  We introduce a weak transversality condition for piecewise C^{1+\alpha} and
piecewise hyperbolic maps which admit a C^{1+\alpha} stable distribution. We
show good bounds on the essential spectral radius of the associated transfer
operators acting on classical anisotropic Sobolev spaces of Triebel-Lizorkin
type. In many cases, we obtain a spectral gap from which we deduce the
existence of finitely many physical measures with basin of total measure. The
analysis 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.
