Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

• Autorzy: Sergiĭ Kolyada , Michał Misiurewicz , L’ubomír Snoha
• Języki publikacji: EN

Abstrakty

EN

The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces $(X_i)^∞_{i = 1}$ and a sequence of continuous maps $(f_i)^∞_{i = 1}$, $f_i : X_i → X_{i+1}$, 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 $2^∞$ and positive topological entropy.

Słowa kluczowe

EN

nonautonomous dynamical system; topological entropy; triangular maps; piecewise monotone maps; $C^∞$ maps

Kategorie tematyczne

• 54H20: Topological dynamics

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1999
• Tom: 160
• Numer: 2
• Strony: 161-181

Daty

wydano

1999

otrzymano

1998-11-11

poprawiono

1999-01-27

Twórcy

autor

Sergiĭ Kolyada

• Institute of Mathematics, Ukrainian Academy of Sciences, Tereshchenkivs'ka 3, 252601 Kiev, Ukraine

autor

Michał Misiurewicz

• Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216, U.S.A.

autor

L’ubomír Snoha

• Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia

Bibliografia

• [AKM] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319.
• [ALM] L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Sci., Singapore, 1993.
• [BEL] F. Balibrea, F. Esquembre and A. Linero, Smooth triangular maps of type $2^∞$ with positive topological entropy, Internat. J. Bifur. Chaos 5 (1995), 1319-1324.
• [BC] L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer, Berlin, 1992.
• [B] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414.
• [CE] P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Progr. in Phys. 1, Birkhäuser, Boston, 1980.
• [D] E. I. Dinaburg, Connection between various entropy characterizations of dynamical systems, Izv. Akad. Nauk SSSR 35 (1971), 324-366 (in Russian).
• [G] M. Gromov, Entropy, homology and semialgebraic geometry (after Y. Yomdin), Astérisque (Séminaire Bourbaki, 1985-86, no. 663) 145-146 (1987), 225-240.
• [Ka] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. IHES 51 (1980), 137-174.
• [Kl] P. E. Kloeden, On Sharkovsky's cycle coexistence ordering, Bull. Austral. Math. Soc. 20 (1979), 171-177.
• [Ko] S. F. Kolyada, On dynamics of triangular maps of the square, Ergodic Theory Dynam. Systems 12 (1992), 749-768.
• [KS] S. Kolyada and L'. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996), 205-233.
• [dMvS] W. de Melo and S. van Strien, One-Dimensional Dynamics, Series of Modern Surveys in Math., Springer, Berlin, 1993.
• [M] M. Misiurewicz, Attracting set of positive measure for a $C^∞$ map of an interval, Ergodic Theory Dynam. Systems 2 (1982), 405-415.
• [MS] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), 45-63.
• [Y] Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), 285-300.