Fix a surface X of negative curvature and for L > 0, let G(L) denote the set of closed geodesics in X of length at most L.
The statistical behavior of the homology of a closed geodesic alpha, chosen randomly uniformly from G(L), has been a subject of interest since the 80s. Local limit theorems for the homology were first established by Sarnak-Phillips and Katsuda-Sunada, and later generalized by Lalley, Pollicott-Sharp, and others.
In this talk we inspect the same question for S(L), the set of simple closed geodesics in X of length at most L. We were able to obtain CLT-type results for the norm of homology of a randomly chosen curve in S(L). We discuss two main steps which reduce the desired CLT to a CLT for the Kontsevich-Zorich cocycle, obtained by Forni-Saqban.
Joint work with F. Arana-Herrera.