American Institute of Mathematical Sciences

Advanced
May  2008, 20(2): 275-311. doi: 10.3934/dcds.2008.20.275

Entropy sets, weakly mixing sets and entropy capacity

François Blanchard 1, and  Wen Huang 2,

1. 

LAMA (CNRS and Université Paris-Est), 5 boulevard Descartes, 77454 Marne-la-Vallée cedex 2, France
2. 

Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

Received  August 2006 Revised  May 2007 Published  November 2007

Entropy sets are defined both topologically and for a measure. The set of topological entropy sets is the union of the sets of entropy sets for all invariant measures. For a topological system $(X,T)$ and an invariant measure $\mu$ on $(X,T)$, let $H(X,T)$ (resp. $H^\mu(X,T)$) be the closure of the set of all entropy sets (resp. $\mu$- entropy sets) in the hyperspace $2^X$. It is shown that if $h_{\text{top}}(T)>0$ (resp. $h_\mu(T)>0$), the subsystem $(H(X,T),\hat{T})$ (resp. $(H^\mu(X,T),\hat{T}))$ of $(2^X,\hat{T})$ has an invariant measure with full support and infinite topological entropy.
    Weakly mixing sets and partial mixing of dynamical systems are introduced and characterized. It is proved that if $h_{\text{top}}(T)>0$ (resp. $h_\mu(T)>0$) the set of all weakly mixing entropy sets (resp. $\mu$-entropy sets) is a dense $G_\delta$ in $H(X,T)$ (resp. $H^\mu(X,T)$). A Devaney chaotic but not partly mixing system is constructed.
     Concerning entropy capacities, it is shown that when $\mu$ is ergodic with $h_\mu(T)>0$, the set of all weakly mixing $\mu$-entropy sets $E$ such that the Bowen entropy $h(E)\ge h_\mu(T)$ is residual in $H^\mu(X,T)$. When in addition $(X,T)$ is uniquely ergodic the set of all weakly mixing entropy sets $E$ with $h(E)=h_{\text{top}}(T)$ is residual in $H(X,T)$.
Keywords: partial mixing, Entropy set, entropy capacity..
Mathematics Subject Classification: Primary: 37B05; Secondary: 54H2.
Citation: François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275
[1]

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

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
[3]

Kendry J. Vivas, Víctor F. Sirvent. Metric entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022010
[4]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
[5]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[6]

Michael Brandenbursky, Michał Marcinkowski. Entropy and quasimorphisms. Journal of Modern Dynamics, 2019, 15: 143-163. doi: 10.3934/jmd.2019017
[7]

Wenxiang Sun, Cheng Zhang. Zero entropy versus infinite entropy. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1237-1242. doi: 10.3934/dcds.2011.30.1237
[8]

Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018
[9]

José M. Amigó, Karsten Keller, Valentina A. Unakafova. On entropy, entropy-like quantities, and applications. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3301-3343. doi: 10.3934/dcdsb.2015.20.3301
[10]

Ping Huang, Ercai Chen, Chenwei Wang. Entropy formulae of conditional entropy in mean metrics. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5129-5144. doi: 10.3934/dcds.2018226
[11]

Boris Kruglikov, Martin Rypdal. Entropy via multiplicity. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 395-410. doi: 10.3934/dcds.2006.16.395
[12]

Nicolas Bedaride. Entropy of polyhedral billiard. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 89-102. doi: 10.3934/dcds.2007.19.89
[13]

Karl Petersen, Ibrahim Salama. Entropy on regular trees. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4453-4477. doi: 10.3934/dcds.2020186
[14]

Vladimír Špitalský. Local correlation entropy. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5711-5733. doi: 10.3934/dcds.2018249
[15]

Baolin He. Entropy of diffeomorphisms of line. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4753-4766. doi: 10.3934/dcds.2017204
[16]

Luiza H. F. Andrade, Rui F. Vigelis, Charles C. Cavalcante. A generalized quantum relative entropy. Advances in Mathematics of Communications, 2020, 14 (3) : 413-422. doi: 10.3934/amc.2020063
[17]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319
[18]

Benjamin Weiss. Entropy and actions of sofic groups. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3375-3383. doi: 10.3934/dcdsb.2015.20.3375
[19]

Lluís Alsedà, David Juher, Deborah M. King, Francesc Mañosas. Maximizing entropy of cycles on trees. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3237-3276. doi: 10.3934/dcds.2013.33.3237
[20]

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

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (256)
  • HTML views (0)
  • Cited by (41)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy