On Large Deviations for Some Dynamical Systems and for Gibbs States at Zero Temperature

Abstract

In this article we analyze two issues related with large deviations in dynamical systems: 1. We show that the level-2 large deviation principle established by Comman and Rivera-Letelier[1], is satisfied by maps with a specification property and, with some additional condition, also by those with the almost property product, which is weaker than specification. The earlier mentioned authors proved that their principle, which is a generalization of previous results by Kifer, is verified by a class of hyperbolic rational maps. 2. In a previous article[5] we have considered a family of Gibbs states {μq } q≥1, which had an accumulation point μ∞ (zero temperature limit since the interpretation of q as the inverse of the temperature). We proved that μ∞ is a maximizing measure for more general systems than symbolics. In a recent article by Lopes and Mengue[4] were considered similar families of states and proved, for symbolic dynamics, that an accumulation point of the family was maximizing. Also they established a large deviation principle. In this note we show how to use the results of our previous work to describe a large deviation process in a more general context than[4].