Good Banach spaces for piecewise hyperbolic maps via interpolation
Viviane Baladi
D.M.A., UMR 8553, École Normale Supérieure, 75005 Paris, FranceSébastien Gouëzel
IRMAR, Université de Rennes 1, 35042 Rennes, France
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Abstract
We introduce a weak transversality condition for piecewise and piecewise hyperbolic maps which admit a stable distribution. We show bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel–Lizorkin type which are better than previously known estimates (when our assumption on the stable distribution holds). In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) but gives a new result on bounded multipliers. Our method applies also to piecewise expanding maps and to Anosov diffeomorphisms, giving a unifying picture of several previous results on a simpler scale of Banach spaces.
Cite this article
Viviane Baladi, Sébastien Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 4, pp. 1453–1481
DOI 10.1016/J.ANIHPC.2009.01.001