For a class of partially hyperbolic ${C}^{k}$, $k>1$ diffeomorphisms with circle center leaves we prove the existence and finiteness of physical (or Sinai–Ruelle–Bowen) measures, whose basins cover a full Lebesgue measure subset of the ambient manifold. Our conditions hold for an open and dense subset of all ${C}^{k}$ partially hyperbolic skew-products on compact circle bundles.Our arguments blend ideas from the theory of Gibbs states for diffeomorphisms with mostly contracting center direction together with recent progress in the theory of cocycles over hyperbolic systems that call into play geometric properties of invariant foliations such as absolute continuity. Recent results show that absolute continuity of the center foliation is often a rigid property among volume preserving systems. We prove that this is not at all the case in the dissipative setting, where absolute continuity can even be robust.

@article{AIHPC_2013__30_5_845_0, author = {Viana, Marcelo and Yang, Jiagang}, title = {Physical measures and absolute continuity for one-dimensional center direction}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {845--877}, publisher = {Elsevier}, volume = {30}, number = {5}, year = {2013}, doi = {10.1016/j.anihpc.2012.11.002}, mrnumber = {3103173}, zbl = {06295444}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2012.11.002/} }

TY - JOUR AU - Viana, Marcelo AU - Yang, Jiagang TI - Physical measures and absolute continuity for one-dimensional center direction JO - Annales de l'I.H.P. Analyse non linéaire PY - 2013 SP - 845 EP - 877 VL - 30 IS - 5 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2012.11.002/ DO - 10.1016/j.anihpc.2012.11.002 LA - en ID - AIHPC_2013__30_5_845_0 ER -

%0 Journal Article %A Viana, Marcelo %A Yang, Jiagang %T Physical measures and absolute continuity for one-dimensional center direction %J Annales de l'I.H.P. Analyse non linéaire %D 2013 %P 845-877 %V 30 %N 5 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2012.11.002/ %R 10.1016/j.anihpc.2012.11.002 %G en %F AIHPC_2013__30_5_845_0

Viana, Marcelo; Yang, Jiagang. Physical measures and absolute continuity for one-dimensional center direction. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 5, pp. 845-877. doi : 10.1016/j.anihpc.2012.11.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2012.11.002/

[1] SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. École Norm. Sup. 33 (2000), 1-32 | EuDML | Numdam | MR | Zbl

,[2] Statistical Analysis of Non-uniformly Expanding Dynamical Systems, Lecture Notes 24th Braz. Math. Colloq., IMPA, Rio de Janeiro (2003) | MR | Zbl

,[3] Random perturbations of nonuniformly expanding maps, Geometric Methods in Dynamics. I Astérisque 286 (2003), xvii | Numdam | MR | Zbl

, ,[4] Hyperbolic times: frequency versus integrability, Ergodic Theory Dynam. Systems 24 (2004), 329-346 | MR | Zbl

, ,[5] Stochastic stability of non-uniformly hyperbolic diffeomorphisms, Stoch. Dyn. 7 (2007), 299-333 | MR | Zbl

, , ,[6] SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000), 351-398 | MR | Zbl

, , ,[7] Markov structures for non-uniformly expanding maps on compact manifolds in arbitrary dimension, Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 26-31 | EuDML | MR | Zbl

, , ,[8] Markov structures and decay of correlations for non-uniformly expanding dynamical systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 817-839 | EuDML | Numdam | MR | Zbl

, , ,[9] Statistical stability for robust classes of maps with non-uniform expansion, Ergodic Theory Dynam. Systems 22 (2002), 1-32 | MR | Zbl

, ,[10] Robust ergodic properties in partially hyperbolic dynamics, Trans. Amer. Math. Soc. 362 (2010), 1831-1867 | MR | Zbl

,[11] Geodesic flows on closed Riemannian manifolds of negative curvature, Proc. Steklov Math. Inst. 90 (1967), 1-235 | MR | Zbl

,[12] On the regularization of conservative maps, Acta Math. 205 (2010), 5-18 | MR | Zbl

,[13] Cocycles over partially hyperbolic maps, www.impa.br/~viana/ (2011) | Zbl

, , ,[14] Extremal Lyapunov exponents: an invariance principle and applications, Invent. Math. 181 (2010), 115-178 | MR | Zbl

, ,[15] Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows, www.impa.br/~viana/ (2011) | Zbl

, , ,[16] A. Avila, M. Viana, A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity II, in preparation.

[17] Removing zero central Lyapunov exponents, Ergodic Theory Dynam. Systems 23 (2003), 1655-1670 | MR | Zbl

, ,[18] Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu 1 (2002), 513-541 | MR | Zbl

, , ,[19] Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia Math. Sci. vol. 102, Springer-Verlag (2005) | MR | Zbl

, , ,[20] Généricité dʼexposants de Lyapunov non-nuls pour des produits déterministes de matrices, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 579-624 | EuDML | Numdam | MR | Zbl

, , ,[21] SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math. 115 (2000), 157-193 | MR | Zbl

, ,[22] Partially hyperbolic dynamical systems, Izv. Acad. Nauk. SSSR 1 (1974), 177-212 | MR | Zbl

, ,[23] Introduction to Dynamical Systems, Cambridge University Press (2002) | MR | Zbl

