4 STEVEN ZELDITCH

E /i (f) and the quotients /iT(f) = (f

r

(l,T) _ 1 f

r

(f,T). The basic result of

L T

7

1 1 1

V

Bowen is:

(0.1) Theorem ([B]) lim /*T(f) = /*(f),

T-HD

l

where fi is normalized Haar measure: /*(f) =

v o

i/r\n\ f d/»- It is not hard

/*T(f)

to show that Bowen's theorem implies that theindividual terms /^ »

/j(f) forall buta sparse subsequence (relative tocounting measure ontheset

of lengths).

Bowen's equidistribution theorem was actually proved for Anosov flows

arising one-parameter subgroups ofLie group G ona compact quotient T\G/K (an

algebraic Anosov flow). It can begeneralized toAxiom A flows, restricted to

basic sets, inthe form: Closed orbits are uniformly distributed relative to

the measure ofmaximal entropy. The article [P] contains a proof, together

with fuller historical remarks andreferences.

Our approach (asin [Z.2]) isto study the sums /*T(f) bymeansof

Selberg-type trace formulae. Todothis, weneed f tobeanautomorphic form

in the sense above. Forsuch forms f,wegive asymptotic expansions for /im(f)

which are reminiscent oftheexpansions ofHuber, Selberg and many othersfor

the case f = 1. This istheso-called

(0.2) Prime Geodesic Theorem: Let T c PSL2(R) bea cofinite subgroup and let

fr(T) = S L . Then:

1 L7 T 7