Precise asymptotics of the length spectrum for finite-geometry Riemann surfaces. (English) Zbl 1073.37021
The author builds on the work of L. Guillopé and M. Zworski [Ann. Math. (2) 145, 597–660 (1997; Zbl 0898.58054) and Geom. Funct. Anal. 9, 1156–1168 (1999; Zbl 0947.58022)] concerning the wave trace formula and the bounds on the number of resonances. He proves a prime orbit theorem for an arbitrary Riemann surface of finite geometry and infinite volume. That is, let \(m = \Gamma\setminus H^2\) be a geometrically finite Riemann surface such that \(\delta>{1\over 2}\) and \(f\geq 1\), then as \(T\to+\infty\),
\[
N(T)= \text{li}(e^{\delta T})+ \sum^p_{k-1} \text{li}(e^{\alpha_kT})+ O(e^{(\delta/2+ 1/4)T}),
\]
where the exponents \(1/2< \alpha \leq\alpha_{p-1}\leq\cdots < \alpha_0=\delta\) are related to the point spectrum of \(\Delta_M\) included in \((0,1/4)\) by the formula \(\alpha_k(1- \alpha_k)=\lambda_k\).
Reviewer: Chiung-Jue Sung (Minhsiung)
MSC:
37C27 | Periodic orbits of vector fields and flows |
37C35 | Orbit growth in dynamical systems |
37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
37F30 | Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010) |
58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |