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

Authors:
B. Schweizer and J. SmĂtal

Journal:
Trans. Amer. Math. Soc. **344** (1994), 737-754

MSC:
Primary 58F13; Secondary 54H20, 58F08

DOI:
https://doi.org/10.1090/S0002-9947-1994-1227094-X

MathSciNet review:
1227094

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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 $F \in \Sigma (f)$.

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© Copyright 1994
American Mathematical Society