  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 $\mathbf D^{x_0}_\alpha.$
