Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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)$.

References [Enhancements On Off] (What's this?)

  • [1] L. S. Block and W. A. Coppel, One-dimensional dynamics, Lecture Notes in Math., vol. 1513, Springer-Verlag, Berlin, 1992. MR 1176513 (93g:58091)
  • [2] R. Bowen, Topological entropy and Axiom A, Global Analysis, Proc. Sympos. Pure Math., vol. 14, Providence, R.I., 1970, pp. 23-41. MR 0262459 (41:7066)
  • [3] P. Collet and J.-P. Eckmann, Iterated maps on the interval as dynamical systems, Birkhäuser, Boston, Mass, 1980. MR 2541754
  • [4] V. V. Fedorenko, A. N. Sharkovsky, and J. Smítal, Characterizations of weakly chaotic maps of the interval, Proc. Amer. Math. Soc. 110 (1990), 141-148. MR 1017846 (91a:58148)
  • [5] N. Franzová and J. Smítal, Positive sequence topological entropy characterizes chaos, Proc. Amer. Math. Soc. 112 (1991), 1083-1086. MR 1062387 (91j:58107)
  • [6] Ch. K. Kenzhegulov and A. N. Sharkovsky, On properties of the set of limit points of an iterated sequence of a continuous function, Volž. Mat. Sb. 3 (1965), 343-348. (Russian) MR 0199316 (33:7464)
  • [7] M. Kuchta and J. Smítal, Two point scrambled set implies chaos, Proc. Europ. Conf. on Iteration Theory, Spain 1987, World Science, Singapore, 1989, pp. 427-430. MR 1085314 (91j:58112)
  • [8] C. Kuratowski, Topologie, Vol. I, Polish Sci. Publ., Warsaw, 1958. MR 0090795 (19:873d)
  • [9] T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992. MR 0385028 (52:5898)
  • [10] M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Math. 27 (1979), 167-169. MR 542778 (81b:58033)
  • [11] M. Misiurewicz and J. Smítal, Smooth chaotic maps with zero topological entropy, Ergodic Theory and Dynamical Systems 8 (1988), 421-424. MR 961740 (90a:58118)
  • [12] Z. Nitecki, Topological dynamics on the interval, Ergodic Theory and Dynamical Systems. II, Proceedings (A. Katok, ed.), Birkhäuser, Boston, Mass, 1982, pp. 1-73. MR 670074 (84g:54051)
  • [13] D. Preiss and J. Smítal, A characterization of non-chaotic maps of the interval stable under small perturbations, Trans. Amer. Math. Soc. 313 (1989), 687-696. MR 997677 (90f:58100)
  • [14] C. Preston, Iterates of piecewise monotone mappings on an interval, Lecture Notes in Math., vol. 1347, Springer-Verlag, Berlin, 1988. MR 969131 (89m:58109)
  • [15] A. N. Sharkovsky, The partially ordered system of attracting sets, Soviet Math. Dokl. 7 (1966), 1384-1386.
  • [16] -, The behavior of a map in a neighborhood of an attracting set, Ukrain. Mat. Ž. 18 (1966), 60--83. (Russian)
  • [17] -, Continuous mapping on the set of $ \omega $-limit sets of iterated sequences, Ukrain. Mat. Ž. 18 (1966), 127-130. (Russian)
  • [18] B. Schweizer and A. Sklar, Probabilistic metric spaces, North-Holland, New York, 1983. MR 790314 (86g:54045)
  • [19] J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), 269-281. MR 849479 (87m:58107)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F13, 54H20, 58F08

Retrieve articles in all journals with MSC: 58F13, 54H20, 58F08

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society