Counting closed geodesics in strata

• 成果类型: Article
• 署名作者: Eskin, Alex; Mirzakhani, Maryam; Rafi, Kasra
• 署名单位: University of Chicago; Stanford University; University of Toronto
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-018-0832-y
• 发表日期: 2019
• 页码: 535-607
• 关键词: connected components; teichmuller flow; ergodic averages; MODULI SPACES; CONSTRUCTION; asymptotics; geometry; differentials; deviation; DYNAMICS

摘要

We compute the asymptotic growth rate of the number N(C,R) of closed geodesics of length R in a connected component C of a stratum of quadratic differentials. We prove that, for any 01, the number of closed geodesics of length at most R such that spends at least -fraction of its time outside of a compact subset of C is exponentially smaller than N(C,R). The theorem follows from a lattice counting statement. For points x,y in the moduli space M(S) of Riemann surfaces, and for 01 we find an upper-bound for the number of geodesic paths of length R in C which connect a point near x to a point near y and spend at least a -fraction of the time outside of a compact subset of C.