A Local Limit Theorem for Closed Geodesics and Homology

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. 0. Introduction Let M be a compact smooth Riemannian manifold with first Betti number k > 0 and with negative sectional curvatures. Suppose also that either dimM = 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 non-trivial multiple of another closed geodesic.) In this paper we shall be interested in how these closed geodesics are distributed with respect to homology. The homology group H1(M,Z) is isomorphic to Z ⊕ Tor, where Tor is the (finite) torsion subgroup. In this paper, it will be convenient to consider the torsion-free part of the homology, H1(M,Z)/Tor. We shall, in fact, assume that an isomorphism has been fixed and write Z instead of H1(M,Z)/Tor. For a typical (prime) closed geodesic γ on M , let l(γ) denote its length and [γ] ∈ H1(M,Z)/Tor = Z the torsion-free part of its homology class. For α ∈ Z, define a counting function π(T, α) = #{γ : l(γ) ≤ T, [γ] = α}. Recently, 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 C0 > 0, independent of α, and Cn(α), n ≥ 1, such that, for any N ≥ 1, we have the asymptotic expansion π(T, α) = e T k/2+1 ( C0 + C1(α) T + C2(α) T 2 + · · ·+ CN (α) TN +O ( 1 TN+1 )) , (0.1) The author was supported by an EPSRC Advanced Research Fellowship. Typeset by AMS-TEX 1 where h > 0 denotes the topological entropy of the geodesic flow over M . (In fact, the expansion in [17] contained 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 Cn(α) on α = (α1, . . . , αk), showing that they may be expressed as polynomials of degree 2n in α1, . . . , αk. In the special case of manifolds of constant negative curvature, the expansion (0.1) was obtained by Phillips and Sarnak [14] and, independently, Katsuda and Sunada [9] obtained the leading term. For manifolds of variable negative curvature (without the pinching condition) the leading term of (0.1) was obtained by Katsuda [8], Lalley [12] and Pollicott [15]. Analogous results for manifolds with cusps have been obtained by Epstein [6] and Babillot and Peigné [3]. In this note, we take a slightly different view and address the question of the behaviour of π(T, α) when α is allowed to vary independently of T . We obtain the following “local limit theorem”. Theorem 1. Let M be a compact smooth Riemannian manifold with first Betti number k > 0 and with negative sectional curvatures. Suppose also that either dimM = 2 or that M is 1/4-pinched. Then there exists a symmetric positive definite real matrix D such that, lim T→∞ ∣∣∣∣hσkT k/2+1 ehT π(T, α)− 1 (2π)k/2 e−〈α,D−1α〉/2T ∣∣∣∣ = 0, uniformly in α ∈ Z, where σ > 0 satisfies σ = detD. Here, 〈·, ·〉 denotes the usual inner product 〈x, y〉 = x1y1+· · ·+xkyk. As a particular consequence, we recover the leading term of the expansion (0.1), with C0 = h−1σ−k(2π)−k/2. Theorem 1 appears not to have been stated even for manifolds of constant negative curvature, although, in that case, the result can be easily deduced from the analysis contained in [14]. Remarks. (i) If we take the torsion part of H1(M,Z) into account then we need to modify Theorem 1 to read lim T→∞ ∣∣∣∣hσkT k/2+1 ehT π(T, α)− 1 (#Tor) (2π)k/2 e−〈αF ,DαF 〉/2T ∣∣∣∣ = 0, uniformly in α ∈ H1(M,Z), where αF ∈ Z denotes the torsion-free part of α ∈ H1(M,Z). (ii) In Kotani’s formula for the term Cn(α)/T in (0.1), the highest power of α makes a contribution 1 (2π)k/2hσk 1 n! ( −〈α,D −1α〉 2T )n . As observed by Kotani in [11], formally summing these contributions gives the expression e−〈α,D −1α〉/2T /(2π)hσ. Theorem 1 should be compared with the results on homology classes varying linearly in T obtained by Lalley [12] and Babillot and Ledrappier [2]. Using these results, one can show that, for δ > 0 sufficiently small, lim T→∞ sup ||α||≤δT ∣∣∣∣ T k/2+1 C(α/T )eH(α/T )T π(T, α)− 1 ∣∣∣∣ = 0, (0.2)