Zeta functions and transfer operators for multidimensional piecewise affine and expanding maps - Buzzi, Jérôme; Keller, Gerhard (Ergodic Theory and Dynamical Systems, 2001)
Let X ⊂ ℝ2 be a finite union of bounded polytopes and let T : X → X be piecewise affine and eventually expanding. Then the Perron-Frobenius operator £ of T is quasicompact as an operator on the space of functions of bounded variation on ℝ2 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.
UnivIS-Import:2015-03-09:Pub.2001.nat.dma.pma29.zetafu