The Variance of Arithmetic Measures Associated to Closed Geodesics on the Modular Surfaces
نویسندگان
چکیده
Contents 1. Introduction 2 1.1. Equidistribution theorems for closed geodesics 2 1.2. The modular surface 3 1.3. Results 5 1.4. Remarks 7 1.5. Plan of the paper 9 1.6. Acknowledgments 9 2. Background on periods 9 2.1. The upper half-plane and its unit tangent bundle 9 2.2. Quotients 11 2.3. A correspondence with binary quadratic forms 12 2.4.
منابع مشابه
The Variance of Arithmetic Measures Associated to Closed Geodesics on the Modular Surface
We determine the variance for the fluctuations of the arithmetic measures obtained by collecting all closed geodesics on the modular surface with the same discriminant and ordering them by the latter. This arithmetic variance differs by subtle factors from the variance that one gets when considering individual closed geodesicswhen ordered by their length. The arithmetic variance is the same one...
متن کاملThe length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces
We examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M . In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using techniques from analytic number theory, we address the following problems: Is the commensurability class of...
متن کاملNumerical Treatment of Geodesic Differential Equations on Two Dimensional Surfaces
This paper presents a brief instructions to nd geodesics equa-tions on two dimensional surfaces in R3. The resulting geodesic equations are solved numerically using Computer Program Matlab, the geodesics are dis-played through Figures.
متن کاملAn arithmetic property of the set of angles between closed geodesics on hyperbolic surfaces of finite type
For a hyperbolic surface S of finite typewe consider the set A(S) of angles between closed geodesics on S. Our main result is that there are only finitely many rational multiples of π in A(S).
متن کاملNon-simple Geodesics in Hyperbolic 3-manifolds
Chinburg and Reid have recently constructed examples of hyperbolic 3manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds. For hyperbolic 3-manifolds which contain an immersed totally geodesic surf...
متن کامل