Zeta functions and transfer operators for multidimensional piecewise affine and expanding maps

Let X subset of R-2 be a finite union of bounded polytopes and let T : X -- > X be piecewise affine and eventually expanding. Then the Perron-Frobenius operator C of T is quasicompact as an operator on the space of functions o f bounded variation on R-2 and its isolated eigenvalues (including multipli cities) are just the reciprocals of the poles of the dynamical zeta functio n of T. In higher dimensions the result remains true under an additional ge nerically satisfied transversality assumption.