Invariant relations for the Bowen-Series transform

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