On the Hannay-Ozorio de Almeida sum formula
نویسندگان
چکیده
The Hannay-Ozorio de Almeida sum formula is a useful principle in the study of the distribution of closed orbits for Hamiltonian flows [7]. Roughly speaking, it asserts that an appropriately weighted sum of measures supported on periodic orbits converges to the physical measure as the periods become large. This formula was originally introduced and used in the study of Quantum Chaos. In particular, Berry used the so-called diagonal approximation and the Hannay-Ozorio de Almeida sum rule to determine the asymptotics of the spectral form factor, which is the Fourier transform of the two-point correlation function for the eigenvalues of the Laplacian [1], [8], [6]. The traditional setting is in the context of Hamiltonian flows, which include the canonical example of geodesic flows on negatively curved manifolds. Let us now give a brief description of the formula in the context of a C attracting hyperbolic flows φt : Λ → Λ, where the attractor Λ is contained in a Riemannian manifold M . Let τ denote a (prime) periodic orbit and let λ(τ) denote its least period. Let f : Λ → R be a continuous function, then we can introduce a weighted period λf(τ) = ∫ λ(τ) 0 f(φtxτ )dt, where xτ ∈ τ . In particular, if we define the expansion coefficient E : Λ → R by
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