A construction of multiply Xiong chaotic set with full topological entropy for \(\beta\)-transformation
Given a topological dynamical system \((X, f)\), a subset \(E\subset X\) containing at least two points is defined to be a Xiong chaotic set if for any subset \(F\subset E\) and any continuous function \(g: F \rightarrow X\), there exists an increasing sequence \(\{q_{k}\}^{\infty}_{k=1}\) of positive integers such that for every \(x\in F\), \(\lim\limits_{k\rightarrow +\infty}f^{q_k}(x)=g(x)\). Two significant results here obtained are as follows. (1) For positively expansive systems with specification property, the authors construct a multiply Xiong chaotic set with full topological entropy everywhere; (2) Although a \(\beta\)-transformation is neither continuous nor satisfies the specification property for all \(\beta>1\), it is possible to construct a Xiong chaotic set with full topological entropy.
scientific article; zbMATH DE number 7374583
scientific article