Banach spaces for piecewise cone hyperbolic maps

Details Banach spaces for piecewise cone hyperbolic maps Banach spaces for piecewise cone hyperbolic maps

2009-07-08

arxiv.org/abs/0907.1402v2

Mathematics

Dynamical Systems

38 pages v2: weakened the transversality condition and added details on physical measures

Scientific paper

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 and our previous work. 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).

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