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

Abstract

Let X\subset\mathbb{R}^2 be a finite union of bounded polytopes and let T:X\to X be piecewise affine and eventually expanding. Then the Perron–Frobenius operator \mathcal{L} of T is quasicompact as an operator on the space of functions of bounded variation on \mathbb{R}^2 and its isolated eigenvalues (including multiplicities) are just the reciprocals of the poles of the dynamical zeta function of T. In higher dimensions the result remains true under an additional generically satisfied transversality assumption.