Journal of Modern Dynamics (JMD)

Banach spaces for piecewise cone-hyperbolic maps

Pages: 91 - 137, Issue 1, January 2010      doi:10.3934/jmd.2010.4.91

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Viviane Baladi - D.M.A., UMR 8553, École Normale Supérieure, 75005 Paris, France (email)
Sébastien Gouëzel - IRMAR, CNRS UMR 6625, Université de Rennes 1, 35042 Rennes, France (email)

Abstract: We consider piecewise cone-hyperbolic systems satisfying a bunching condition, and we obtain a bound on the essential spectral radius of the associated weighted transfer operators acting on anisotropic Sobolev spaces. The bunching condition is always satisfied in dimension two, and our results give a unifying treatment of the work of Demers-Liverani [9] and our previous work [2]. When the complexity is subexponential, our bound implies a spectral gap for the transfer operator corresponding to the physical measures in many cases (for example if $T$ preserves volume, or if the stable dimension is equal to $1$ and the unstable dimension is not zero).

Keywords:  hyperbolic systems with singularities, transfer operators, spectral gap, anisotropic Sobolev spaces, physical/SRB measures, exponential decay of correlations, piecewise hyperbolic systems.
Mathematics Subject Classification:  Primary: 37C30; Secondary: 37D50, 46B99, 46E99.

Received: July 2009;      Revised: March 2010;      Available Online: May 2010.