A central limit theorem for periodic orbits of hyperbolic flows. Issue 1 (2nd January 2021) - Cantrell, Stephen; Sharp, Richard
Abstract : We consider a counting problem in the setting of hyperbolic dynamics. Let ϕ t : Λ → Λ be a weak-mixing hyperbolic flow. We count the proportion of prime periodic orbits of ϕ t, with length less than T, that satisfy an averaging condition related to a Hölder continuous function f : Λ → R . We show, assuming an approximability condition on ϕ, that as T → ∞, we obtain a central limit theorem. The proof uses transfer operator estimates due to Dolgopyat to provide the bounds on complex functions that we need to carry out our analysis. We can then use contour integration to obtain the asymptotic behaviour which gives the central limit theorem.