A local limit theorem for closed geodesics and homology

Abstract

Abstract. In this paper, we study the distribution of closed geodesics on a compact negatively curved manifold. We concentrate on geodesics lying in a prescribed homology class and, under certain conditions, obtain a local limit theorem to describe the asymptotic behaviour of the associated counting function as the homology class varies. IntroductionLet M be a compact smooth Riemannian manifold with first Betti number k > 0 and with negative sectional curvatures. Suppose also that either dim M = 2 or that M is 1/4-pinched, i.e., the sectional curvatures all lie in an interval [−κ, −κ/4], for some κ > 0. Such a manifold contains a countable infinity of prime closed geodesics.(We say that a closed geodesic is prime if it is not a nontrivial multiple of another closed geodesic.) In this paper we are interested in how these closed geodesics are distributed with respect to homology.The homology group H 1 (M, Z) is isomorphic to Z k ⊕Tor, where Tor is the (finite) torsion subgroup. In this paper, it will be convenient to consider the torsion-free part of the homology, H 1 (M, Z)/Tor. We shall, in fact, assume that an isomorphism has been fixed and write Z k instead of H 1 (M, Z)/Tor. For a typical (prime) closed geodesic γ on M , let l(γ) denote its length and [γ] ∈ H 1 (M, Z)/Tor = Z k the torsion-free part of its homology class. For α ∈ Z k , define a counting functionRecently, several papers have studied the asymptotics of this function as T → ∞. In particular, Anantharaman [1] and Pollicott and Sharp [17] have shown that there exist constants C 0 > 0, independent of α, and C n (α), n ≥ 1, such that, for any N ≥ 1, we have the asymptotic expansionwhere h > 0 denotes the topological entropy of the geodesic flow over M . (In fact, the expansion in [17] contains some extra terms corresponding to powers of T −1/2 ; a more careful analysis, as carried out in [11], shows that these terms vanish.) Furthermore, Kotani [11], has studied the dependence of the coefficients C n (α) on

Anosov flows, growth rates on covers and group extensions of subshifts

“…The extension to Anosov flows was made by Katsuda and Sunada [22] under the assumption that the winding cycle for the measure of maximal entropy vanishes. Results giving more detailed information about the asymptotic behaviour are contained in [1,25,33] and [40].…”

Equidistribution of geodesics on homology classes and analogues for free groups

“…The proof of Theorem 1.1 uses the Selberg trace formula with characters as used in [17]. We combine this approach with ideas from [24], where the stationary phase argument used in [17] is simplified to make more transparent the dependence on the homology class. This idea seems to go back at least to [19].…”

“…In particular the leading term, in contrast to the lower order terms, does not depend on α. The dependence on α in the lower order terms has been considered in [12,24], but the results are not strong enough to handle equidistribution for sets of positive density by simply summing up asymptotics. For sets of positive natural density Theorems 1.1 gives precise information about the asymptotic behavior of π A (T ).…”

“…The fact that we are considering surfaces of fixed negative sectional curvature −1 is not essential. If M has variable negative curvature we can combine the ideas of this paper with the ideas developed by Sharp [24] to prove Theorem 1.1 in this case. Instead of taking [17, (2.37) Lemma 2.1, 2.2] as a starting point as we do in this paper, one may take [24, Propositions 1, 2, and Lemma 1] as a starting point and use variations of our techniques to prove such a result (see [6]).…”

Uniform estimates for closed geodesics and homology on finite area hyperbolic surfaces

“…In particular, they have understood the dependence of the asymptotics on the ranks of the cusps. A version of Theorem 1 for compact variable negative curvature surfaces was obtained in [18]; however, in the constant curvature case the result may be more easily deduced directly from the analysis in [15].…”