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

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=(α0α1⋯αl−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, μP(α). When the hole encompasses a point of a short periodic orbit, the entropy difference is significantly smaller than μP(α). This discrepancy is described by the formula which relates the length of the short periodic orbit, the Parry measure μP(α), and the topological entropies of the two chaotic sets.