Measures of chaos and a spectral decomposition of dynamical systems on the interval. (English) Zbl 0812.58062

Various notions of chaos of dynamical systems on the interval have been proposed by different authors. Among them there is the definition of chaotic functions given by Li and Yorke and another notion based on the topological entropy. In this paper the authors use the upper and lower distance distribution functions associated to a continuous function on the interval to measure the degree of chaos of the function.
Let \(f\) be a continuous function on the interval. For \(x, y \in [0,1]\), the upper and lower distribution functions, \(F^ \ast_{xy}\) 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. Both functions are nondecreasing and may be viewed as cumulative probability distribution functions.
Among the set of all lower distribution functions associated to a given \(f\), the authors pick out a subset, \(\Sigma(f)\), called the spectrum of \(f\). The definition of the spectrum is based on initial results on dynamical systems concerning the notion of scrambled set and periodic decompositions. Thanks to a very simple example the authors are able to illustrate the ideas behind the use of this definition and to measure chaos.
The main results relate previous notions of chaos with the one proposed here: 1) If \(f\) has positive topological entropy, then the spectrum is nonempty and finite, and any element of the spectrum is zero on an interval \([0,\varepsilon]\) (thus relating the spectrum with the notion of chaos of Li and Yorke). 2) If \(f\) has zero topological entropy, then the spectrum is just the constant unit function.


37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37A30 Ergodic theorems, spectral theory, Markov operators
54C70 Entropy in general topology
26A18 Iteration of real functions in one variable
37B99 Topological dynamics
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[1] L. S. Block and W. A. Coppel, Dynamics in one dimension, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992.
[2] Rufus Bowen, Topological entropy and axiom \?, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 23 – 41.
[3] Pierre Collet and Jean-Pierre Eckmann, Iterated maps on the interval as dynamical systems, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2009. Reprint of the 1980 edition. · Zbl 1192.37003
[4] V. V. Fedorenko, A. N. Šarkovskii, and J. Smítal, Characterizations of weakly chaotic maps of the interval, Proc. Amer. Math. Soc. 110 (1990), no. 1, 141 – 148. · Zbl 0728.26008
[5] N. Franzová and J. Smítal, Positive sequence topological entropy characterizes chaotic maps, Proc. Amer. Math. Soc. 112 (1991), no. 4, 1083 – 1086. · Zbl 0735.26005
[6] A. N. Šarkovskiĭ and H. K. Kenžegulov, On properties of the set of limit points of an iterative sequence of a continuous function, Volž. Mat. Sb. Vyp. 3 (1965), 343 – 348 (Russian).
[7] M. Kuchta and J. Smítal, Two-point scrambled set implies chaos, European Conference on Iteration Theory (Caldes de Malavella, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 427 – 430.
[8] Casimir Kuratowski, Topologie. Vol. I, Monografie Matematyczne, Tom 20, Państwowe Wydawnictwo Naukowe, Warsaw, 1958 (French). 4ème éd. · Zbl 0078.14603
[9] T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985 – 992. · Zbl 0351.92021
[10] Michał Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 2, 167 – 169 (English, with Russian summary). · Zbl 0459.54031
[11] M. Misiurewicz and J. Smítal, Smooth chaotic maps with zero topological entropy, Ergodic Theory Dynam. Systems 8 (1988), no. 3, 421 – 424. · Zbl 0689.58028
[12] Zbigniew Nitecki, Topological dynamics on the interval, Ergodic theory and dynamical systems, II (College Park, Md., 1979/1980), Progr. Math., vol. 21, Birkhäuser, Boston, Mass., 1982, pp. 1 – 73. · Zbl 0461.58007
[13] D. Preiss and J. Smítal, A characterization of nonchaotic continuous maps of the interval stable under small perturbations, Trans. Amer. Math. Soc. 313 (1989), no. 2, 687 – 696. · Zbl 0698.58033
[14] Chris Preston, Iterates of piecewise monotone mappings on an interval, Lecture Notes in Mathematics, vol. 1347, Springer-Verlag, Berlin, 1988. · Zbl 0684.58002
[15] A. N. Sharkovsky, The partially ordered system of attracting sets, Soviet Math. Dokl. 7 (1966), 1384-1386. · Zbl 0167.51502
[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 Series in Probability and Applied Mathematics, North-Holland Publishing Co., New York, 1983. · Zbl 0546.60010
[19] J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), no. 1, 269 – 282. · Zbl 0639.54029
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