§ 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
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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