Invariant relations for the Bowen-Series transform

Abstract

Consider the Bowen-Series transform T T associated with an even corners fundamental domain of finite volume for some Fuchsian group Γ \Gamma . We prove a generic invariance result that abstracts Series’ orbit-equivalence theorem to families of relations on the unit circle. Two applications of this result are developed. We first prove that T T satisfies a strong-orbit equivalence property, which allows to identify its hyperbolic periodic orbits with primitive hyperbolic conjugacy classes of Γ \Gamma . Then, we show thanks to the invariance theorem that the eigendistributions for the eigenvalue 1 1 of the transfer operator of T T with spectral parameter s ∈ C s \in \mathbb {C} are in bijection with smooth bounded eigenfunctions for the eigenvalue s ( 1 − s ) s(1-s) of the hyperbolic Laplacian on the quotient D / Γ \mathbb {D} / \Gamma .