February  2008, 22(1&2): 215-234. doi: 10.3934/dcds.2008.22.215

Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure

1. 

Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States

2. 

Department of Mathematics, Wake Forest University, Winston Salem, NC 27109

3. 

Department of Mathematics, The Graduate Center and Queens College of CUNY, Flushing, NY 11367, United States

Received  July 2007 Revised  December 2007 Published  June 2008

Let $M^{n}$ be a compact $C^{\infty}$ Riemannian manifold of dimension $n\geq 2$. Let $\text{Diff}^{\r }(M^{n})$ be the space of all $C^{\r }$ diffeomorphisms of $M^{n}$, where $ 1 < r \le \infty$. For a $C^{\r }$ diffeomorphism $f$ in $\text{Diff}^{\r }(M^{n})$ with a hyperbolic attractor $\Lambda_{f}$ on which $f$ is topologically transitive, let $U(f)$ be the $C^{1}$ open set of $\text{Diff}^{\r }(M^{n})$ such that each element in $U(f)$ can be connected to $f$ by finitely many $C^{1}$ structural stability balls in $\text{Diff}^{\r }(M^{n})$. Then by the structural stability, any element $g$ in $U(f)$ has a hyperbolic attractor $\Lambda_{g}$ and $g|\Lambda_{g}$ is topologically conjugate to $f|\Lambda_{f}$. Therefore, the topological entropy $h(g|\Lambda_{g})$ is a constant function when it is restricted to $U(f)$. However, the metric entropy $h_{\mu}(g)$ with respect to the SRB measure $\mu=\mu_{g}$ can vary. We prove that the infimum of the metric entropy $h_{\mu}(g)$ on $U(f)$ is zero.
Citation: Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215
[1]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of volume preserving Anosov systems. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4767-4783. doi: 10.3934/dcds.2017205

[2]

Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341

[3]

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

[4]

Dominic Veconi. SRB measures of singular hyperbolic attractors. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3415-3430. doi: 10.3934/dcds.2022020

[5]

Eleonora Catsigeras, Heber Enrich. SRB measures of certain almost hyperbolic diffeomorphisms with a tangency. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 177-202. doi: 10.3934/dcds.2001.7.177

[6]

Petr Kůrka. On the measure attractor of a cellular automaton. Conference Publications, 2005, 2005 (Special) : 524-535. doi: 10.3934/proc.2005.2005.524

[7]

Zhicong Liu. SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1545-1562. doi: 10.3934/dcds.2013.33.1545

[8]

Zeng Lian, Peidong Liu, Kening Lu. Existence of SRB measures for a class of partially hyperbolic attractors in banach spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3905-3920. doi: 10.3934/dcds.2017164

[9]

Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192

[10]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076

[11]

François Ledrappier, Seonhee Lim. Volume entropy of hyperbolic buildings. Journal of Modern Dynamics, 2010, 4 (1) : 139-165. doi: 10.3934/jmd.2010.4.139

[12]

Xufeng Guo, Gang Liao, Wenxiang Sun, Dawei Yang. On the hybrid control of metric entropy for dominated splittings. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5011-5019. doi: 10.3934/dcds.2018219

[13]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325

[14]

Dong Chen. Positive metric entropy in nondegenerate nearly integrable systems. Journal of Modern Dynamics, 2017, 11: 43-56. doi: 10.3934/jmd.2017003

[15]

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

[16]

Erik M. Bollt, Joseph D. Skufca, Stephen J . McGregor. Control entropy: A complexity measure for nonstationary signals. Mathematical Biosciences & Engineering, 2009, 6 (1) : 1-25. doi: 10.3934/mbe.2009.6.1

[17]

Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159

[18]

Yuntao Zang. An upper bound of the measure-theoretical entropy. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022052

[19]

Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673

[20]

Miaohua Jiang. Derivative formula of the potential function for generalized SRB measures of hyperbolic systems of codimension one. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 967-983. doi: 10.3934/dcds.2015.35.967

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]