A symbolic proof of a theorem of Margulis on geodesic arcs on negatively curved manifolds.

Introduction. Let V be a compact manifold with negative sectional curvatures and, given two points x, y [element of] V, we denote by [rho] xy(t) the number of geodesic segments with length at most t > 0, which start at the point x and finish at point y (cf. Figure 1 [a]). In 1970, Margulis announced the result that

[Mathematical Expression Omitted]

where h > 0 is the topological entropy of the associated geodesic flow and [C.sub.xy] is a constant (which is independent of t, but may depend on the points x,y [element of] V) [5].

Unfortunately, although the proof of this result appears (in Russian) in Margulis' thesis, it has never been published. The purpose of this note is to give a proof of a more general result. It is more natural to formulate this generalization in the broader context of hyperbolic flows [phi]t : M [right arrow] M on a compact manifold, restricted to a hyperbolic set A. Let us assume that the unstable bundle for [phi] [vertical bar] A has dimension k and the stable bundle has dimension l. Let [C.sub.1], [C.sub.2], [subset] M be submanifolds (without boundary) of dimensions l and k, respectively. Finally, we denote by [PC.sub.1] [C.sub.2] (t) the number of [phi]-orbit segments with length at most t > 0 which start in [C.sub.1] and finish in [C.sub.2] (cf. Figure 1 [b]). Our main result is the following.

Theorem. Provided [C.sub.1] and [C.sub.2] are sufficiently small, and our flow is weak mixing, we have that

[Mathematical Expression Omitted]

where h = h([phi]) > 0 is the topological entropy of the flow [phi] [vertical bar] A and C = C([C.sub.1], [C.sub.2]) > 0 is independent of t.

In this theorem, "sufficiently small" should be understood as small with respect to the expansiveness constant.

Margulis' result can easily be recovered from the above theorem by taking [phi] to be the geodesic flow on the sphere bundle M = SV of the manifold V and the choices [C.sub.1] = [S.sub.x] V and [C.sub.2] = [S.sub.y] V. Margulis also gives an explicit solution expression for [C.sub.xy], for geodesic flows, and we provide the analogous expression for the constant C = C([C.sub.1], [C.sub.2]), for hyperbolic flows. In particular, we recover Margulis' observation that for locally symmetric manifolds the constants [C.sub.xy] are independent of the points x, y [element of] M.

Our proof follows the same general lines as that of the proof of the analogous result for closed orbits of Axiom A flows in [6]. However, in the present case we need to develop some additional techniques.

Acknowledgment. I was introduced to this problem by Laurent Guillope. I am grateful to him for several illuminating conversations, and for telling me about his own unpublished work.

1. Preliminaries. Let [phi]: M [right arrow] M be a differentiable flow on a compact manifold. We call closed [phi]-invariant set A [subset] M a basic set if

(i) [phi] [vertical bar] A is hyperbolic i.e. There exists a continuous splitting T [conjunction] M = [E.sup.o] + [El.sup.u] + [E.sup.s] where [E.sup.o] is tangent to the flow direction and constants C, [lambda] > 0 such that

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted]

(ii) A contains a dense orbit

(iii) the closed orbits in [phi] [vertical bar] A are dense

(iv) There exists a neighborhood A [subset] U such that [conjunction] = [Mathematical Expression Omitted]

We want to model this by a symbolic flow, and we proceed as follows. Let A be a k x k transitive matrix with entries 0 or 1 and then define

[Mathematical Expression Omitted]

with the metric

[Mathematical Expression Omitted]

(where [delta](i,j) = 1 if i = j, and 0 otherwise).

We call the homeomorphism [sigma] : [X.sub.A] [right arrow] ([Mathematical Expression Omitted] the subshift of finite type. Given a Holder function f : [X.sub.A] [right arrow] [R.sup.+] we defined

[Mathematical Expression Omitted]

where we identify x,f(x)) [is nearly equal to] ([sigma] x, 0). We define the suspended flow by [Mathematical Expression Omitted] (x, u + t), subject to the identifications.

The main technical result we shall need is the following.

Proposition 1. Given [element of] > 0, there exists a suspended flow [Mathematical Expression Omitted] with [double vertical bar] f [double vertical bar] [infinity] [element of] and a continuous surjective map [pi] : [X.sub.A.sup.f] [right arrow] such that

(i) [Mathematical Expression Omitted], for all t [elemenat of] R

(ii) [T.sub.i] = [pi] {x [element of] [X.sub.A] : [x.sub.0] = i} is contained in a transverse section of diameter at most [element of], for i = 1,...., k

(iii) [phi] is one-one on a dense Baire set

(iv) [phi] is an isomorphism relative to the measures of maximal entropy

(v) We can assume that the function f is a function of the future" (i.e. f(x) = f(y), whenever [x.sub.n] = …