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

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

MathSciNet review:
1227094

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Abstract: Let be continuous. For , the upper and lower (distance) distribution functions, and , are defined for any as the lim sup and lim inf as of the average number of times that the distance between the trajectories of *x* and *y* is less than *t* during the first *n* iterations. The spectrum of *f* is the system of lower distribution functions which is characterized by the following properties: (1) The elements of are mutually incomparable; (2) for any , there is a perfect set such that and for any distinct *u*, ; (3) if *S* is a scrambled set for *f*, then there are *F*, *G* in and a decomposition ( may be empty) such that if *u*, and if *u*, . Our principal results are: (1) If *f* has positive topological entropy, then is nonempty and finite, and any is zero on an interval , where (and hence any is a scrambled set in the sense of Li and Yorke). (2) If *f* has zero topological entropy, then where .

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 , where .

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

Article copyright:
© Copyright 1994
American Mathematical Society