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

Authors: B. Schweizer and J. Smítal
Journal: Trans. Amer. Math. Soc. 344 (1994), 737-754
MSC: Primary 58F13; Secondary 54H20, 58F08
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 $ \vert{f^i}(x) - {f^i}(y)\vert$ 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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