Selberg zeta functions, cuspidal accelerations, and existence of strict transfer operator approaches

TL;DR

This paper constructs a family of transfer operators for geometrically finite hyperbolic orbisurfaces, linking their Fredholm determinants to the Selberg zeta function, and provides an algorithmic method to achieve uniform expansion.

Contribution

It introduces a uniform, algorithmic construction of transfer operators for hyperbolic orbisurfaces that accurately model the geodesic flow and relate to the Selberg zeta function.

Findings

Transfer operator families are nuclear of order zero on Banach spaces.

The construction includes finite-dimensional twists with non-expanding cusp monodromy.

The approach yields a faithful, uniformly expanding dynamical system model.

Abstract

For geometrically finite non-compact developable hyperbolic orbisurfaces (including those of infinite volume), we provide transfer operator families whose Fredholm determinants are identical to the Selberg zeta function. Our proof yields an algorithmic and uniform construction. This construction is initiated with an externally provided cross section for the geodesic flow on the considered orbisurface that yields a highly faithful, but non-uniformly expanding discrete dynamical system modelling the geodesic flow. Through a number of algorithmic steps of reduction, extension, translation, induction and acceleration, we turn this cross section into one that yields a still highly faithful, but now uniformly expanding discrete dynamical system. The arising transfer operator family is nuclear of order zero on suitable Banach spaces. In addition, finite-dimensional twists with non-expanding cusp monodromy can be included.