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 $f:\mathcal{X}⟶\mathcal{X}$ be a continuous weakly mixing map. Let $\beta =\sup \{\rho (x,y):\{x,y\}\subset \mathcal{X}\}$. 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 $\{F_n^m((a_1,a_2,\cdots ,a_n)):m\in \mathcal{M}\}$ is dense in $\mathcal{X}\times \mathcal{X}\times \cdots \times \mathcal{X}$ ($n$ terms), where $F_n((a_1,a_2,\cdots ,a_n))=(f(a_1),f(a_2),\cdots ,f(a_n))$; (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 $\hat{\mathbb{S}}={\bigcup }_{i=0}^{\infty }{f}^i(\mathbb{S})$, $\{x,c\}$ is a ($\beta /2$)-scrambled set of $f$. Furthermore, if $f$ has a fixed point and $\delta ={\inf }_{n\ge 1}\{\sup \{\rho (f^n(x),x):x\in \mathcal{X}\}\}\ge 0$, then the above Cantor sets ${\mathcal{S}}^{(1)},{\mathcal{S}}^{(2)},\cdots$ can be chosen to satisfy the additional property that $\hat{\mathbb{S}}={\bigcup }_{i=0}^{\infty }{f}^i(\mathbb{S})$ 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.

24 pages. arXiv admin note: text overlap with arXiv:1701.02589

References in corpus (3)

• General definitions of chaos for continuous and discrete-time processes
• About Stable Periodic Helixes, L-iteration and Chaos Generated by Unbounded Functions
• Continuous Transitive Maps on the Interval Revisited

Cited by in corpus (1)

• Continuous Transitive Maps on the Interval Revisited