Expanding Measures
By Vilton Pinheiro. 2008
48 citations recorded by the source corpus.
We prove that any C^{1+} 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 for maps of the interval with negative Schwarzian derivative. We also show how to construct an induced Markov map F such that every expanding probability of the initial transformation lifts to an invariant probability of F. The induced time is bounded at each point by the corresponding first hyperbolic time (the first time the dynamics exhibits hyperbolic behavior). In particular, F may be used to study decay of correlations and others statistical properties of the initial map, relative to any expanding probability.
Subfields: Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/665044be9969fde325ac1a971c04762a90cc7db7
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In Mathematical Physics
- Jiagang Yang
- Marcelo Viana
- Artur Avila
- F. R. Hertz
- José F. Alves
- C. Bonatti
- Dmitry Dolgopyat
- A. Zorich
Citations in the corpus
- Physical measures and absolute continuity for one-dimensional center direction (2013, background): “Alves, Luzzatto and Pinheiro [7,8] constructed Markov structures and used them to study fine ergodic properties of these maps.”
- SRB measures for partially hyperbolic systems whose central direction is weakly expanding (2014, unknown): “…but the same kind of questions, and the Palis conjecture, also apply to endomorphisms—see for example [9, 10, 17, 24, 32, 31] for the setting of one-dimensional maps, [45] in two dimensions, and [5, 6, 40, 41] for the construction of SRB measures in certain higher-dimensional settings.”