We show that for a large class of maps on manifolds of arbitrary finite dimension, the existence of a Gibbs–Markov–Young structure (with Lebesgue as the reference measure) is a necessary as well as sufficient condition for the existence of an invariant probability measure which is absolutely continuous measure (with respect to Lebesgue) and for which all Lyapunov exponents are positive.

Keywords: Positive Lyapunov exponents, Gibbs–Markov–Young structure

@article{AIHPC_2013__30_1_101_0, author = {Alves, Jos\'e F. and Dias, Carla L. and Luzzatto, Stefano}, title = {Geometry of expanding absolutely continuous invariant measures and the liftability problem}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {101--120}, publisher = {Elsevier}, volume = {30}, number = {1}, year = {2013}, doi = {10.1016/j.anihpc.2012.06.004}, mrnumber = {3011293}, zbl = {06154084}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.06.004/} }

TY - JOUR AU - Alves, José F. AU - Dias, Carla L. AU - Luzzatto, Stefano TI - Geometry of expanding absolutely continuous invariant measures and the liftability problem JO - Annales de l'I.H.P. Analyse non linéaire PY - 2013 SP - 101 EP - 120 VL - 30 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2012.06.004/ DO - 10.1016/j.anihpc.2012.06.004 LA - en ID - AIHPC_2013__30_1_101_0 ER -

%0 Journal Article %A Alves, José F. %A Dias, Carla L. %A Luzzatto, Stefano %T Geometry of expanding absolutely continuous invariant measures and the liftability problem %J Annales de l'I.H.P. Analyse non linéaire %D 2013 %P 101-120 %V 30 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2012.06.004/ %R 10.1016/j.anihpc.2012.06.004 %G en %F AIHPC_2013__30_1_101_0

Alves, José F.; Dias, Carla L.; Luzzatto, Stefano. Geometry of expanding absolutely continuous invariant measures and the liftability problem. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 1, pp. 101-120. doi : 10.1016/j.anihpc.2012.06.004. http://www.numdam.org/articles/10.1016/j.anihpc.2012.06.004/

[1] Group extensions of Gibbs–Markov maps, Probab. Theory Related Fields 123 (2002), 38-40 | MR | Zbl

, ,[2] Strong statistical stability of non-uniformly expanding maps, Nonlinearity 17 (2004), 1193-1215 | MR | Zbl

,[3] SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. École Norm. Sup. (4) 33 (2000), 1-32 | EuDML | Numdam | MR | Zbl

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

, , ,[5] Markov structures and decay of correlations for non-uniformly expanding dynamical systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 no. 6 (2005), 817-839 | EuDML | Numdam | MR | Zbl

, , ,[6] The multiplicative ergodic theorem of Oseledets, http://www.mat.puc-rio.br/~jairo/docs/oseledets.pdf

,[7] Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970), 725-747 | MR | Zbl

,[8] Almost isomorphism for countable state Markov shifts, J. Reine Angew. Math. 592 (2006), 23-47 | MR | Zbl

, , ,[9] Subshifts on an infinite alphabet, Ergodic Theory Dynam. Systems 19 no. 5 (1999), 1175-1200 | MR | Zbl

,[10] Markov extensions and lifting measures for complex polynomials, Ergodic Theory Dynam. Systems 27 no. 3 (2007), 743-768 | MR | Zbl

, ,[11] Decay of correlations in one-dimensional dynamics, Ann. Sci. École Norm. Sup. (4) 36 no. 4 (2003), 621-646 | EuDML | Numdam | MR | Zbl

, , ,[12] Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time, Discrete Contin. Dyn. Syst. 12 no. 4 (2005), 639-656 | MR | Zbl

, ,[13] Decay of correlations and dispersing billiards, J. Statist. Phys. 94 no. 3–4 (1999), 513-556 | MR | Zbl

,[14] Billiards with polynomial mixing rates, Nonlinearity 18 no. 4 (2005), 1527-1553 | MR | Zbl

, ,[15] Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-like maps, Discrete Contin. Dyn. Syst. 15 no. 1 (2006), 159-176 | MR | Zbl

,[16] Statistical properties of one-dimensional maps with critical points and singularities, Stoch. Dyn. 6 no. 4 (2006), 423-458 | MR | Zbl

, , ,[17] N. Dobbs, Critical points, cusps and induced expansion in dimension one, PhD thesis, 2006.

[18] Continuity of SRB measure and entropy for Benedicks–Carleson quadratic maps, Nonlinearity 18 no. 2 (2005), 831-854 | MR | Zbl

,[19] Central limit theorem and stable laws for intermittent maps, Probab. Theory Related Fields 128 no. 1 (2004), 82-122 | MR | Zbl

,[20] Decay of correlations for non-uniformly expanding systems, Bull. Soc. Math. France 134 no. 1 (2006), 1-31 | EuDML | Numdam | MR | Zbl

,[21] Topological entropy of a countable Markov chain, Dokl. Akad. Nauk SSSR 187 (1969), 715-718 | MR

,[22] Slowly mixing systems and intermittency maps, Ergodic Theory Dynam. Systems 25 no. 1 (2005), 133-159 | MR | Zbl

,[23] Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl. vol. 54, Cambridge University Press, Cambridge (1995) | MR | Zbl

, ,[24] Lifting measures to Markov extensions, Monatsh. Math. 108 no. 2–3 (1989), 183-200 | EuDML | MR | Zbl

,[25] Markov partitions and shadowing for non-uniformly hyperbolic systems with singularities, Ergodic Theory Dynam. Systems 12 no. 3 (1992), 487-508 | MR | Zbl

, ,[26] Pesinʼs entropy formula for endomorphisms, Nagoya Math. J. 150 (1998), 197-209 | MR

,[27] Almost sure invariance principle for non-uniformly hyperbolic systems, Comm. Math. Phys. 260 no. 1 (2005), 131-146 | MR | Zbl

, ,[28] Large and moderate deviations for slowly mixing dynamical systems, Proc. Amer. Math. Soc. 137 no. 5 (2009), 1735-1741 | MR | Zbl

,[29] Lifting measures to inducing schemes, Ergodic Theory Dynam. Systems 28 no. 2 (2008), 553-574 | MR | Zbl

, , ,[30] Thermodynamics of Inducing Schemes and Liftability of Measures, Fields Inst. Commun. vol. 51, Fields Institute for Research in Mathematical Sciences, Toronto (2007) | MR | Zbl

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

,[32] Expanding measures, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 no. 6 (2011), 889Đ 939 | Numdam | MR | Zbl

,[33] Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 no. 6 (1999), 1565-1593 | MR | Zbl

,[34] Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc. 131 no. 6 (2003), 1751-1758 | MR | Zbl

,[35] Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Prilozen 2 no. 1 (1968), 64-89 | MR

,[36] An Introduction to Ergodic Theory, Graduate Texts in Mathematics vol. 79, Springer-Verlag, New York, Berlin (1982) | MR | Zbl

,[37] Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. 147 (1998), 585-650 | MR | Zbl

,[38] Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153-188 | MR | Zbl

,[39] Invariant measures for general(ized) induced transformations, Proc. Amer. Math. Soc. 133 no. 8 (2005), 2283-2295 | MR | Zbl

,*Cited by Sources: *