An example of unbounded chaos

TL;DR

This paper studies a continuous extension of a specific function on a compact space, demonstrating complex chaotic behavior including topological mixing, dense irrational periodic points, positive entropy, and uncountable scrambled sets, revealing the true nature of chaos.

Contribution

It introduces a continuous extension of a function exhibiting unbounded chaos, proving its topological mixing, entropy, and existence of invariant scrambled sets, thus deepening understanding of chaotic dynamics.

Findings

The map is topologically mixing with positive entropy.

It has dense irrational periodic points.

It contains uncountable invariant scrambled sets.

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 $\phi(x)$.