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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 .

**[1]**L. S. Block and W. A. Coppel,*Dynamics in one dimension*, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992. MR**1176513****[2]**Rufus Bowen,*Topological entropy and axiom 𝐴*, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 23–41. MR**0262459****[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. MR**2541754****[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. MR**1017846**, https://doi.org/10.1090/S0002-9939-1990-1017846-5**[5]**N. Franzová and J. Smítal,*Positive sequence topological entropy characterizes chaotic maps*, Proc. Amer. Math. Soc.**112**(1991), no. 4, 1083–1086. MR**1062387**, https://doi.org/10.1090/S0002-9939-1991-1062387-3**[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). MR**0199316****[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. MR**1085314****[8]**Casimir Kuratowski,*Topologie. Vol. I*, Monografie Matematyczne, Tom 20, Państwowe Wydawnictwo Naukowe, Warsaw, 1958 (French). 4ème éd. MR**0090795****[9]**T. Y. Li and James A. Yorke,*Period three implies chaos*, Amer. Math. Monthly**82**(1975), no. 10, 985–992. MR**385028**, https://doi.org/10.2307/2318254**[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). MR**542778****[11]**M. Misiurewicz and J. Smítal,*Smooth chaotic maps with zero topological entropy*, Ergodic Theory Dynam. Systems**8**(1988), no. 3, 421–424. MR**961740**, https://doi.org/10.1017/S0143385700004557**[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. MR**670074****[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. MR**997677**, https://doi.org/10.1090/S0002-9947-1989-0997677-9**[14]**Chris Preston,*Iterates of piecewise monotone mappings on an interval*, Lecture Notes in Mathematics, vol. 1347, Springer-Verlag, Berlin, 1988. MR**969131****[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*-*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. MR**790314****[19]**J. Smítal,*Chaotic functions with zero topological entropy*, Trans. Amer. Math. Soc.**297**(1986), no. 1, 269–282. MR**849479**, https://doi.org/10.1090/S0002-9947-1986-0849479-9

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

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

Article copyright:
© Copyright 1994
American Mathematical Society