Invariant relations for the Bowen-Series transform

Abstract

Abstract. Consider the Bowen-Series transform T associated with an even corners fundamental domain of finite volume for some Fuchsian group Γ. 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 satisfies a strong-orbit equivalence property, which allows to identify its hyperbolic periodic orbits with primitive hyperbolic conjugacy classes of Γ. Then, we show thanks to the invariance theorem that the eigendistributions for the eigenvalue 1 of the transfer operator of T with spectral parameter s ∈ C are in bijection with smooth bounded eigenfunctions for the eigenvalue s(1 − s) of the hyperbolic Laplacian on the quotient D/Γ.

Duality between eigenfunctions and eigendistributions of Ruelle and Koopman operators via an integral kernel

“…Distributions related to eigenfunctions and the involution kernel appeared in [18], [19], [20] and [14] (plus, in a non rigorous form, in [8] and [9]). Moreover, distributions have been intensively studied, starting with [3], through anisotropic 1 We will use the two notations y → M e A ⋆ (ya) ϕ(ya)da and M e A ⋆ (•a) ϕ(•a)da indistinctly to indicate the needed function.…”