Devaney’s chaos implies existence of $s$-scrambled sets

Abstract:

Let $X$ be a complete metric space without isolated points, and let $f:X\to X$ be a continuous map. In this paper we prove that if $f$ is transitive and has a periodic point of period $p$, then $f$ has a scrambled set $S=\bigcup _{n=1}^{\infty }C_{n}$ consisting of transitive points such that each $C_{n}$ is a synchronously proximal Cantor set, and $\bigcup _{i=0}^{p-1}f^{i}(S)$ is dense in $X$. Furthermore, if $f$ is sensitive (for example, if $f$ is chaotic in the sense of Devaney), with $2s$ being a sensitivity constant, then this $S$ is an $s$-scrambled set.