Abstract
AbstractGiven any smooth Anosov map, we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that, in the case of expanding maps, it reduces exactly to the usual space of functions of bounded variation which has proved to be particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations, which provides new information that could prove useful in treating hyperbolic systems with singularities.
Highlights
Starting with the paper [BKL], there has been a growing interest in the possibility of developing a functional analytic setting allowing the direct study of the transfer operator of a hyperbolic dynamical system
It is natural to construct Banach spaces that generalize bounded variation (BV) and are adapted to the study of the transfer operator associated with hyperbolic maps
The Banach space Our goal is to develop a space in the spirit of BV for the study of the statistical properties of a dynamical system (M, T, μ), where M is a compact Cε−r h (C r) manifold, T is uniformly hyperbolic and μ is the Sinai–Ruelle–Bowen (SRB) measure
Summary
Starting with the paper [BKL], there has been a growing interest in the possibility of developing a functional analytic setting allowing the direct study of the transfer operator of a hyperbolic dynamical system. It is natural to construct Banach spaces that generalize BV and are adapted to the study of the transfer operator associated with hyperbolic maps. We define Banach spaces B0,q and B1,q that, when the stable direction is absent, reduce to L1 and BV, respectively (see Remark 2.18). Note that a similar estimate could be obtained using the spectral properties on spaces already existing in the literature and by deducing the behaviour for BV densities using an approximation argument This would produce a less sharp result (in particular, it would allow only θ > σeαss for some α < 1). Spectral properties of the annealed transfer operator associated with the above random map follows from this work and stability results can be obtained using the current setting and the framework of [KL1].
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