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

Abstract

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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Cited by in corpus (24)

• Sensitive dependence of Gibbs measures at low temperatures
• Duality Theorems in Ergodic Transport
• Phase Transitions in One-dimensional Translation Invariant Systems: a Ruelle Operator Approach
• Combined mean-field and semiclassical limits of large fermionic systems
• Ruelle Operator for Continuous Potentials and DLR-Gibbs Measures
• Spectral Properties of the Ruelle Operator on the Walters Class over Compact Spaces
• The Analyticity of a Generalized Ruelle's Operator
• Applications of Variable Discounting Dynamic Programming to Iterated Function Systems and Related Problems
• Gibbs States and Gibbsian Specifications on the space ${\mathbb{R}}^{\mathbb{N}}$
• The Ruelle operator for symmetric $\beta$-shifts
• Generic properties of invariant measures of full-shift systems over perfect separable metric spaces
• A thermodynamic formalism for continuous time Markov chains with values on the Bernoulli Space: entropy, pressure and large deviations
• Entropy, pressure, ground states and calibrated sub-actions for linear dynamics
• Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism
• A formula for the Entropy of the Convolution of Gibbs probabilities on the circle
• On the selection of subaction and measure for perturbed potentials
• Uniqueness and statistical properties of the Gibbs state on general one-dimensional lattice systems with markovian structure
• On involution kernels and large deviations principles on $\beta$-shifts
• Existence of Gibbs states and maximizing measures on a general one-dimensional lattice system with markovian structure
• Pressure Derivative on Uncountable Alphabet Setting: a Ruelle Operator Approach
• Entropy, Pressure and Duality for Gibbs plans in Ergodic Transport
• Spectral Triples on Thermodynamic Formalism and Dixmier Trace Representations of Gibbs: theory and examples
• Product type potential on the $XY$ model: selection of maximizing probability and a large deviation principle
• Sup-sums principles for F-divergence, Kullback--Leibler divergence, and new definition for t-entropy