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

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dc.creatorLopes, Artur O.
dc.creatorMohr, Joana
dc.creatorSouza, Rafael R.
dc.creatorThieullen, Philippe
dc.date2008-06-05
dc.date2009-01-06
dc.date.accessioned2026-07-07T12:24:17Z
dc.date.available2026-07-07T12:24:17Z
dc.descriptionWe 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 $μ_\infty$ on $ [0,1]^\mathbb{N}$ that maximize the value $ \int A d μ,$ among all stationary Markov probabilities $μ$ 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 $μ_β$ which weakly converges to $μ_\infty$. The probabilities $μ_β$ 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, $\frac{\partial^2 A}{\partial_x \partial_y} (x,y) \neq 0$, for all $(x,y) \in [0,1]^2$, we show the graph property.
dc.identifierhttps://arxiv.org/abs/0806.1012
dc.identifierhttp://arxiv.org/abs/0806.1012
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/214275
dc.subjectDynamical Systems
dc.subjectProbability
dc.subject37A35; 37A50; 37A60; 60J10
dc.titleNegative Entropy, Zero temperature and stationary Markov Chains on the interval
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