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FORMULAE FOR RESIDUES OF DYNAMICAL ZETA FUNCTIONS

In [9], Jorgenson and Kramer considered geometric estimates for (0.2) using spectral methods. By contrast, I used we used dynamical methods introduced by Ruelle. We present the proof of this theorem, along with the explicit expressions for a k and a k in section 2, based on a dynamical viewpoint contained in a fundamental article of Ruelle [13]. Let f : M → R be a real analytic function (for example, an eigenfunction of the Laplacian) and write λf (τ) = ∫ λ(τ) 0 f(φtxτ )dt for the weighting of the orbit by f . In 1972, Bowen proved a natural equidistribution result for the closed geodesics that covers this case [3], i.e., ∑ λ(τ)≤T λf (τ) ∑ λ(τ)≤T λ(τ) → ∫ fdμ, as T → +∞,