# American Institute of Mathematical Sciences

January  2010, 4(1): 91-137. doi: 10.3934/jmd.2010.4.91

## Banach spaces for piecewise cone-hyperbolic maps

 1 D.M.A., UMR 8553, École Normale Supérieure, 75005 Paris 2 IRMAR, CNRS UMR 6625, Université de Rennes 1, 35042 Rennes, France

Received  July 2009 Revised  March 2010 Published  May 2010

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).
Citation: Viviane Baladi, Sébastien Gouëzel. Banach spaces for piecewise cone-hyperbolic maps. Journal of Modern Dynamics, 2010, 4 (1) : 91-137. doi: 10.3934/jmd.2010.4.91
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