Full Topological entropy of Xiong chaotic sets beyond the specification property

Abstract

In this paper, we show that for a shift dynamical system $(X_{\mathcal{L}}, σ)$ with a weakened form of the specification property, there exists a Xiong chaotic set $C$ of $X_{\mathcal{L}}$ with full topological entropy everywhere (i.e. the intersection of $C$ and arbitrary non-empty open subset of $X_{\mathcal{L}}$ has full topological entropy). Moreover, $C$ satisfies that for any non-empty subset $A$ and any continuous map $F: A\rightarrow X_{\mathcal{L}},$ there exists an increasing sequence $\{p_{k}\}_{k=1}^{+\infty}$ of positive integers such that $\lim\limits_{k\rightarrow +\infty}σ^{p_{k}}(x)=F(x)$ holds for any $x\in A.$