Geometry of expanding absolutely continuous invariant measures and the liftability problem
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 1, pp. 101-120.

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.

DOI : https://doi.org/10.1016/j.anihpc.2012.06.004
Classification : 37A05,  37C40,  37D25
Mots clés : 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},
zbl = {06154084},
mrnumber = {3011293},
language = {en},
url = {http://www.numdam.org/item/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, Tome 30 (2013) no. 1, pp. 101-120. doi : 10.1016/j.anihpc.2012.06.004. http://www.numdam.org/item/AIHPC_2013__30_1_101_0/

[1] J. Aaronson, M. Denker, Group extensions of Gibbs–Markov maps, Probab. Theory Related Fields 123 (2002), 38-40 | MR 1906436 | Zbl 1028.37003

[2] J.F. Alves, Strong statistical stability of non-uniformly expanding maps, Nonlinearity 17 (2004), 1193-1215 | MR 2069701 | Zbl 1061.37020

[3] J.F. Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. École Norm. Sup. (4) 33 (2000), 1-32 | EuDML 82508 | Numdam | MR 1743717 | Zbl 0955.37012

[4] J.F. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000), 351-398 | MR 1757000 | Zbl 0962.37012

[5] J.F. Alves, S. Luzzatto, V. Pinheiro, 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 78680 | Numdam | MR 2172861 | Zbl 1134.37326

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

[7] R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970), 725-747 | MR 277003 | Zbl 0208.25901

[8] M. Boyle, J. Buzzi, R. Gómez, Almost isomorphism for countable state Markov shifts, J. Reine Angew. Math. 592 (2006), 23-47 | MR 2222728 | Zbl 1094.37006

[9] X. Bressaud, Subshifts on an infinite alphabet, Ergodic Theory Dynam. Systems 19 no. 5 (1999), 1175-1200 | MR 1721615 | Zbl 0958.37003

[10] H. Bruin, M. Todd, Markov extensions and lifting measures for complex polynomials, Ergodic Theory Dynam. Systems 27 no. 3 (2007), 743-768 | MR 2322177 | Zbl 1130.37370

[11] H. Bruin, S. Luzzatto, S. Van Strien, Decay of correlations in one-dimensional dynamics, Ann. Sci. École Norm. Sup. (4) 36 no. 4 (2003), 621-646 | EuDML 82613 | Numdam | MR 2013929 | Zbl 1039.37021

[12] J. Buzzi, V. Maume-Deschamps, 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 2129364 | Zbl 1075.37002

[13] N. Chernov, Decay of correlations and dispersing billiards, J. Statist. Phys. 94 no. 3–4 (1999), 513-556 | MR 1675363 | Zbl 1047.37503

[14] N. Chernov, H.-K. Zhang, Billiards with polynomial mixing rates, Nonlinearity 18 no. 4 (2005), 1527-1553 | MR 2150341 | Zbl 1143.37314

[15] K. Díaz-Ordaz, 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 2191390 | Zbl 1115.37039

[16] K. Díaz-Ordaz, M.P. Holland, S. Luzzatto, Statistical properties of one-dimensional maps with critical points and singularities, Stoch. Dyn. 6 no. 4 (2006), 423-458 | MR 2285510 | Zbl 1130.37362

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

[18] J.M. Freitas, Continuity of SRB measure and entropy for Benedicks–Carleson quadratic maps, Nonlinearity 18 no. 2 (2005), 831-854 | MR 2122687 | Zbl 1067.37007

[19] S. Gouëzel, Central limit theorem and stable laws for intermittent maps, Probab. Theory Related Fields 128 no. 1 (2004), 82-122 | MR 2027296 | Zbl 1038.37007

[20] S. Gouëzel, Decay of correlations for non-uniformly expanding systems, Bull. Soc. Math. France 134 no. 1 (2006), 1-31 | EuDML 272509 | Numdam | MR 2233699 | Zbl 1111.37018

[21] B.M. Gurevic, Topological entropy of a countable Markov chain, Dokl. Akad. Nauk SSSR 187 (1969), 715-718 | MR 263162

[22] M. Holland, Slowly mixing systems and intermittency maps, Ergodic Theory Dynam. Systems 25 no. 1 (2005), 133-159 | MR 2122916 | Zbl 1073.37044

[23] A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl. vol. 54, Cambridge University Press, Cambridge (1995) | MR 1326374 | Zbl 0878.58020

[24] G. Keller, Lifting measures to Markov extensions, Monatsh. Math. 108 no. 2–3 (1989), 183-200 | EuDML 178444 | MR 1026617 | Zbl 0712.28008

[25] T. Krüger, S. Troubetzkoy, Markov partitions and shadowing for non-uniformly hyperbolic systems with singularities, Ergodic Theory Dynam. Systems 12 no. 3 (1992), 487-508 | MR 1182660 | Zbl 0756.58038

[26] P.-D. Liu, Pesinʼs entropy formula for endomorphisms, Nagoya Math. J. 150 (1998), 197-209 | MR 1633167

[27] I. Melbourne, M. Nicol, Almost sure invariance principle for non-uniformly hyperbolic systems, Comm. Math. Phys. 260 no. 1 (2005), 131-146 | MR 2175992 | Zbl 1084.37024

[28] I. Melbourne, Large and moderate deviations for slowly mixing dynamical systems, Proc. Amer. Math. Soc. 137 no. 5 (2009), 1735-1741 | MR 2470832 | Zbl 1167.37020

[29] Ya. Pesin, S. Senti, K. Zhang, Lifting measures to inducing schemes, Ergodic Theory Dynam. Systems 28 no. 2 (2008), 553-574 | MR 2408392 | Zbl 1154.37302

[30] Y. Pesin, K. Zhang, Thermodynamics of Inducing Schemes and Liftability of Measures, Fields Inst. Commun. vol. 51, Fields Institute for Research in Mathematical Sciences, Toronto (2007) | MR 2388701 | Zbl 1139.37016

[31] V. Pinheiro, Sinai–Ruelle–Bowen measures for weakly expanding maps, Nonlinearity 19 (2006), 1185-1200 | MR 2222364 | Zbl 1100.37014

[32] V. Pinheiro, Expanding measures, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 no. 6 (2011), 889Đ 939 | Numdam | MR 2859932 | Zbl 1254.37026

[33] O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 no. 6 (1999), 1565-1593 | MR 1738951 | Zbl 0994.37005

[34] O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc. 131 no. 6 (2003), 1751-1758 | MR 1955261 | Zbl 1009.37003

[35] Y. Sinai, Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Prilozen 2 no. 1 (1968), 64-89 | MR 233038

[36] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics vol. 79, Springer-Verlag, New York, Berlin (1982) | MR 648108 | Zbl 0475.28009

[37] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. 147 (1998), 585-650 | MR 1637655 | Zbl 0945.37009

[38] L.-S. Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153-188 | MR 1750438 | Zbl 0983.37005

[39] R. Zweimüller, Invariant measures for general(ized) induced transformations, Proc. Amer. Math. Soc. 133 no. 8 (2005), 2283-2295 | MR 2138871 | Zbl 1119.28011