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We introduce a weak transversality condition for piecewise C^{1+\alpha} andpiecewise hyperbolic maps which admit a C^{1+\alpha} stable distribution. Weshow good bounds on the essential spectral radius of the associated transferoperators acting on classical anisotropic Sobolev spaces of Triebel-Lizorkintype. In many cases, we obtain a spectral gap from which we deduce theexistence of finitely many physical measures with basin of total measure. Theanalysis relies on standard techniques (in particular complex interpolation)and applies also to piecewise expanding maps and to Anosov diffeomorphisms,giving a unifying picture of several previous results.

Good Banach spaces for piecewise hyperbolic maps via interpolation