Negative Entropy, Zero temperature and stationary Markov Chains on the interval

Abstract

We analyze some properties of maximizing stationary Markov probabilities on the Bernoulli space $[0,1{]}^{\mathbb{N}}$, More precisely, we consider ergodic optimization for a continuous potential $A$, where $A:[0,1{]}^{\mathbb{N}}\to \mathbb{R}$ which depends only on the two first coordinates. We are interested in finding stationary Markov probabilities ${\mu }_{\infty }$ on $[0,1{]}^{\mathbb{N}}$ that maximize the value $\int Ad\mu ,$ among all stationary Markov probabilities $\mu$ on $[0,1{]}^{\mathbb{N}}$. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential $A$. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities ${\mu }_{\beta }$ which weakly converges to ${\mu }_{\infty }$. The probabilities ${\mu }_{\beta }$ are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure. Under the hypothesis of $A$ being $C^2$ and the twist condition, that is, $\frac{{\partial }^{2}A}{{\partial }_x{\partial }_y}(x,y)\ne 0$, for all $(x,y)\in [0,1{]}^2$, we show the graph property.

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Cited by in corpus (6)

• Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature
• The Analyticity of a Generalized Ruelle's Operator
• 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
• Piecewise Analytic Subactions for Analytic Dynamics
• Ergodic Transport Theory and Piecewise Analytic Subactions for Analytic Dynamics