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, p. 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
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},
     publisher = {Elsevier},
     volume = {30},
     number = {1},
     year = {2013},
     pages = {101-120},
     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, Volume 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/

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