Geodesic length spectrum on compact Riemann surfaces
The variation of the geodesic length spectrum of a compact Riemannian surface of genus 2 is studied via Fenchel-Nielsen coordinates, the structure of Fuchsian groups and Markov symbolic dynamics on the boundary of the Poincaré model of the Lobachevsky geometry. Basing on the correspondence between the geometry of the surface and the algebraic structure of the fundamental group, a precise relation between closed geodesics and periodic sequences in the symbolic system is deduced. This yields a way to compute the length spectra and implies formulae for exponential growth rate of the fundamental group with a particular choice of the generators and the first terms of the growth series.
scientific article; zbMATH DE number 5791202
scientific article