January  2015, 35(1): 341-352. doi: 10.3934/dcds.2015.35.341

A note on partially hyperbolic attractors: Entropy conjecture and SRB measures

1. 

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871

2. 

Department of Mathematics, Brigham Young University, Provo, Utah 84602

Received  September 2013 Revised  June 2014 Published  August 2014

In this note we show that, for a class of partially hyperbolic $C^r$ ($r \geq 1$) diffeomorphisms, (1) Shub's entropy conjecture holds true; (2) SRB measures exist as zero-noise limits.
Citation: Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341
References:
[1]

J. F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding,, Invent. Math., 140 (2000), 351. doi: 10.1007/s002220000057.

[2]

P. Billingsley, Convergence of Probability Measures,, John Wiley & Sons, (1968).

[3]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity,, Encyclopaedia of Mathematical Sciences, (2005).

[4]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting,, Israel J. Math., 115 (2000), 157. doi: 10.1007/BF02810585.

[5]

R. Bowen, Entropy-expansive maps,, Trans. Amer. Math. Soc., 164 (1972), 323. doi: 10.1090/S0002-9947-1972-0285689-X.

[6]

B. Cowieson and L.-S. Young, SRB measures as zero-noise limits,, Ergod. Th. Dynam. Syst., 25 (2005), 1115. doi: 10.1017/S0143385704000604.

[7]

D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems,, Invent. Math., 155 (2004), 389. doi: 10.1007/s00222-003-0324-5.

[8]

J. Franks and M. Misiurewicz, Topological methods in dynamics,, in Handbook of Dynamical Systems, (2002), 547. doi: 10.1016/S1874-575X(02)80009-1.

[9]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511809187.

[10]

R. Z. Khasminskii, The averaging principle for parabolic and elliptic differential equations and Markov processes with small diffusion,, Theor. Probab. Appl., 8 (1963), 3.

[11]

Y. Kifer, Ergodic Theory of Random Transformations,, Birkhäuser, (1986). doi: 10.1007/978-1-4684-9175-3.

[12]

Y. Kifer and B. Weiss, Generating partitions for random transformations,, Ergod. Th. Dynam. Syst., 22 (2002), 1813. doi: 10.1017/S0143385702000755.

[13]

Y. Kifer and P.-D. Liu, Random dynamics,, in Handbook of Dynamical Systems, (2006), 379. doi: 10.1016/S1874-575X(06)80030-5.

[14]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Part I: Characterization of measures satisfying Pesin's formula,, Ann. Math., 122 (1985), 509. doi: 10.2307/1971328.

[15]

F. Ledrappier and L.-S. Young, Entropy formula for random transformations,, Probab. Th. Rel. Fields, 80 (1988), 217. doi: 10.1007/BF00356103.

[16]

G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphism away from tangencies,, J. Euro. Math. Soc., 15 (2013), 2043. doi: 10.4171/JEMS/413.

[17]

P.-D. Liu, (Survey) Dynamics of random transformations: Smooth ergodic theory,, Ergod. Th. Dynam. Syst., 21 (2001), 1279. doi: 10.1017/S0143385701001614.

[18]

A. Manning, Topological entropy and the first homology group,, in Dynamical systems-Warwick 1974, (1974), 185.

[19]

W. Marzantowicz and F. Przytycki, Estimates of the topological entropy from below for continuous self-maps on some compact manifolds,, Discrete Contin. Dyn. Syst., 21 (2008), 501. doi: 10.3934/dcds.2008.21.501.

[20]

M. Misiurewicz, Topological conditional entropy,, Studia Mathematica, 55 (1976), 175.

[21]

Ya. B. Pesin and Ya. G. Sinai, Gibbs measures for partially hyperbolic attractors,, Ergod. Th. Dynam. Syst., 2 (1982), 417. doi: 10.1017/S014338570000170X.

[22]

D. Ruelle, An inequality for the entropy of differentiable maps,, Bol. Soc. Bras. Math., 9 (1978), 83. doi: 10.1007/BF02584795.

[23]

D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms,, Topology, 14 (1975), 319. doi: 10.1016/0040-9383(75)90016-6.

[24]

M. Shub, Dynamical systems, filtrations and entropy,, Bull. Amer. Math. Soc., 80 (1974), 27. doi: 10.1090/S0002-9904-1974-13344-6.

[25]

M. Shub and R. Williams, Entropy and stability,, Topology, 14 (1975), 329. doi: 10.1016/0040-9383(75)90017-8.

[26]

R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center,, Topology Appl., 157 (2010), 29. doi: 10.1016/j.topol.2009.04.053.

[27]

K. Yano, A remark on the topological entropy of homeomorphism,, Invent. Math., 59 (1980), 215. doi: 10.1007/BF01453235.

[28]

Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285. doi: 10.1007/BF02766215.

show all references

References:
[1]

J. F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding,, Invent. Math., 140 (2000), 351. doi: 10.1007/s002220000057.

[2]

P. Billingsley, Convergence of Probability Measures,, John Wiley & Sons, (1968).

[3]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity,, Encyclopaedia of Mathematical Sciences, (2005).

[4]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting,, Israel J. Math., 115 (2000), 157. doi: 10.1007/BF02810585.

