On the Chaos in Continuous Weakly Mixing Maps

Abstract

Let $\mathcal X$ be an infinite locally compact separable metric space with metric $\rho$ and let $f : \mathcal X \longrightarrow \mathcal X$ be a continuous weakly mixing map. Let $\beta = \sup \big\{ \rho(x, y): \{x, y \} \subset \mathcal X \big\}$. In this note, we show (Theorem 4) that, for any countably infinite set $\{x_1, x_2, \cdots\}$ of points in $\mathcal X$ with compact orbit closures $\overline{O_f(x_i)}$'s, there exist an infinite set $\mathcal M$ of positive integers and countably infinitely many pairwise disjoint Cantor sets ${\mathcal S}^{(1)}, {\mathcal S}^{(2)}, \cdots$ of totally transitive points of $f$ such that (1) for any integers $\ell \ge 1$ and $n \ge 1$, $\ell!$ divides all sufficiently large integers in $\mathcal M$ and for any distinct points $a_1, a_2, \cdots, a_n$ in ${\mathbb S} = \bigcup_{j=1}^\infty \, {\mathcal S}^{(j)}$, the set $\{ F_n^m\big((a_1, a_2, \cdots, a_n)\big): m \in \mathcal M \}$ is dense in $\mathcal X \times \mathcal X \times \cdots \times \mathcal X$ ($n$ terms), where $F_n\big((a_1, a_2, \cdots, a_n)\big) = \big(f(a_1), f(a_2), \cdots, f(a_n)\big)$; (2) ${\mathbb S}$ is a dense $\beta$-scrambled set of $f^n$ for all $n \ge 1$; (3) for any $x$ in $\{x_1, x_2, \cdots\}$ and any $c$ in $\widehat {\mathbb S} = \bigcup_{i=0}^\infty \, f^i({\mathbb S})$, $\{ x, c \}$ is a (${\beta}/2$)-scrambled set of $f$. Furthermore, if $f$ has a fixed point and $\delta = \inf_{n \ge 1} \big\{ \sup\{ \rho(f^n(x), x): x \in \mathcal X \} \big\} \ge 0$, then the above Cantor sets ${\mathcal S}^{(1)}, {\mathcal S}^{(2)}, \cdots$ can be chosen to satisfy the additional property that $\widehat {\mathbb S} = \bigcup_{i=0}^\infty f^i({\mathbb S})$ is a dense {\it invariant} $\delta$-scrambled set of $f^n$ for all $n \ge 1$. For continuous mixing maps on $\mathcal X$, we have a stronger result (Theorem 5). A notion of chaos is also introduced.