Precise counting results for closed orbits of Anosov flows. (English) Zbl 0992.37026

Anantharaman, Nalini

Ann. Sci. Éc. Norm. Supér. (4) 33, No. 1, 33-56 (2000).

The author studies the problem of counting closed geodesics on a negatively curved surface \(S\) according to their length and under the following homological constraint: Let \(\xi\in H_1(S,\mathbb{R})\), \(\alpha\in H_1 (S,\mathbb{Z})\) be fixed homology classes, let \(\delta>0\) and for \(T>0\) define \(\pi (\xi,\alpha, \delta,T)\) to be the number of closed geodesics \(\gamma\) of length \(\ell(\gamma) \in[T,T+ \delta]\) on \(S\) and such that the integer part \([\gamma]\) of the homology class of \(\gamma\) equals \(\alpha+ [T\xi]\). The main result of the paper is a full asymptotic expansion for the function \(\pi(\xi, \alpha,\delta,T)\) as \(T\to\infty\).
The leading term of this expansion was given earlier by M. Babillot and F. Ledrappier [Ergodic Theory Dyn. Syst. 18, 17-39 (1998; Zbl 0915.58074)]. The nontrivial case is when the class \(\xi\) is contained in the interior of the compact convex subset \(C\) of \(H_1(S,\mathbb{R})\) which is defined via integration of closed 1-forms with respect to the Borel probability measures on the unit tangent bundle \(T^1S\) of \(S\) which are invariant under the geodesic flow. There is a natural analytic function \(H\) on \(C\) which assigns to a class \(\xi\) the supremum of the measure theoretic entropies of all measures defining \(\xi\). The leading term of the expansion for \(\pi(\xi, \alpha,\delta,T)\) whenever \(\xi\in\overset \circ C\) is then given in terms of exponentials of \(H\) and its derivatives.
The present paper determines in a similar way the other terms of the expansion. The main tool is the use of transfer operators to study the Fourier-Laplace transform of the function \(\pi\). The author then uses as the major new ingredient a recent spectral theorem of D. Dolgopyat [Ann. Math. (2) 147, 357-390 (1998; Zbl 0911.58029)]. The details of the argument are rather subtle.

Reviewer: Ursula Hamenstädt (Bonn)

MSC:

37D40

Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)

Keywords:

closed orbits of Anosov flows; negatively curved surface; homological constraint; number of closed geodesics; asymptotic expansion; transfer operators

Citations:

Zbl 0915.58074; Zbl 0911.58029

References:

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