Asymptotic Estimation Of The Closed Orbit Of The Hyperbolic Flow On The Riemann Surface

Ergodic theory and topological dynamical system is one of the most promising branch of the 20th century,and it is widely used in other fields of mathematics,such as number theory,functional analysis and topology.The distribution and its asymptotic estimation of hyperbolic flow cycle are important research directions.Distribution of the periodic orbit of the hyperbolic flow on Riemann surface is a classic problem.Closed geodesics are special periodic of hyperbolic flow.The distribution of periodic of hyperbolic flow has already become an important research direction.Based on the distribution of periodic orbits and the theory of gradual estimation,this paper mainly solves the problem of the double periodic orbit counting in the negative cross section curvature manifolds.The length of the periodic orbit difference respectively are fixed in the interval[z—a,z + a]and discrete interval[z + ?na,z + ?nb],the word length is less than the setting value,we studies distribution of the periodic orbit of the hyperbolic flow on Riemann surface.By studying symbolic dynamics,Markov section,by use of the sequence of function and Zeta function research,using Fourier inverse transform,we obtained the estimation formulae of periodic orbits of hyperbolic flow in pairs.