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
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