Banachspaces for piecewise cone-hyperbolic maps

Abstract

We consider piecewise cone-hyperbolic systems satisfying abunching condition, and we obtain a bound on the essentialspectral radius of the associated weighted transfer operatorsacting on anisotropic Sobolev spaces. The bunching condition isalways satisfied in dimension two, and our results give aunifying treatment of the work of Demers-Liverani[9] and our previous work [2]. When the complexity issubexponential, our bound implies a spectral gap for thetransfer operator corresponding to the physical measures inmany cases (for example if $T$ preserves volume, or if thestable dimension is equal to $1$ and the unstable dimension isnot zero).