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 α = (α 0 α 1 . . . α 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 α approximately equal to the Parry measure corresponding to α, µ P (α). When the hole encompasses a point of a short periodic orbit, the entropy difference is significantly smaller than µ P (α). This discrepancy is described by the formula which relates the length of the short periodic orbit, the Parry measure µ P (α), and the topological entropies of the two chaotic sets. 05.45.+b,05.45.Ac, 05.45.Vx

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