Banach spaces for piecewise cone hyperbolic maps

Abstract. We consider piecewise cone hyperbolic systems satisfying a bunch-ing 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 [DL08] and our pre-vious work [BG09]. 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). 1