We prove that any transformation, possibly with a (non-flat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a famous theorem of Keller (1990) [33] for maps of the interval with negative Schwarzian derivative.Given a non-uniformly expanding set, we also show how to construct a Markov structure such that any invariant measure defined on this set can be lifted. We used these structure to study decay of correlations and others statistical properties for general expanding measures.
@article{AIHPC_2011__28_6_889_0, author = {Pinheiro, Vilton}, title = {Expanding measures}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {889--939}, publisher = {Elsevier}, volume = {28}, number = {6}, year = {2011}, doi = {10.1016/j.anihpc.2011.07.001}, mrnumber = {2859932}, zbl = {1254.37026}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2011.07.001/} }
TY - JOUR AU - Pinheiro, Vilton TI - Expanding measures JO - Annales de l'I.H.P. Analyse non linéaire PY - 2011 SP - 889 EP - 939 VL - 28 IS - 6 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2011.07.001/ DO - 10.1016/j.anihpc.2011.07.001 LA - en ID - AIHPC_2011__28_6_889_0 ER -
Pinheiro, Vilton. Expanding measures. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 6, pp. 889-939. doi : 10.1016/j.anihpc.2011.07.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2011.07.001/
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