On the topological entropy of the set with a special shadowing time

Abstract

Let $(X,d,T)$ be a topological dynamical system with specification property. For $\alpha \in {\mathbb{R}}^{+}$ and any $x_0\in X$, define $${\mathbf{D}}_{\alpha }^{x_0}:=\{x\in X:\underset{\epsilon \to 0}{\lim }\underset{n\to \infty }{\mathrm{limsup}}\frac{\max \{t\in \mathbb{N}:T^n(x)\in B_t(x_0,\epsilon )\}}{n}\ge \alpha \}.$$ Then we have $h_{top}^B(T,{\mathbf{D}}_{\alpha }^{x_0})=\frac{h_{top}(T)}{1+\alpha }$, where $h_{top}^B(T,{\mathbf{D}}_{\alpha }^{x_0})$ denotes the Bowen topological entropy of ${\mathbf{D}}_{\alpha }^{x_0}.$

26 pages. arXiv admin note: text overlap with arXiv:0807.2123 by other authors