# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 351-398. doi: 10.1007/s002220000057. [2] P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. [3] C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia of Mathematical Sciences, 102, Springer-Verlag, 2005. [4] C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193. doi: 10.1007/BF02810585. [5] R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331. 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), 1115-1138. doi: 10.1017/S0143385704000604. [7] D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math., 155 (2004), 389-449. doi: 10.1007/s00222-003-0324-5. [8] J. Franks and M. Misiurewicz, Topological methods in dynamics, in Handbook of Dynamical Systems, Vol. 1A (eds. B. Hasselblatt and A. Katok), North-Holland, Amsterdam, 2002, 547-598. 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, Cambridge, 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), 3-25. [11] Y. Kifer, Ergodic Theory of Random Transformations, Birkhäuser, Boston, 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), 1813-1830. doi: 10.1017/S0143385702000755. [13] Y. Kifer and P.-D. Liu, Random dynamics, in Handbook of Dynamical Systems, Vol. 1B (eds. B. Hasselblatt and A. Katok), Elsevier B. V., Amsterdam, 2006, 379-499. 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), 509-539. doi: 10.2307/1971328. [15] F. Ledrappier and L.-S. Young, Entropy formula for random transformations, Probab. Th. Rel. Fields, 80 (1988), 217-240. 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), 2043-2060. doi: 10.4171/JEMS/413. [17] P.-D. Liu, (Survey) Dynamics of random transformations: Smooth ergodic theory, Ergod. Th. Dynam. Syst., 21 (2001), 1279-1319. doi: 10.1017/S0143385701001614. [18] A. Manning, Topological entropy and the first homology group, in Dynamical systems-Warwick 1974, Lecture Notes in Math., 468, Springer-Verlag, 1975, 185-190. [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), 501-512. doi: 10.3934/dcds.2008.21.501. [20] M. Misiurewicz, Topological conditional entropy, Studia Mathematica, 55 (1976), 175-200. [21] Ya. B. Pesin and Ya. G. Sinai, Gibbs measures for partially hyperbolic attractors, Ergod. Th. Dynam. Syst., 2 (1982), 417-438. doi: 10.1017/S014338570000170X. [22] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Math., 9 (1978), 83-87. doi: 10.1007/BF02584795. [23] D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms, Topology, 14 (1975), 319-327. doi: 10.1016/0040-9383(75)90016-6. [24] M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6. [25] M. Shub and R. Williams, Entropy and stability, Topology, 14 (1975), 329-338. 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), 29-34. doi: 10.1016/j.topol.2009.04.053. [27] K. Yano, A remark on the topological entropy of homeomorphism, Invent. Math., 59 (1980), 215-220. doi: 10.1007/BF01453235. [28] Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.

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##### 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), 351-398. doi: 10.1007/s002220000057. [2] P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. [3] C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia of Mathematical Sciences, 102, Springer-Verlag, 2005. [4] C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193. doi: 10.1007/BF02810585. [5] R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331. 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), 1115-1138. doi: 10.1017/S0143385704000604. [7] D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math., 155 (2004), 389-449. doi: 10.1007/s00222-003-0324-5. [8] J. Franks and M. Misiurewicz, Topological methods in dynamics, in Handbook of Dynamical Systems, Vol. 1A (eds. B. Hasselblatt and A. Katok), North-Holland, Amsterdam, 2002, 547-598. 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, Cambridge, 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), 3-25. [11] Y. Kifer, Ergodic Theory of Random Transformations, Birkhäuser, Boston, 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), 1813-1830. doi: 10.1017/S0143385702000755. [13] Y. Kifer and P.-D. Liu, Random dynamics, in Handbook of Dynamical Systems, Vol. 1B (eds. B. Hasselblatt and A. Katok), Elsevier B. V., Amsterdam, 2006, 379-499. 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), 509-539. doi: 10.2307/1971328. [15] F. Ledrappier and L.-S. Young, Entropy formula for random transformations, Probab. Th. Rel. Fields, 80 (1988), 217-240. 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), 2043-2060. doi: 10.4171/JEMS/413. [17] P.-D. Liu, (Survey) Dynamics of random transformations: Smooth ergodic theory, Ergod. Th. Dynam. Syst., 21 (2001), 1279-1319. doi: 10.1017/S0143385701001614. [18] A. Manning, Topological entropy and the first homology group, in Dynamical systems-Warwick 1974, Lecture Notes in Math., 468, Springer-Verlag, 1975, 185-190. [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), 501-512. doi: 10.3934/dcds.2008.21.501. [20] M. Misiurewicz, Topological conditional entropy, Studia Mathematica, 55 (1976), 175-200. [21] Ya. B. Pesin and Ya. G. Sinai, Gibbs measures for partially hyperbolic attractors, Ergod. Th. Dynam. Syst., 2 (1982), 417-438. doi: 10.1017/S014338570000170X. [22] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Math., 9 (1978), 83-87. doi: 10.1007/BF02584795. [23] D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms, Topology, 14 (1975), 319-327. doi: 10.1016/0040-9383(75)90016-6. [24] M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6. [25] M. Shub and R. Williams, Entropy and stability, Topology, 14 (1975), 329-338. 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), 29-34. doi: 10.1016/j.topol.2009.04.053. [27] K. Yano, A remark on the topological entropy of homeomorphism, Invent. Math., 59 (1980), 215-220. doi: 10.1007/BF01453235. [28] Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.
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