Measures of chaos and a spectral decomposition of dynamical systems on the interval

Abstract:

Let $f:[0,1] \to [0,1]$ be continuous. For $x,y \in [0,1]$, the upper and lower (distance) distribution functions, $F_{xy}^\ast$ and ${F_{xy}}$, are defined for any $t \geq 0$ as the lim sup and lim inf as $n \to \infty$ of the average number of times that the distance $|{f^i}(x) - {f^i}(y)|$ between the trajectories of x and y is less than t during the first n iterations. The spectrum of f is the system $\Sigma (f)$ of lower distribution functions which is characterized by the following properties: (1) The elements of $\Sigma (f)$ are mutually incomparable; (2) for any $F \in \Sigma (f)$, there is a perfect set ${P_F} \ne \emptyset$ such that ${F_{uv}} = F$ and $F_{uv}^\ast \equiv 1$ for any distinct u, $v \in {P_F}$; (3) if S is a scrambled set for f, then there are F, G in $\Sigma (f)$ and a decomposition $S = {S_F} \cup {S_G}$ (${S_G}$ may be empty) such that ${F_{uv}} \geq F$ if u, $v \in {S_F}$ and ${F_{uv}} \geq G$ if u, $v \in {S_G}$. Our principal results are: (1) If f has positive topological entropy, then $\Sigma (f)$ is nonempty and finite, and any $F \in \Sigma (f)$ is zero on an interval $[0,\varepsilon ]$, where $\varepsilon > 0$ (and hence any ${P_F}$ is a scrambled set in the sense of Li and Yorke). (2) If f has zero topological entropy, then $\Sigma (f) = \{ F\}$ where $F \equiv 1$. It follows that the spectrum of f provides a measure of the degree of chaos of f. In addition, a useful numerical measure is the largest of the numbers $\int _0^1 {(1 - F(t))dt}$, where $F \in \Sigma (f)$.