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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, China |
2. | School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China, China |
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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