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

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 $h(T;\overline{\hbox{Orb}_+(x)}) = 0$, where $\hbox{Orb}_+(x) = \{T^n(x) | n \in \mathbb{Z}_+\}$ is the forward orbit of x under T and Orb₊(x) is the closure. Let ε⁰(X, T) denote the set of all zero-entropy points. Naturally, one would like to ask the following important question:

How big is ε⁰(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 ε⁰(X, T) vanishes.