§ 0. Introduction

Ve will be concerned in this paper with a pair of closely related

problems of asymptotic analysis on a (non-compact) finite area hyperbolic

surface T\h. Here, h is the hyperbolic plane and T is a cofinite discrete

subgroup of G = PSL2(R). The compact case was studied in [Z.2], of which this

paper is a continuation.

The first problem is to determine the precise asymptotic distribution of

closed geodesies in the unit tangent bundle. For general algebraic Anosov

flows on compact quotients, it was proved by Bowen in 1972 [B] that closed

orbits tend on average to become uniformly distributed with respect to Haar

(or, Liouville) measure as the period tends to infinity. Bowen's theorem has

since been generalized and refined by Parry, Pollicott and others, using the

method of symbolic dynamics. Here, we approach Bowen's equidistribution

theory of closed geodesies through harmonic analysis on T\G (i.e.

representation theory and trace formulae). Our method yields asymptotic

expansions and sharp remainder terms which are inaccessible to dynamical

methods.

The general problem of relating harmonic analysis on T\G to dynamical

properties of the geodesic flow is of course well known. Suffice it to recall

that the ergodic and mixing properties of the flow have long been studied this

way. Unlike these properties, the equidistribution of closed geodesies is not

a spectral invariant of the flow. Hence, it requires something more than pure

representation theory, It turns out that the additional concern is the dual

asymptotic behavior of the eigenfunctions of the Laplace operator as the

eigenvalue tends to infinity. Due to the presence of continuous spectrum in

non-compact quotients, the asymptotic behavior of eigenfunctions is a good

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