Minimal sets and $\omega$-chaos in expansive systems with weak specification property

Lidong Wang; Hui Wang; Guifeng Huang

doi:10.3934/dcds.2015.35.1231

Article Contents

2015, Volume 35, Issue 3: 1231-1238. Doi: 10.3934/dcds.2015.35.1231

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Minimal sets and $\omega$-chaos in expansive systems with weak specification property

1.
School of Science, Dalian Nationalities University, Dalian 116600
2.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024

Received: December 31, 2013

Revised: June 30, 2014

Published: February 2015

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Abstract

In this paper, we first discuss minimal sets in the compact dynamical system with weak specification property. On the basis of this discussion, we show that an expansive system with weak specification property displays a stronger form of $\omega$-chaos.

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Mathematics Subject Classification: Primary: 37B05; Secondary: 54H20.

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References

[1]	J. Bobok, On multidimensional $\omega$-chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 737-740.doi: 10.1142/S0218127406015118.
[2]	R. Bowen, Periodic points and measures for axiom a diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.
[3]	R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer-Verlag, Berlin, 1975.doi: 10.1007/978-3-540-77695-6.
[4]	J. Doleželová, Distributionally scrambled invariant sets in a compact metric space, Nonlinear Analysis: Theory, Methods and Applications, 79 (2013), 80-84.doi: 10.1016/j.na.2012.11.005.
[5]	G. L. Forti, Various notions of chaos for discrete dynamical systems. A brief survey, Aequationes math., 70 (2005), 1-13.doi: 10.1007/s00010-005-2771-0.
[6]	W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, American Mathematical Soc. Colloquium Publications, Providence, 1955.
[7]	W. H. Gottschalk, Minimal sets: An introduction to topological dynamics, Bull. Amer. Math. Soc., 64 (1958), 336-351.doi: 10.1090/S0002-9904-1958-10223-2.
[8]	J. L. García Guirao and M. Lampart, Relations between distributional, Li-Yorke and $\omega$ chaos, Chaos Solitons Fractals, 28 (2006), 788-792.doi: 10.1016/j.chaos.2005.08.005.
[9]	N. T. A. Haydn and D. Ruelle, Equivalence of gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification, Commun. Math. Phys., 148 (1992), 155-167.doi: 10.1007/BF02102369.
[10]	A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.doi: 10.1017/CBO9780511809187.
[11]	M. Lampart, Two kinds of chaos and relations between them, Acta Math. Univ. Comenian, 72 (2003), 119-127.
[12]	M. Lampart and P. Oprocha, Shift spaces, $\omega$-chaos and specification property, Topology and its Applications, 156 (2009), 2979-2985.doi: 10.1016/j.topol.2009.04.063.
[13]	S. H. Li, $\omega$-chaos and topological entropy, Trans. Amer. Math. Soc., 339 (1993), 243-249.doi: 10.2307/2154217.
[14]	E. Mihailescu, Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms, Discrete Contin. Dyn. Syst., 32 (2012), 2485-2502.doi: 10.3934/dcds.2012.32.2485.
[15]	P. Oprocha and M. Štefánková, Specification property and distributional chaos almost everywhere, Proc. Amer. Math. Soc., 136 (2008), 3931-3940.doi: 10.1090/S0002-9939-08-09602-0.
[16]	P. Oprocha, Distributional chaos revisited, Trans. Amer. Math. Soc., 361 (2009), 4901-4925.doi: 10.1090/S0002-9947-09-04810-7.
[17]	D. Ruelle, Statistical mechanics on a compact set with $Z^p$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc., 187 (1973), 237-251.
[18]	D. Ruelle, Thermodynamic formalism of maps satisfying positive expansiveness and specification, Nonlinearity, 5 (1992), 1223-1236.doi: 10.1088/0951-7715/5/6/002.
[19]	K. Sakai, N. Sumi and K. Yamamoto, Diffeomorphisms satisfying the specification property, Proc. Amer. Math. Soc., 138 (2010), 315-321.doi: 10.1090/S0002-9939-09-10085-0.
[20]	K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc., 190 (1974), 285-299.doi: 10.1090/S0002-9947-1974-0352411-X.
[21]	J. Smíal and M. Štefánková, Omega-chaos almost everywhere, Discrete Contin. Dyn. Syst., 9 (2003), 1323-1327.doi: 10.3934/dcds.2003.9.1323.
[22]	F. Takens and E. Verbitski, Multifractal analysis of local entropies for expansive homeomorphisms with specification, Commun. Math. Phys., 203 (1999), 593-612.doi: 10.1007/s002200050627.
[23]	H. Wang and L. Wang, The weak specification property and distributional chaos, Nonlinear Analysis: Theory, Methods, & Applications, 91 (2013), 46-50.doi: 10.1016/j.na.2013.06.007.
[24]	K. Yamamoto, On the weaker forms of the specification property and their applications, Proc. Amer. Math. Soc., 137 (2009), 3807-3814.doi: 10.1090/S0002-9939-09-09937-7.