duality Theorem. We also analyze uniqueness of the optimal plan under generic assumptions on $c$. We investigate the existence of a dual pair of Lipschitz functions which realizes the present dual Kantorovich problem under the assumption that the cost is Lipschitz continuous. For continuous costs $c$ the corresponding results in the Classical Transport Theory and in Ergodic Transport Theory can be, eventually, different. We also consider the problem of approximating the optimal plan $π$ by convex combinations of plans such that the support projects in periodic orbits.

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arXiv2010

Zeta measures and Thermodynamic Formalism for temperature zero

We address the analysis of the following problem: given a real Hölder potential $f$ defined on the Bernoulli space and $μ_f$ its equilibrium state, it is known that this shift-invariant probability can be weakly approximated by probabilities in periodic orbits associated to certain zeta functions. Given a Hölder function $f>0$ and a value $s$ such that $0 0$. We do not assume here the maximizing probability for $f$ is unique.

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