[5]

R. Bowen, Entropy-expansive maps,, Trans. Amer. Math. Soc., 164 (1972), 323. doi: 10.1090/S0002-9947-1972-0285689-X.

[6]

B. Cowieson and L.-S. Young, SRB measures as zero-noise limits,, Ergod. Th. Dynam. Syst., 25 (2005), 1115. doi: 10.1017/S0143385704000604.

[7]

D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems,, Invent. Math., 155 (2004), 389. doi: 10.1007/s00222-003-0324-5.

[8]

J. Franks and M. Misiurewicz, Topological methods in dynamics,, in Handbook of Dynamical Systems, (2002), 547. doi: 10.1016/S1874-575X(02)80009-1.

[9]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511809187.

[10]

R. Z. Khasminskii, The averaging principle for parabolic and elliptic differential equations and Markov processes with small diffusion,, Theor. Probab. Appl., 8 (1963), 3.

[11]

Y. Kifer, Ergodic Theory of Random Transformations,, Birkhäuser, (1986). doi: 10.1007/978-1-4684-9175-3.

[12]

Y. Kifer and B. Weiss, Generating partitions for random transformations,, Ergod. Th. Dynam. Syst., 22 (2002), 1813. doi: 10.1017/S0143385702000755.

[13]

Y. Kifer and P.-D. Liu, Random dynamics,, in Handbook of Dynamical Systems, (2006), 379. doi: 10.1016/S1874-575X(06)80030-5.

[14]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Part I: Characterization of measures satisfying Pesin's formula,, Ann. Math., 122 (1985), 509. doi: 10.2307/1971328.

[15]

F. Ledrappier and L.-S. Young, Entropy formula for random transformations,, Probab. Th. Rel. Fields, 80 (1988), 217. doi: 10.1007/BF00356103.

[16]

G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphism away from tangencies,, J. Euro. Math. Soc., 15 (2013), 2043. doi: 10.4171/JEMS/413.

[17]

P.-D. Liu, (Survey) Dynamics of random transformations: Smooth ergodic theory,, Ergod. Th. Dynam. Syst., 21 (2001), 1279. doi: 10.1017/S0143385701001614.

[18]

A. Manning, Topological entropy and the first homology group,, in Dynamical systems-Warwick 1974, (1974), 185.

[19]

W. Marzantowicz and F. Przytycki, Estimates of the topological entropy from below for continuous self-maps on some compact manifolds,, Discrete Contin. Dyn. Syst., 21 (2008), 501. doi: 10.3934/dcds.2008.21.501.

[20]

M. Misiurewicz, Topological conditional entropy,, Studia Mathematica, 55 (1976), 175.

[21]

Ya. B. Pesin and Ya. G. Sinai, Gibbs measures for partially hyperbolic attractors,, Ergod. Th. Dynam. Syst., 2 (1982), 417. doi: 10.1017/S014338570000170X.

[22]

D. Ruelle, An inequality for the entropy of differentiable maps,, Bol. Soc. Bras. Math., 9 (1978), 83. doi: 10.1007/BF02584795.

[23]

D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms,, Topology, 14 (1975), 319. doi: 10.1016/0040-9383(75)90016-6.

[24]

M. Shub, Dynamical systems, filtrations and entropy,, Bull. Amer. Math. Soc., 80 (1974), 27. doi: 10.1090/S0002-9904-1974-13344-6.

[25]

M. Shub and R. Williams, Entropy and stability,, Topology, 14 (1975), 329. doi: 10.1016/0040-9383(75)90017-8.

[26]

R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center,, Topology Appl., 157 (2010), 29. doi: 10.1016/j.topol.2009.04.053.

[27]

K. Yano, A remark on the topological entropy of homeomorphism,, Invent. Math., 59 (1980), 215. doi: 10.1007/BF01453235.

[28]

Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285. doi: 10.1007/BF02766215.

[1]

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

[2]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215

[3]

Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869

[4]

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

[5]

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

[6]

Lorenzo J. Díaz, Todd Fisher, M. J. Pacifico, José L. Vieitez. Entropy-expansiveness for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4195-4207. doi: 10.3934/dcds.2012.32.4195

[7]

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

[8]

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

[9]

Luis Barreira, Yakov Pesin and Jorg Schmeling. On the pointwise dimension of hyperbolic measures: a proof of the Eckmann-Ruelle conjecture. Electronic Research Announcements, 1996, 2: 69-72.

[10]

Wael Bahsoun, Paweł Góra. SRB measures for certain Markov processes. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 17-37. doi: 10.3934/dcds.2011.30.17

[11]

Radu Saghin. Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3789-3801. doi: 10.3934/dcds.2014.34.3789

[12]

Keith Burns, Dmitry Dolgopyat, Yakov Pesin, Mark Pollicott. Stable ergodicity for partially hyperbolic attractors with negative central exponents. Journal of Modern Dynamics, 2008, 2 (1) : 63-81. doi: 10.3934/jmd.2008.2.63

[13]

Carlos H. Vásquez. Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents. Journal of Modern Dynamics, 2009, 3 (2) : 233-251. doi: 10.3934/jmd.2009.3.233

[14]

Qiuxia Liu, Peidong Liu. Topological stability of hyperbolic sets of flows under random perturbations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 117-127. doi: 10.3934/dcdsb.2010.13.117

[15]

Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123

[16]

Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435

[17]

Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037

[18]

Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419

[19]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271

[20]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]