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Stable ergodicity for partially hyperbolic attractors with negative central exponents

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Burns, Keith, Dolgopyat, Dmitry, Pesin, Yakov and Pollicott, Mark. (2008) Stable ergodicity for partially hyperbolic attractors with negative central exponents. Journal of Modern Dynamics, Vol.2 (No.1). pp. 63-81. ISSN 1930-5311

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Abstract

We establish stable ergodicity of diffeomorphisms with partially hyperbolic attractors whose Lyapunov exponents along the central direction are all negative with respect to invariant SRB-measures.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Diffeomorphisms, Lyapunov exponents, Ergodic theory, Exponential functions
Journal or Publication Title: Journal of Modern Dynamics
Publisher: American Institute of Mathematical Sciences
ISSN: 1930-5311
Date: January 2008
Volume: Vol.2
Number: No.1
Number of Pages: 19
Page Range: pp. 63-81
Status: Not Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Description: Paper presented at: Conference on Lie Groups - Dynamics, Rigidity, Arithmetic in honor of Gregory Margulis 60th Birthday, Yale University, New Haven, CT, Feb 24-27, 2006
Type of Event: Conference
References: [1] J. Alves, C. Bonatti andM. Viana, SRBmeasures for partially hyperbolic systems whose central direction ismostly expanding, Inv.Math., 140 (2000), 351–398. [2] L. Barreira and Ya. Pesin, Lyapunov exponents and smooth ergodic theory, Univ. Lect. Series, Amer.Math. Soc., 23 2001. [3] C. Bonatti and L. J. Diaz, Persistent nonhyperbolic transitive diffeomorphims, Ann.Math., 143 (1996), 357–396. [4] C. Bonatti, L. J. Diaz and R. Ures, Minimality of strong stable and unstable foliations for par- tially hyperbolic diffeomorphisms, J. Inst.Math. Jussieu, 1 (2002), 513–541. [5] C. Bonatti, L. Diaz and M. Viana, “Dynamics Beyond Uniform Hyperbolicity. A global Geometric and Probabilistic Perspective," Encyclopaedia ofMathematical Sciences, 102. Mathematical Physics, III. Springer-Verlag, Berlin, 2005. [6] C. Bonatti andM. Viana, SRBmeasures for partially hyperbolic systemswhose central direction ismostly contracting, Israel J.Math., 115 (2000), 157–193. [7] K. Burns, D. Dolgopyat and Ya. Pesin, Partially Hyperbolic Diffeomorphisms With Non-Zero Exponents, J. Statist. Phys., 108 (2002), 927–942. [8] K. Burns, C. Pugh, M. Shub and A. Wilkinson, Recent results about stable ergodicity, 327– 366, in “Smooth Ergodic Theory and Its Applications," A. Katok, R. de la Llave, Ya. Pesin and H.Weiss eds., Proc. Symp. PureMath., Amer.Math. Soc., 2001. [9] D. Dolgopyat, On dynamics of mostly contracting diffeomorphisms, Comm. in Math. Phys., 213 (2000), 181–201. [10] D. Dolgopyat, Limit Theorems for partially hyperbolic systems, Transactions of the AMS, 356 (2004), 1637–1689. [11] D. Dolgopyat,On differentiability of SRB states for partially hyperbolic systems, Invent.Math., 155 (2004), 389–449. [12] D. Dolgopyat, Averaging and invariantmeasures,MoscowMath. J., 5 (2005), 537–576. [13] M. Hirsch, C. Pugh and M. Shub, “Invariant Manifolds," Springer Lecture Notes on Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. [14] A. Katok and B. Hasselblatt, “Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and its Applications, Cambridge University Press, London – New York, 54, 1995. [15] G.Margulis, On some aspects of the theory of Anosov systems, Springer, 2004. [16] Ya. Pesin and Ya. Sinai, Gibbs measures for partially hyperbolic attractors, Ergod. Theory and Dyn. Syst., 2 (1982), 417–438. [17] D. Ruelle, Perturbation theory for Lyapunov exponents of a toral map: extension of a result of Shub andWilkinson, Israel J.Math., 134 (2003), 345–361. [18] Y. Sinai, Gibbs measures in ergodic theory, Russ.Math. Surv., 27 (1972), 21–64. [19] M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents, Invent. Math., 139 (2000), 495–508. [20] M. Viana, Multidimensional nonhyperbolic attractors, Inst. Hautes Études Sci. Publ. Math., 85 (1997), 63–96.
URI: http://wrap.warwick.ac.uk/id/eprint/30182

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