Hausdorff dimensions of zero-entropy sets of dynamical systems with positive entropy

摘要

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)))over bar> = 0, where Orb(+)(x) = {T-n(x) vertical bar n is an element of Z(+)} is the forward orbit of x under T and <(Orb(+)(x))over bar> is the closure. Let epsilon(0) (X, T) denote the set of all zero-entropy points. Naturally, one would like to ask the following important question: How big is epsilon(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 epsilon(0)(X, T) vanishes.