An example of unbounded chaos

Abstract

Let $\phi(x) = |1 - \frac 1x|$ for all $x > 0$. Then we extend $\phi(x)$ in the usual way to become a continuous map from the compact topological (but not metric) space $[0, \infty]$ onto itself which also maps the set of irrational points in $(0, \infty)$ onto itself. In this note, we show that (1) on $[0, \infty]$, $\phi(x)$ is topologically mixing, has dense irrational periodic points, and has topological entropy $\log \lambda$, where $\lambda$ is the unique positive zero of the polynomial $x^3 - 2x -1$; (2) $\phi(x)$ has bounded uncountable {\it invariant} 2-scrambled sets of irrational points in $(0, 3)$; (3) for any countably infinite set $X$ of points (rational or irrational) in $(0, \infty)$, there exists a dense unbounded uncountable {\it invariant} $\infty$-scrambled set $Y$ of irrational transitive points in $(0, \infty)$ such that, for any $x \in X$ and any $y \in Y$, we have $\limsup_{n \to \infty} |\phi^n(x) - \phi^n(y)| = \infty$ and $\liminf_{n \to \infty} |\phi^n(x) - \phi^n(y)| = 0$. This demonstrates the true nature of chaos for $\phi(x)$.