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

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March 2015, 35(3): 1231-1238. doi: 10.3934/dcds.2015.35.1231

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

Lidong Wang ^1, , Hui Wang ^2, and Guifeng Huang ^2,

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

Received January 2014 Revised July 2014 Published October 2014

Abstract
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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.

Keywords: specification property, expansive map, Minimal set, $\omega$-chaos., $\omega$-limit set.

Mathematics Subject Classification: Primary: 37B05; Secondary: 54H2.

Citation: Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231

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.

show all references

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.

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