Scrambled sets on compact metric spaces
Let (X,d) be a compact metric space and C(X) be the space of all continuous functions f:X\(\to X\). For \(f\in C(X)\), an uncountable subset S of X is said to be a scrambled set of (X,d,f) iff for all x,y\(\in S\), \(x\neq y\), and for some \(\delta >0\) \[ \lim_{n\to \infty} \inf d(f^ n(x),f^ n(y))=0,\quad \lim_{n\to \infty} \sup d(f^ n(x),f^ n(y))>\delta. \] If \(f\in C(X)\) has a point of period \(k\cdot 2^ m\) for some odd integer \(k\geq 3\) and for some positive integer m, then there exists a scrambled set of (I,d,f). The set of functions in C(I) which possess a scrambled set contains an open dense subset of C(I).
scientific article; zbMATH DE number 3942488