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 $$ \mathbf D^{x_0}_\alpha :=\Big\{x\in X: \lim\limits_{\epsilon\to 0}\limsup\limits_{n\to\infty}\dfrac{\max\{t\in\mathbb N:~T^n(x)\in B_{t}( x_0,\epsilon)\}}{n}\geq \alpha\Big\}.$$ Then we have $h_{top}^{B} ( T, \mathbf D^{x_0}_\alpha )= \dfrac{h_{top}(T)}{1+\alpha} $, where $h_{top}^{B} ( T, \mathbf D^{x_0}_\alpha )$ denotes the Bowen topological entropy of $\mathbf D^{x_0}_\alpha.$