Mixing sets, positive entropy homeomorphisms and non-Suslinean continua

Abstract

Let $\{h_{n}\}_{n\in \mathbb{N}}$ be a sequence of self maps on a metric space $X$. We say that $Q\subset X$ is a mixing set on $\{h_{n}\}_{n\in \mathbb{N}}$ if for every $V\subset Q$ such that $\text{int}_{Q}(V)\not =\emptyset$ and every $\unicode[STIX]{x1D716}>0$ there exists $N=N(V,\unicode[STIX]{x1D716})$ such that $\text{d}_{H}(Q,h_{n}(V))<\unicode[STIX]{x1D716}$ for all $n\geq N$, where $\text{d}_{H}$ is the Hausdorff metric. It is shown that if $Q$ is a non-degenerate mixing set for a sequence of homeomorphisms on a continuum, then the continuum must be non-Suslinean. This is generalized to the notion of a $\unicode[STIX]{x1D719}$-mixing set. As a corollary, it is shown that a continuum must be non-Suslinean in order to admit a positive entropy homeomorphism.