Fredholm Determinant for Piecewise Linear Transformations on a Plane

TLDR: In this paper, the spectrum of a piecewise linear expanding map on a finite union of polygons in a convex linear space is determined by Fredholm matrices, and new signed symbolic dynamics are defined by using screens.

Abstract: Certain piecewise linear expanding maps on a finite union of polygons in $\mathbf{R}^2$ are considered. The Perron-Frobenius operator associated with a map is considered on a locally convex linear space which is an extension of the space of bounded variation functions, and the spectrum of it is determined by Fredholm matrices. New signed symbolic dynamics are defined by using screens, and the Fredholm matrices are constructed by renewal equations on this signed symbolic dynamics.