Hyperbolic geometry and Blaschke-functions

Abstract. By the Fermat-principle, assuming a refractive index of \(1/(1 - |z|^2) \ (z \in \mathbb{D})\), light traverses along hyperbolic lines. These lines can be described by Blaschke-functions and are also the solutions to a variational problem. In this paper we describe a generalization of the above problem, by taking a refractive index of \((1 - |B_{\mathbf a}(z)|^2)/|B^{'}_{\mathbf a}(z)|\), where \(B_{\mathbf a}\) denotes a Blaschke-product. We show that the solutions of the generalized problem are inverse images of Blaschke-products who induce an \(n\)-fold geometry on \(\mathbb{D}\). In the special case of \(n=2\), we provide the explicit form of the solutions.