, ,[24] Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Statist. Phys. 108 (2002), 927-942 | MR | Zbl

, , ,[25] Stable ergodicity for partially hyperbolic attractors with negative central exponents, J. Mod. Dyn. 2 (2008), 63-81 | MR | Zbl

, , , ,[26] Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center, Discrete Contin. Dyn. Syst. 22 (2008), 75-88 | MR | Zbl

, , , , ,[27] Stable ergodicity of skew products, Ann. Sci. École Norm. Sup. 32 (1999), 859-889 | EuDML | Numdam | MR | Zbl

, ,[28] On the ergodicity of partially hyperbolic systems, Ann. of Math. 171 (2010), 451-489 | MR | Zbl

, ,[29] Weakly expanding skew-products of quadratic maps, Ergodic Theory Dynam. Systems 23 (2003), 1401-1414 | MR | Zbl

, , ,[30] Backward inducing and exponential decay of correlations for partially hyperbolic attractors with mostly contracting central direction, Israel J. Math. 130 (2002), 29-75 | MR

,[31] On dynamics of mostly contracting diffeomorphisms, Comm. Math. Phys. 213 (2000), 181-201 | MR | Zbl

,[32] On differentiability of SRB states for partially hyperbolic systems, Invent. Math. 155 (2004), 389-449 | MR | Zbl

,[33] Stable accessibility is ${C}^{1}$ dense, Astérisque 287 (2003), 33-60 | Numdam | MR | Zbl

, ,[34] Decay of correlations for nonuniformly expanding systems, Bull. Soc. Math. France 134 (2006), 1-31 | EuDML | Numdam | MR | Zbl

,[35] Creation of blenders in the conservative setting, Nonlinearity 23 (2010), 211-223 | MR | Zbl

, , , ,[36] Some results on the integrability of the center bundle for partially hyperbolic diffeomorphisms, Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, Fields Inst. Commun. vol. 51, Amer. Math. Soc. (2007), 103-109 | MR | Zbl

, , ,[37] Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math. 172 (2008), 353-381 | MR | Zbl

, , ,[38] Non-absolutely continuous foliations, Israel J. Math. 160 (2007), 173-187 | MR | Zbl

, ,[39] Invariant Manifolds, Lect. Notes in Math. vol. 583, Springer-Verlag (1977) | MR | Zbl

, , ,[40] Lyapunov exponents, entropy and periodic points of diffeomorphisms, Publ. Math. IHES 51 (1980), 137-173 | EuDML | Numdam | MR | Zbl

,[41] Contributions to the stability conjecture, Topology 17 (1978), 383-396 | MR | Zbl

,[42] A proof of the ${C}^{1}$ stability conjecture, Publ. Math. IHES 66 (1988), 161-210 | EuDML | Numdam | MR | Zbl

,[43] An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one, Topology 40 (2001), 259-278 | MR | Zbl

, ,[44] On the Bernoulli nature of systems with some hyperbolic structure, Ergodic Theory Dynam. Systems 18 (1998), 441-456 | MR | Zbl

, ,[45] Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems 2 (1982), 417-438 | MR | Zbl

, ,[46] Families of invariant manifolds corresponding to non-zero characteristic exponents, Math. USSR. Izv. 10 (1976), 1261-1302

,[47] V. Pinheiro, Expanding measures, preprint, 2008. | Numdam | MR

[48] Sinai–Ruelle–Bowen measures for weakly expanding maps, Nonlinearity 19 (2006), 1185-1200 | MR | Zbl

,[49] Ergodic attractors, Trans. Amer. Math. Soc. 312 (1989), 1-54 | MR | Zbl

, ,[50] Stably ergodic dynamical systems and partial hyperbolicity, J. Complexity 13 (1997), 125-179 | MR | Zbl

, ,[51] C. Pugh, M. Viana, A. Wilkinson, Absolute continuity of foliations, in preparation.

[52] On the fundamental ideas of measure theory, Amer. Math. Soc. Transl. 10 (1952), 1-52, Mat. Sbornik 25 (1949), 107-150 | MR

,[53] Absolutely singular dynamical foliations, Comm. Math. Phys. 219 (2001), 481-487 | MR | Zbl

, ,[54] Geometric expansion, Lyapunov exponents and foliations, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 689-704 | EuDML | Numdam | MR | Zbl

, ,[55] Global Stability of Dynamical Systems, Springer-Verlag (1987) | MR

,[56] Expanding endomorphisms of the circle revisited, Ergodic Theory Dynam. Systems 5 (1985), 285-289 | MR | Zbl

, ,[57] Pathological foliations and removable zero exponents, Invent. Math. 139 (2000), 495-508 | MR | Zbl

, ,[58] Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817 | MR | Zbl

,[59] Physical measures for partially hyperbolic surface endomorphisms, Acta Math. 194 (2005), 37-132 | MR | Zbl

,[60] Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents, J. Mod. Dyn. 3 (2009), 233-251 | MR | Zbl

,[61] Statistical stability for diffeomorphisms with dominated splitting, Ergodic Theory Dynam. Systems 27 (2007), 253-283 | MR | Zbl

,[62] Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Ann. of Math. 167 (2008), 643-680 | MR | Zbl

,[63] M. Viana, J. Yang, Towards a theory of maps with mostly contracting center, in preparation.

*Cited by Sources: *