Pairs of closed geodesics in metric graphs

Abstract In this paper we study the counting pairs of closed geodesics on metric graphs. We obtain an asymptotic for the number of pairs of closed geodesics ordered by their word lengths on the metric graph, such that the difference of their geometric lengths lies in an interval that is allowed to shrink at a specific rate and positioned arbitrarily on the real line to obtain a uniform result. This result gives the same asymptotic if pairs of closed geodesics are ordered by their word length of the corresponding conjugacy class with respect to a free generating set of the fundamental group of the metric graph. The techniques we use in our study include coding metric graphs by subshifts of finite type and the concepts from the thermodynamic formalism that appear in the ergodic theory of these systems. In particular, we use the spectral properties of transfer operators and their relationship to pressure and entropy.