On the nature of chaos

Abstract

Based on newly discovered properties of the shift map (Theorem 1), we believe that chaos should involve not only nearby points can diverge apart but also faraway points can get close to each other. Therefore, we propose to call a continuous map $f$ from an infinite compact metric space $(X, d)$ to itself chaotic if there exists a positive number $δ$ such that for any point $x$ and any nonempty open set $V$ (not necessarily an open neighborhood of $x$) in $X$ there is a point $y$ in $V$ such that $\limsup_{n \to \infty} d(f^n(x), f^n(y)) \ge δ$ and $\liminf_{n \to \infty} d(f^n(x), f^n(y)) = 0$.