Good Banach spaces for piecewise hyperbolic maps via interpolation

Viviane Baladi (1) , Sébastien Gouëzel (2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 IRMAR - Institut de Recherche Mathématique de Rennes

We introduce a weak transversality condition for piecewise C^{1+\alpha} and piecewise hyperbolic maps which admit a C^{1+\alpha} stable distribution. We show good bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel-Lizorkin type. 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) and applies also to piecewise expanding maps and to Anosov diffeomorphisms, giving a unifying picture of several previous results.

Domaines: Systèmes dynamiques [math.DS]

Viviane Baladi, Sébastien Gouëzel. Good Banach spaces for piecewise hyperbolic maps via interpolation. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2009, 26 (4), pp.1453-1481. ⟨10.1016/j.anihpc.2009.01.001⟩. ⟨hal-00186985⟩