Parry measure and the topological entropy of chaotic repellers embedded within chaotic attractors

Abstract

We study the topological entropy of chaotic repellers formed by those points in a given chaotic attractor that never visit some small forbidden hole-region in the phase space. The hole is a set of points in the phase space that have a sequence $\alpha=(\alpha_0\alpha_1...\alpha_{l-1})$ as the first $l$ letters in their itineraries. We point out that the difference between the topological entropies of the attractor and the embedded repeller is for most choices of $\alpha$ approximately equal to the Parry measure corresponding to $\alpha$, $\mu_P(\alpha)$. When the hole encompasses a point of a short periodic orbit, the entropy difference is significantly smaller than $\mu_P(\alpha)$. This discrepancy is described by the formula which relates the length of the short periodic orbit, the Parry measure $\mu_P(\alpha)$, and the topological entropies of the two chaotic sets.