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

Banach spaces for piecewise cone-hyperbolic maps

Viviane Baladi 1, and  Sébastien Gouëzel 2,

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).
Keywords: exponential decay of correlations, hyperbolic systems with singularities, anisotropic Sobolev spaces, physical/SRB measures, transfer operators, piecewise hyperbolic systems., spectral gap.
Mathematics Subject Classification: Primary: 37C30; Secondary: 37D50, 46B99, 46E9.
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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