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

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- by B. Schweizer and J. Smítal
- Trans. Amer. Math. Soc.
**344**(1994), 737-754 - DOI: https://doi.org/10.1090/S0002-9947-1994-1227094-X
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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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## Bibliographic Information

- © Copyright 1994 American Mathematical Society
- 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