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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random & Comput. Dynamics, 4 (1996), 205-233.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. N. Šarkovskii, Attracting sets containing no cycles, Ukrain. Mat. Ž., 20 (1968), 136-142.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random & Comput. Dynamics, 4 (1996), 205-233.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. N. Šarkovskii, Attracting sets containing no cycles, Ukrain. Mat. Ž., 20 (1968), 136-142.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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