American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Qualitative analysis of a Lotka-Volterra competition system with advection
  • DCDS Home
  • This Issue
  • Next Article
    Center manifolds and attractivity for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions
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

[1]

Changjing Zhuge, Xiaojuan Sun, Jinzhi Lei. On positive solutions and the Omega limit set for a class of delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2487-2503. doi: 10.3934/dcdsb.2013.18.2487
[2]

Jaroslav Smítal, Marta Štefánková. Omega-chaos almost everywhere. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1323-1327. doi: 10.3934/dcds.2003.9.1323
[3]

Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671
[4]

Francisco Balibrea, J.L. García Guirao, J.I. Muñoz Casado. A triangular map on $I^{2}$ whose $\omega$-limit sets are all compact intervals of $\{0\}\times I$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 983-994. doi: 10.3934/dcds.2002.8.983
[5]

Nancy Guelman, Jorge Iglesias, Aldo Portela. Examples of minimal set for IFSs. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5253-5269. doi: 10.3934/dcds.2017227
[6]

Andrew D. Barwell, Chris Good, Piotr Oprocha, Brian E. Raines. Characterizations of $\omega$-limit sets in topologically hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1819-1833. doi: 10.3934/dcds.2013.33.1819
[7]

Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1103-1114. doi: 10.3934/dcdss.2020065
[8]

Bruce Kitchens, Michał Misiurewicz. Omega-limit sets for spiral maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 787-798. doi: 10.3934/dcds.2010.27.787
[9]

James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667
[10]

Alexander Blokh, Michał Misiurewicz. Dense set of negative Schwarzian maps whose critical points have minimal limit sets. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 141-158. doi: 10.3934/dcds.1998.4.141
[11]

José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781
[12]

Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979
[13]

Kazumine Moriyasu, Kazuhiro Sakai, Kenichiro Yamamoto. Regular maps with the specification property. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2991-3009. doi: 10.3934/dcds.2013.33.2991
[14]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[15]

Liangwei Wang, Jingxue Yin, Chunhua Jin. $\omega$-limit sets for porous medium equation with initial data in some weighted spaces. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 223-236. doi: 10.3934/dcdsb.2013.18.223
[16]

Olexiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Chain recurrence and structure of $ \omega $-limit sets of multivalued semiflows. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2197-2217. doi: 10.3934/cpaa.2020096
[17]

José Ginés Espín Buendía, Víctor Jiménez Lopéz. A topological characterization of the $\omega$-limit sets of analytic vector fields on open subsets of the sphere. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1143-1173. doi: 10.3934/dcdsb.2019010
[18]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519
[19]

Irene Márquez-Corbella, Edgar Martínez-Moro. Algebraic structure of the minimal support codewords set of some linear codes. Advances in Mathematics of Communications, 2011, 5 (2) : 233-244. doi: 10.3934/amc.2011.5.233
[20]

María Isabel Cortez, Samuel Petite. Realization of big centralizers of minimal aperiodic actions on the Cantor set. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2891-2901. doi: 10.3934/dcds.2020153

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences