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

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

2002-03-13

arxiv.org/abs/nlin/0203022v1

Nonlinear Sciences

Chaotic Dynamics

Submitted to Physica D

Scientific paper

10.1016/S0167-2789(02)00622-X

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.

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