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

TL;DR

This paper investigates the Bowen topological entropy of a set characterized by a specific shadowing time condition in a topological dynamical system with the specification property, deriving a precise entropy formula.

Contribution

It provides a new explicit formula for the Bowen topological entropy of sets defined by shadowing time constraints in systems with the specification property.

Findings

The Bowen topological entropy of the set is exactly h_{top}(T)/(1+α).

The result applies to systems with the specification property.

It links shadowing time behavior to entropy in dynamical systems.

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