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. Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[5]

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

[6]

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

[7]

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

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

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

[12]

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

[13]

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

[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. Google Scholar

[15]

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

[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. Google Scholar

[17]

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

[18]

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

[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. Google Scholar

[20]

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

[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. Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[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. Google Scholar

[27]

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

[28]

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

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. Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[5]

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

[6]

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

[7]

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

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

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

[12]

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

[13]

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

[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. Google Scholar

[15]

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

[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. Google Scholar

[17]

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

[18]

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

[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. Google Scholar

[20]

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

[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. Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[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. Google Scholar

[27]

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

[28]

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

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