On the Chaos in Continuous Weakly Mixing Maps

TL;DR

This paper demonstrates complex chaotic behavior in continuous weakly mixing maps on infinite locally compact spaces, constructing dense scrambled sets with specific divisibility and invariance properties, and introduces a new chaos notion.

Contribution

It provides new constructions of dense scrambled sets with divisibility and invariance properties in weakly mixing maps, extending understanding of chaos in such systems.

Findings

Existence of dense $eta$-scrambled sets for all iterates

Construction of invariant $eta$-scrambled sets under certain conditions

Stronger chaos results for mixing maps on $ ext{X}$

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.