Piecewise linear functions with almost all points eventually periodic

Abstract:

Let $f:[0,1] \to [0,1]$ be continuous, and let ${f^p}$ denote the pth iterate of /. Li and Yorke [2] proved that if there is a point $x \in [0,1]$ such that ${f^3}(x) = x$ but $f(x) \ne x$, then f is chaotic in the sense that f has periodic points of arbitrarily large period, and uncountably many points which are not even asymptotically periodic. But this chaos can be measure theoretically trivial. For each $p \geqslant 3$ we construct a continuous, piecewise linear function $f:[0,1] \to [0,1]$ such that f is chaotic, but almost every point of $[0,1]$ has eventual period p. The condition “eventual period p” cannot be replaced by “period p". We prove that if ${f^p}(x) = x$ for almost all $x \in [0,1]$, then ${f^2}(x) = x$ for all $x \in [0,1]$. Moreover, we describe a normal form for all such “square roots of the identity."