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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Measures of chaos and a spectral decomposition of dynamical systems on the interval
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by B. Schweizer and J. Smítal PDF
Trans. Amer. Math. Soc. 344 (1994), 737-754 Request permission

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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Additional 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