A generalized push-forward construction of Sinai-Ruelle-Bowen measures
Grigorii Dvorkin
Abstract
For Anosov diffeomorphisms the geometric approach for constructing Sinai-Ruelle-Bowen (SRB) measures works by pushing forward under the dynamics the leaf volume on a local unstable manifold or any measure that is equivalent to the leaf volume. Such a measure is called a reference measure. In applications one often needs to use a reference measure that is absolutely continuous with respect to leaf volume. In this paper we show that a reference measure can be chosen from a substantially more general class of measures which are the restriction of the leaf volume to any measurable subset of a local unstable manifold of positive leaf volume. Moreover, we do this in a more general settings of diffeomorphisms which are effectively hyperbolic on a set of positive volume. Such diffeomorphisms were introduced by Climenhaga, Dolgopyat and Pesin in CDP where they used the geometric approach to construct SRB measures. Our results applies also to diffeomorphisms that are uniformly partially hyperbolic or more generally possess a dominated splitting.
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