The Rényi entropy function and the large deviation of short return times

Abstract

We consider the Rényi entropy function for weakly ψ-mixing systems. The first main result proves existence and regularity properties. The second main result of the paper is to get the decay rate for the large deviation of the return time to cylinder sets. We show it to be exponential with a rate given by the Rényi entropy function. Finally we also obtain bounds for the free energy.

Speed of convergence for laws of rare events and escape rates

“…So, if γ decays exponentially fast, then the fourth term should decay as γ (R ε ). The quantity R ε has been studied in [8,4,6,27,42], for example, and we generally expect it to behave like − log(P(A ε )), which means that the fourth term should also decay like a power of P(A ε ). As a consequence the behaviour of Γ ε should be ruled by a trade-off between the first, the third and the fourth terms of Γ ε .…”

Rényi Entropies and Large Deviations for the First Match Function

“…This produces a more natural symmetry around the vertical axis. This is the case of some works in the literature including [7], [22], [29]. 2 Through this paper logarithms will be taken in base e.…”

“…Ruelle's book [38]) to get the existence and the smoothness property of the function H R (β) [12]. Recently its existence was proved in [22] for weakly ψ-mixing processes. In general, its existence is not known.…”

“…Luczak and Szpankowski [30] shows that the Rényi entropies exists for β > 0 under the ψ-mixing condition plus the constraint ψ(0) < 1. Haydn and Vaienti [22] in a recent paper proved the large deviation for the first return and the existence of the Rényi entropy for a class of system they introduce, called weakly ψ-mixing processes. These results do not cover our case since, for instance, all such processes enjoy the property of exponential decay of the measure of all the n-cylinders, while our hypothesis allows to have cylinders with polynomial decay as we will see in the examples.…”

“…We provide several examples that show its generality. Different from previous papers [7], [22] that assume the (uniform) exponential decay of the measure of any n-string, our condition allows us to consider processes that admit strings with sub-exponential or even algebraic measures. We present some examples that verify our condition but do not verify previous conditions used in the literature.…”