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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

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 & Continuous Dynamical Systems - A, 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.  doi: 10.1142/S0218127406015118.  Google Scholar

[2]

R. Bowen, Periodic points and measures for axiom a diffeomorphisms,, Trans. Amer. Math. Soc., 154 (1971), 377.   Google Scholar

[3]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Springer-Verlag, (1975).  doi: 10.1007/978-3-540-77695-6.  Google Scholar

[4]

J. Doleželová, Distributionally scrambled invariant sets in a compact metric space,, Nonlinear Analysis: Theory, 79 (2013), 80.  doi: 10.1016/j.na.2012.11.005.  Google Scholar

[5]

G. L. Forti, Various notions of chaos for discrete dynamical systems. A brief survey,, Aequationes math., 70 (2005), 1.  doi: 10.1007/s00010-005-2771-0.  Google Scholar

[6]

W. H. Gottschalk and G. A. Hedlund, Topological Dynamics,, American Mathematical Soc. Colloquium Publications, (1955).   Google Scholar

[7]

W. H. Gottschalk, Minimal sets: An introduction to topological dynamics,, Bull. Amer. Math. Soc., 64 (1958), 336.  doi: 10.1090/S0002-9904-1958-10223-2.  Google Scholar

[8]

J. L. García Guirao and M. Lampart, Relations between distributional, Li-Yorke and $\omega$ chaos,, Chaos Solitons Fractals, 28 (2006), 788.  doi: 10.1016/j.chaos.2005.08.005.  Google Scholar

[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.  doi: 10.1007/BF02102369.  Google Scholar

[10]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511809187.  Google Scholar

[11]

M. Lampart, Two kinds of chaos and relations between them,, Acta Math. Univ. Comenian, 72 (2003), 119.   Google Scholar

[12]

M. Lampart and P. Oprocha, Shift spaces, $\omega$-chaos and specification property,, Topology and its Applications, 156 (2009), 2979.  doi: 10.1016/j.topol.2009.04.063.  Google Scholar

[13]

S. H. Li, $\omega$-chaos and topological entropy,, Trans. Amer. Math. Soc., 339 (1993), 243.  doi: 10.2307/2154217.  Google Scholar

[14]

E. Mihailescu, Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms,, Discrete Contin. Dyn. Syst., 32 (2012), 2485.  doi: 10.3934/dcds.2012.32.2485.  Google Scholar

[15]

P. Oprocha and M. Štefánková, Specification property and distributional chaos almost everywhere,, Proc. Amer. Math. Soc., 136 (2008), 3931.  doi: 10.1090/S0002-9939-08-09602-0.  Google Scholar

[16]

P. Oprocha, Distributional chaos revisited,, Trans. Amer. Math. Soc., 361 (2009), 4901.  doi: 10.1090/S0002-9947-09-04810-7.  Google Scholar

[17]

D. Ruelle, Statistical mechanics on a compact set with $Z^p$ action satisfying expansiveness and specification,, Trans. Amer. Math. Soc., 187 (1973), 237.   Google Scholar

[18]

D. Ruelle, Thermodynamic formalism of maps satisfying positive expansiveness and specification,, Nonlinearity, 5 (1992), 1223.  doi: 10.1088/0951-7715/5/6/002.  Google Scholar

[19]

K. Sakai, N. Sumi and K. Yamamoto, Diffeomorphisms satisfying the specification property,, Proc. Amer. Math. Soc., 138 (2010), 315.  doi: 10.1090/S0002-9939-09-10085-0.  Google Scholar

[20]

K. Sigmund, On dynamical systems with the specification property,, Trans. Amer. Math. Soc., 190 (1974), 285.  doi: 10.1090/S0002-9947-1974-0352411-X.  Google Scholar

[21]

J. Smíal and M. Štefánková, Omega-chaos almost everywhere,, Discrete Contin. Dyn. Syst., 9 (2003), 1323.  doi: 10.3934/dcds.2003.9.1323.  Google Scholar

[22]

F. Takens and E. Verbitski, Multifractal analysis of local entropies for expansive homeomorphisms with specification,, Commun. Math. Phys., 203 (1999), 593.  doi: 10.1007/s002200050627.  Google Scholar

[23]

H. Wang and L. Wang, The weak specification property and distributional chaos,, Nonlinear Analysis: Theory, 91 (2013), 46.  doi: 10.1016/j.na.2013.06.007.  Google Scholar

[24]

K. Yamamoto, On the weaker forms of the specification property and their applications,, Proc. Amer. Math. Soc., 137 (2009), 3807.  doi: 10.1090/S0002-9939-09-09937-7.  Google Scholar

show all references

References:
[1]

