Devaney’s chaos implies existence of 𝑠-scrambled sets

Abstract

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