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On a problem of iteration invariants for distributional chaos. (English) Zbl 1248.37024
Summary: We show that if f is a DC3 continuous map of a compact metric space then also f N is DC3, for every N>0. This solves a problem given by R. Li [ibid. 16, No. 4, 1993–1997 (2011; Zbl 1221.37017)].
MSC:
37C05Smooth mappings and diffeomorphisms
37B05Transformations and group actions with special properties
37D45Strange attractors, chaotic dynamics
References:
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[2]Hric, R.; Málek, M.: Omega-limit sets and distributional chaos on graphs, Topol appl 153, 2469-2475 (2006) · Zbl 1099.37011 · doi:10.1016/j.topol.2005.09.007
[3]Kurková, V.: The sharkovskys program for the classification of triangular maps is almost completed, Nonlinear anal 73, 1663-1669 (2010) · Zbl 1193.37003 · doi:10.1016/j.na.2010.04.075
[4]Oprocha, P.: Distributional chaos revisited, Trans amer math soc 361, 4901-4925 (2009) · Zbl 1179.37017 · doi:10.1090/S0002-9947-09-04810-7
[5]Oprocha, P.; Štefánková, M.: Specification property and distributional chaos almost everywhere, Proc amer math soc 135, 1931-1940 (2008)
[6]Paganoni, L.; Smítal, J.: Strange distributively chaotic triangular maps, Chaos soliton fract 26, 581-589 (2005) · Zbl 1081.37005 · doi:10.1016/j.chaos.2005.01.026
[7]Li, R.: A note on the three versions of distributional chaos, Commun nonlinear sci numer simulat 16, 1993-1997 (2011) · Zbl 1221.37017 · doi:10.1016/j.cnsns.2010.08.014
[8]Schweizer, B.; Smítal, J.: Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans amer math soc 344, 737-854 (1994) · Zbl 0812.58062 · doi:10.2307/2154504
[9]Smítal, J.; Štefánková, M.: Distributional chaos for triangular maps, Chaos soliton fract 21, 1125-1128 (2004) · Zbl 1060.37037 · doi:10.1016/j.chaos.2003.12.105
[10]Wang, L.; Huang, G.; Huan, S.: A note on schweizer – smítal chaos, Nonlinear anal 68, 1682-1686 (2008) · Zbl 1142.37309 · doi:10.1016/j.na.2006.12.048