Negative entropy, zero temperature and Markov chains on the interval

Abstract

We consider ergodic optimization for the shift map on the modified Bernoulli space σ: [0, 1]^ℕ → [0, 1]^ℕ, where [0, 1] is the unit closed interval, and the potential A: [0, 1]^ℕ → ℝ considered depends on the two first coordinates of [0, 1]^ℕ. We are interested in finding stationary Markov probabilities µ_∞ on [0, 1]^ℕ that maximize the value ∫ Adµ, among all stationary (i.e. σ-invariant) probabilities µ on [0, 1]^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_∞. 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 in Thermodynamic Formalism. As the potential A depends only on the first two coordinates, instead of the probability µ on [0, 1]^ℕ, we can consider its projection ν on [0, 1]². We look at the problem in both ways. If µ_∞ is the maximizing probability on [0, 1]^ℕ, we also have that its projection ν _∞ is maximizing for A. The hypothesis about stationarity on the maximization problem can also be seen as a transhipment problem. 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 of the maximizing probability ν on [0, 1]². Moreover, the graph is monotonous. An important result we get is: the maximizing probability is unique generically in Mañé’s sense. Finally, we exhibit a separating sub-action for A.