J. Bobok, On multidimensional $\omega$-chaos,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 737.  doi: 10.1142/S0218127406015118.  Google Scholar

[2]

R. Bowen, Periodic points and measures for axiom a diffeomorphisms,, Trans. Amer. Math. Soc., 154 (1971), 377.   Google Scholar

[3]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Springer-Verlag, (1975).  doi: 10.1007/978-3-540-77695-6.  Google Scholar

[4]

J. Doleželová, Distributionally scrambled invariant sets in a compact metric space,, Nonlinear Analysis: Theory, 79 (2013), 80.  doi: 10.1016/j.na.2012.11.005.  Google Scholar

[5]

G. L. Forti, Various notions of chaos for discrete dynamical systems. A brief survey,, Aequationes math., 70 (2005), 1.  doi: 10.1007/s00010-005-2771-0.  Google Scholar

[6]

W. H. Gottschalk and G. A. Hedlund, Topological Dynamics,, American Mathematical Soc. Colloquium Publications, (1955).   Google Scholar

[7]

W. H. Gottschalk, Minimal sets: An introduction to topological dynamics,, Bull. Amer. Math. Soc., 64 (1958), 336.  doi: 10.1090/S0002-9904-1958-10223-2.  Google Scholar

[8]

J. L. García Guirao and M. Lampart, Relations between distributional, Li-Yorke and $\omega$ chaos,, Chaos Solitons Fractals, 28 (2006), 788.  doi: 10.1016/j.chaos.2005.08.005.  Google Scholar

[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.  doi: 10.1007/BF02102369.  Google Scholar

[10]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511809187.  Google Scholar

[11]

M. Lampart, Two kinds of chaos and relations between them,, Acta Math. Univ. Comenian, 72 (2003), 119.   Google Scholar

[12]

M. Lampart and P. Oprocha, Shift spaces, $\omega$-chaos and specification property,, Topology and its Applications, 156 (2009), 2979.  doi: 10.1016/j.topol.2009.04.063.  Google Scholar

[13]

S. H. Li, $\omega$-chaos and topological entropy,, Trans. Amer. Math. Soc., 339 (1993), 243.  doi: 10.2307/2154217.  Google Scholar

[14]

E. Mihailescu, Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms,, Discrete Contin. Dyn. Syst., 32 (2012), 2485.  doi: 10.3934/dcds.2012.32.2485.  Google Scholar

[15]

P. Oprocha and M. Štefánková, Specification property and distributional chaos almost everywhere,, Proc. Amer. Math. Soc., 136 (2008), 3931.  doi: 10.1090/S0002-9939-08-09602-0.  Google Scholar

[16]

P. Oprocha, Distributional chaos revisited,, Trans. Amer. Math. Soc., 361 (2009), 4901.  doi: 10.1090/S0002-9947-09-04810-7.  Google Scholar

[17]

D. Ruelle, Statistical mechanics on a compact set with $Z^p$ action satisfying expansiveness and specification,, Trans. Amer. Math. Soc., 187 (1973), 237.   Google Scholar

[18]

D. Ruelle, Thermodynamic formalism of maps satisfying positive expansiveness and specification,, Nonlinearity, 5 (1992), 1223.  doi: 10.1088/0951-7715/5/6/002.  Google Scholar

[19]

K. Sakai, N. Sumi and K. Yamamoto, Diffeomorphisms satisfying the specification property,, Proc. Amer. Math. Soc., 138 (2010), 315.  doi: 10.1090/S0002-9939-09-10085-0.  Google Scholar

[20]

K. Sigmund, On dynamical systems with the specification property,, Trans. Amer. Math. Soc., 190 (1974), 285.  doi: 10.1090/S0002-9947-1974-0352411-X.  Google Scholar

[21]

J. Smíal and M. Štefánková, Omega-chaos almost everywhere,, Discrete Contin. Dyn. Syst., 9 (2003), 1323.  doi: 10.3934/dcds.2003.9.1323.  Google Scholar

[22]

F. Takens and E. Verbitski, Multifractal analysis of local entropies for expansive homeomorphisms with specification,, Commun. Math. Phys., 203 (1999), 593.  doi: 10.1007/s002200050627.  Google Scholar

[23]

H. Wang and L. Wang, The weak specification property and distributional chaos,, Nonlinear Analysis: Theory, 91 (2013), 46.  doi: 10.1016/j.na.2013.06.007.  Google Scholar

[24]

K. Yamamoto, On the weaker forms of the specification property and their applications,, Proc. Amer. Math. Soc., 137 (2009), 3807.  doi: 10.1090/S0002-9939-09-09937-7.  Google Scholar

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