American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Decay and local eventual positivity for biharmonic parabolic equations
  • DCDS Home
  • This Issue
  • Next Article
    Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space
November  2008, 21(4): 1103-1128. doi: 10.3934/dcds.2008.21.1103

Generic points in systems of specification and Banach valued Birkhoff ergodic average

Aihua Fan 1, , Lingmin Liao 2, and  Jacques Peyrière 3,

1. 

Department of Mathematics, Wuhan University, 430072 Wuhan, China
2. 

CNRS UMR 6140, Université de Picardie Jules Verne, 33, Rue Saint Leu, 80039 Amiens Cedex 1, France
3. 

Université Paris-Sud, CNRS UMR 8628, Mathématique bât. 425, 91405 Orsay Cedex, France

Received  May 2007 Revised  April 2008 Published  May 2008

We prove that systems satisfying the specification property are saturated in the sense that the topological entropy of the set of generic points of any invariant measure is equal to the measure-theoretic entropy of the measure. We study Banach valued Birkhoff ergodic averages and obtain a variational principle for its topological entropy spectrum. As application, we examine a particular example concerning with the set of real numbers for which the frequencies of occurrences in their dyadic expansions of infinitely many words are prescribed. This relies on our explicit determination of a maximal entropy measure.
Keywords: Variational principle., Generic points, Topological entropy, Systems of specification.
Mathematics Subject Classification: Primary: 37B40, 37B45; Secondary: 37A0.
Citation: Aihua Fan, Lingmin Liao, Jacques Peyrière. Generic points in systems of specification and Banach valued Birkhoff ergodic average. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1103-1128. doi: 10.3934/dcds.2008.21.1103
[1]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547
[2]

Jinjun Li, Min Wu. Divergence points in systems satisfying the specification property. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 905-920. doi: 10.3934/dcds.2013.33.905
[3]

Jinjun Li, Min Wu. Generic property of irregular sets in systems satisfying the specification property. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 635-645. doi: 10.3934/dcds.2014.34.635
[4]

Xiaojun Huang, Zhiqiang Li, Yunhua Zhou. A variational principle of topological pressure on subsets for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2687-2703. doi: 10.3934/dcds.2020146
[5]

Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086
[6]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288
[7]

Zheng Yin, Ercai Chen. Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6581-6597. doi: 10.3934/dcds.2016085
[8]

Xianfeng Ma, Ercai Chen. Pre-image variational principle for bundle random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 957-972. doi: 10.3934/dcds.2009.23.957
[9]

João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465
[10]

Silvére Gangloff, Alonso Herrera, Cristobal Rojas, Mathieu Sablik. Computability of topological entropy: From general systems to transformations on Cantor sets and the interval. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4259-4286. doi: 10.3934/dcds.2020180
[11]

Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469
[12]

Welington Cordeiro, Manfred Denker, Michiko Yuri. A note on specification for iterated function systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3475-3485. doi: 10.3934/dcdsb.2015.20.3475
[13]

Ricardo Almeida, Agnieszka B. Malinowska. Fractional variational principle of Herglotz. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2367-2381. doi: 10.3934/dcdsb.2014.19.2367
[14]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[15]

Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363
[16]

Dubi Kelmer. Approximation of points in the plane by generic lattice orbits. Journal of Modern Dynamics, 2017, 11: 143-153. doi: 10.3934/jmd.2017007
[17]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[18]

Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225
[19]

Nassif Ghoussoub. A variational principle for nonlinear transport equations. Communications on Pure & Applied Analysis, 2005, 4 (4) : 735-742. doi: 10.3934/cpaa.2005.4.735
[20]

Richard Miles, Thomas Ward. Directional uniformities, periodic points, and entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3525-3545. doi: 10.3934/dcdsb.2015.20.3525

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (25)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences