Boundary maps and Fenchel-Nielsen coordinates

Abstract

We consider a genus $2$ surface, $M$, of constant negative curvature and we construct a $12$-sided fundamental domain, where the sides are segments of the lifts of closed geodesics on $M$ (which determines the Fenchel-Nielsen-Maskit coordinates). Then we study the linear fractional transformations of the side pairing of the fundamental domain. This construction gives rise to $24$ distinct points on the boundary of the hyperbolic covering space. Their itineraries determine Markov partitions that we use to study the dependence of the Lyapunov exponent and length spectrum of the closed geodesics with the Fenchel-Nielsen coordinates.}