arXiv subjects

E. R. Oliveira

Publications and source records attributed to E. R. Oliveira.

5 recordsLinked to original sources

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.

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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.

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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.

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arXiv2014

On the dynamics of a rational semigroup on a convolution measure algebra

We are going to study the dynamical properties of the rational semigroup $Q_{t}(μ)$ where $Q_{t}(μ)= (1-t) μ* (1- t μ)^{-1},$ for $t \in [0,1)$, that is defined for $μ\in \mathcal{P}(G)$, the set of Borel probabilities over $(G, \cdot)$ an abelian compact topological group where we define the \textbf{convolution}, $ν* μ\in \mathcal{P}(G)$, as usual for a group $\int f d(ν* μ)= \int \int f(xy) dν(x) dμ(y),$ then $(\mathcal{P}(G), *)$ became a $\textbf{convolution measure algebra}$ (CM-algebra). We investigate several properties for this semigroup (as the Stable Manifold Theorem, Asymptotic behavior, invariant sets, differential properties, stationary points, etc) and how they are related with the Choquet-Deny equation. As an application we give a complete description of this semigroup for finite abelian groups.

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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.

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