Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

Kolyada, Sergiĭ; Misiurewicz, Michał; Snoha, L’ubomír

Kolyada, Sergiĭ ; Misiurewicz, Michał ; Snoha, L’ubomír

Fundamenta Mathematicae, Tome 159 (1999), p. 161-181 / Harvested from The Polish Digital Mathematics Library

Résumé

The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces ${(X_{i})}_{i = 1}^{\infty}$ and a sequence of continuous maps ${(f_{i})}_{i = 1}^{\infty}$ , $f_{i} : X_{i} \to X_{i + 1}$ , is defined. If all the spaces are compact real intervals and all the maps are piecewise monotone then, under some additional assumptions, a formula for the entropy of the system is obtained in terms of the number of pieces of monotonicity of $f_{n} ○ . . . ○ f_{2} ○ f_{1}$ . As an application we construct a large class of smooth triangular maps of the square of type $2^{\infty}$ and positive topological entropy.

Publié le : 1999-01-01

Zbl 0936.37004

EUDML-ID : urn:eudml:doc:212386

@article{bwmeta1.element.bwnjournal-article-fmv160i2p161bwm,
     author = {Sergi\u\i\ Kolyada and Micha\l\ Misiurewicz and L'ubom\'\i r Snoha},
     title = {Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval},
     journal = {Fundamenta Mathematicae},
     volume = {159},
     year = {1999},
     pages = {161-181},
     zbl = {0936.37004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv160i2p161bwm}
}

Kolyada, Sergiĭ; Misiurewicz, Michał; Snoha, L’ubomír. Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval. Fundamenta Mathematicae, Tome 159 (1999) pp. 161-181. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv160i2p161bwm/

Bibliographie

[00000] [AKM] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. | Zbl 0127.13102

[00001] [ALM] L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Sci., Singapore, 1993. | Zbl 0843.58034

[00002] [BEL] F. Balibrea, F. Esquembre and A. Linero, Smooth triangular maps of type $2^{\infty}$ with positive topological entropy, Internat. J. Bifur. Chaos 5 (1995), 1319-1324. | Zbl 0886.58044

[00003] [BC] L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer, Berlin, 1992.

[00004] [B] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414. | Zbl 0212.29201

[00005] [CE] P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Progr. in Phys. 1, Birkhäuser, Boston, 1980.

[00006] [D] E. I. Dinaburg, Connection between various entropy characterizations of dynamical systems, Izv. Akad. Nauk SSSR 35 (1971), 324-366 (in Russian).

[00007] [G] M. Gromov, Entropy, homology and semialgebraic geometry (after Y. Yomdin), Astérisque (Séminaire Bourbaki, 1985-86, no. 663) 145-146 (1987), 225-240.

[00008] [Ka] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. IHES 51 (1980), 137-174. | Zbl 0445.58015

[00009] [Kl] P. E. Kloeden, On Sharkovsky's cycle coexistence ordering, Bull. Austral. Math. Soc. 20 (1979), 171-177. | Zbl 0465.58022

[00010] [Ko] S. F. Kolyada, On dynamics of triangular maps of the square, Ergodic Theory Dynam. Systems 12 (1992), 749-768. | Zbl 0784.58038

[00011] [KS] S. Kolyada and L'. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996), 205-233. | Zbl 0909.54012

[00012] [dMvS] W. de Melo and S. van Strien, One-Dimensional Dynamics, Series of Modern Surveys in Math., Springer, Berlin, 1993. | Zbl 0791.58003

[00013] [M] M. Misiurewicz, Attracting set of positive measure for a $C^{\infty}$ map of an interval, Ergodic Theory Dynam. Systems 2 (1982), 405-415. | Zbl 0522.58032

[00014] [MS] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), 45-63. | Zbl 0445.54007

[00015] [Y] Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), 285-300.