Hausdorff Dimensions of Zero-Entropy Sets of Dynamical Systems with Positive Entropy

Abstract

Suppose that (X, T ) is a compact positive entropy dynamical system which we mean that X is a compact metric space and T : X → X is a continuous transformation of X and the topological entropy h(T ) > 0. A point x ∈ X is called a zero-entropy point provided h(T ; Orb+(x)) = 0, where Orb+(x) ={ T n (x) | n ∈ Z+} is the forward orbit of x under T and Orb+(x) is the closure. Let E 0 (X, T ) denote the set of all zero-entropy points. Naturally, one would like to ask the following important question: How big is E 0 (X, T ) for a dynamical system? In this paper, we answer this question. More precisely, we prove that if, furthermore, (X, T ) is locally expanding, then the Hausdorff dimension of E 0 (X, T ) vanishes.