Asymptotic Estimations For Closed Orbits In Homology Classes On Hyperbolic Flows

Ergodic theory and dynamic system is one of the most successful branches of mathematics in the 20 th century and it has extremely widespread application in other mathematical fields,such as Functional Theory,Combinatorial Mathematics,Computational Mathematics and so on.The most basically prime theorem and the approximate of orbits on manifolds attracted great attention from plenty of scholars and lots of important developments are achived.The distribution of closed orbits for hyperbolic flow is very active recently.This paper is based on the asymptotic expansions of hyperbolic flow and homology classes on hyperbolic manifolds.The asymptotic counting results concerning pairs of prime geodesics with fixed homology difference is studied.Major work is to discuss the asymptotic expansion about the homology difference ? sastisfy the specific condition.The asymptotic estimations for closed orbits on hyperbolic flows,we alaways establish ? function,and discussion its analyticity.This paper is based on the method of Sarnak and by use of Selberg trace formula,we obtain a full asymptotic expansion for pairs of prime closed orbits with fixed homology difference include error term.The results in this paper are partly extended and improved the existed theorems.