Twisted Dirac operators and dynamical zeta functions

2015/07/21 by Polyxeni Spilioti, Spilioti, Polyxeni
Mathematics · #FOS: Mathematics #Spectral Theory (math.SP) #math.SP

paper · pdf · doi:10.48550/arxiv.1507.05932

This paper is part of the author's phd thesis and is a follow-up of the paper arXiv:1506.04672 for the case, where the irreducible representation σof M in not invariant under the action of the restricted Weyl group

arxiv created 2015/09/25 · arxiv updated 2015/09/29

Abstract

In this paper, we consider the dynamical zeta functions of Ruelle and Selberg associated with the geodesic flow of a compact hyperbolic odd dimensional manifold X. These functions are initially defined on one complex variable s in some right half-plane of ℂ. Our goal is the continue meromorphically the dynamical zeta functions to the whole complex plane, using the Selberg trace formula for arbitrary, not necessarily unitary, representations χ of the fundamental group. First, we prove a trace formula for the integral operator D^\sharp_χ(σ)e^-t(D^\sharp_χ(σ))², induced by the twisted Dirac operator D^\sharp_χ(σ) on X. Then we use these results to establish the meromorphic continuation of the dynamical zeta functions to ℂ.

Citations

• Ruelle and Selberg zeta functions for non-unitary twists

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• Mayer’s transfer operator approach to Selberg’s zeta function