Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval

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June 2016, 36(6): 3435-3443. doi: 10.3934/dcds.2016.36.3435

Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval

Marta Štefánková ^1,

1.	Mathematical Institute, Silesian University, 746 01 Opava

Received April 2015 Revised September 2015 Published December 2015

Abstract
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We consider nonautonomous discrete dynamical systems $\{ f_n\}_{n\ge 1}$, where every $f_n$ is a surjective continuous map $[0,1]\to [0,1]$ such that $f_n$ converges uniformly to a map $f$. It is well-known that $f$ has positive topological entropy iff $\{ f_n\}_{n\ge 1}$ has. On the other hand, for systems with zero topological entropy, $\{ f_n\}_{n\ge 1}$ with very complex dynamics can converge even to the identity map. We study the following question: Which properties of the limit function $f$ are inherited by nonautonomous system $\{ f_n\}_{n\ge 1}$? We show that Li-Yorke chaos, distributional chaos DC1 and, for zero entropy maps, infinite $\omega$-limit sets are inherited by nonautonomous systems and, for zero entropy maps, we give a criterion on $f$ under which $\{ f_n\}_{n\ge 1}$ is DC1. More precisely, our main results are: (i) If $f$ is Li-Yorke chaotic then $ \{ f_n\}_{n\ge 1}$ is Li-Yorke chaotic as well, and the analogous implication is true for distributional chaos DC1; (ii) If $f$ has zero topological entropy then the nonautonomous system inherits its infinite $\omega$-limit sets; (iii) We introduce new notion of a quasi horseshoe, a generalization of horseshoe. It turns out that $\{f_n\}_{n\ge 1}$ exhibits distributional chaos DC1 if $f$ has a quasi horseshoe. The last result is true for maps defined on arbitrary compact metric spaces.

Keywords: distributional chaos, topological entropy., $\omega-$limit sets, Li-Yorke chaos, Nonautonomous dynamical systems.

Mathematics Subject Classification: Primary: 37B55, 37B40, 54H20; Secondary: 37B05, 37B2.

Citation: Marta Štefánková. Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3435-3443. doi: 10.3934/dcds.2016.36.3435

References:

[1]	A. M. Bruckner and J. Smítal, A characterization of $\omega$-limit sets of maps of the interval with zero topological entropy, Ergodic Theory & Dynam. Systems, 13 (1993), 7-19. doi: 10.1017/S0143385700007173.
[2]	J. Cánovas, Li-Yorke chaos in a class of nonautonomous discrete systems, J. Difference Equ. Appl., 17 (2011), 479-486. doi: 10.1080/10236190903049025.
[3]	T. Downarowicz, Positive entropy implies distributional chaos DC2, Proc. Amer. Math. Soc., 142 (2014), 137-149. doi: 10.1090/S0002-9939-2013-11717-X.
[4]	J. Dvořáková, Chaos in nonautonomous discrete dynamical systems, Commun. Nonlin. Sci. Numer. Simulat., 17 (2012), 4649-4652. doi: 10.1016/j.cnsns.2012.06.005.
[5]	V. V. Fedorenko, A. N. Šarkovskii and J. Smítal, Characterizations of weakly chaotic maps of the interval, Proc. Amer. Math. Soc., 110 (1990), 141-148. doi: 10.1090/S0002-9939-1990-1017846-5.
[6]	G.-L. Forti, L. Paganoni and J. Smítal, Dynamics of homeomorphisms on minimal sets generated by triangular mappings, Bull. Austral. Math. Soc., 59 (1999), 1-20. doi: 10.1017/S000497270003255X.
[7]	S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random & Comput. Dynamics, 4 (1996), 205-233.
[8]	M. Kuchta, Shadowing property of continuous maps with zero topological entropy, Proc. Amer. Math. Soc., 119 (1993), 641-648. doi: 10.1090/S0002-9939-1993-1165058-X.
[9]	M. Misiurewicz and J. Smítal, Smooth chaotic maps with zero topological entropy, Ergodic Theory & Dynam. Systems, 8 (1988), 421-424. doi: 10.1017/S0143385700004557.
[10]	A. N. Šarkovskii, Attracting sets containing no cycles, Ukrain. Mat. Ž., 20 (1968), 136-142.
[11]	B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754. doi: 10.1090/S0002-9947-1994-1227094-X.
[12]	J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269-282. doi: 10.1090/S0002-9947-1986-0849479-9.
[13]	J. Smítal and M. Štefánková, Distributional chaos for triangular maps, Chaos, Solitons and Fractals, 21 (2004), 1125-1128. doi: 10.1016/j.chaos.2003.12.105.

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References:

[1]	A. M. Bruckner and J. Smítal, A characterization of $\omega$-limit sets of maps of the interval with zero topological entropy, Ergodic Theory & Dynam. Systems, 13 (1993), 7-19. doi: 10.1017/S0143385700007173.
[2]	J. Cánovas, Li-Yorke chaos in a class of nonautonomous discrete systems, J. Difference Equ. Appl., 17 (2011), 479-486. doi: 10.1080/10236190903049025.
[3]	T. Downarowicz, Positive entropy implies distributional chaos DC2, Proc. Amer. Math. Soc., 142 (2014), 137-149. doi: 10.1090/S0002-9939-2013-11717-X.
[4]	J. Dvořáková, Chaos in nonautonomous discrete dynamical systems, Commun. Nonlin. Sci. Numer. Simulat., 17 (2012), 4649-4652. doi: 10.1016/j.cnsns.2012.06.005.
[5]	V. V. Fedorenko, A. N. Šarkovskii and J. Smítal, Characterizations of weakly chaotic maps of the interval, Proc. Amer. Math. Soc., 110 (1990), 141-148. doi: 10.1090/S0002-9939-1990-1017846-5.
[6]	G.-L. Forti, L. Paganoni and J. Smítal, Dynamics of homeomorphisms on minimal sets generated by triangular mappings, Bull. Austral. Math. Soc., 59 (1999), 1-20. doi: 10.1017/S000497270003255X.
[7]	S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random & Comput. Dynamics, 4 (1996), 205-233.
[8]	M. Kuchta, Shadowing property of continuous maps with zero topological entropy, Proc. Amer. Math. Soc., 119 (1993), 641-648. doi: 10.1090/S0002-9939-1993-1165058-X.
[9]	M. Misiurewicz and J. Smítal, Smooth chaotic maps with zero topological entropy, Ergodic Theory & Dynam. Systems, 8 (1988), 421-424. doi: 10.1017/S0143385700004557.
[10]	A. N. Šarkovskii, Attracting sets containing no cycles, Ukrain. Mat. Ž., 20 (1968), 136-142.
[11]	B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754. doi: 10.1090/S0002-9947-1994-1227094-X.
[12]	J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269-282. doi: 10.1090/S0002-9947-1986-0849479-9.
[13]	J. Smítal and M. Štefánková, Distributional chaos for triangular maps, Chaos, Solitons and Fractals, 21 (2004), 1125-1128. doi: 10.1016/j.chaos.2003.12.105.

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