Geodesic Flow Cycle Track Average

Ergodic theory and dynamical system is one of the most successful branch of mathematics since the 20th century. It also is an important part of nonlinear science.This paper addresses the understanding of asymptotic behaviour of periodic orbits for geodesic flows. With symbolic dynamics, hyperbolic flows can be modelled by suspended flows. Geodesic flows are special hyperbolic flows. By defining transfer operator, discussing its contractivity and using the properties of L-function, we derive the asymptotic expressions of orbits averages for geodesic flows.