Geodesic length spectrum on compact Riemann surfaces

In this paper we use techniques linking combinatorial structures (symbolic dynamics) and algebraic–geometric structures to study the variation of the geodesic length spectrum, with the Fenchel–Nielsen coordinates, which parametrize the surface of genus τ=2$\tau =2$. We explicitly compute length spectra, for all closed orientable hyperbolic genus two surfaces, identifying the exponential growth rate and the first terms of a growth series.