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.

Submitted to Physica D

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