LAUSR

We compute the asymptotic growth rate of the number $$N({{\mathcal {C}}}, R)$$N(C,R) of closed geodesics of length $$\le R$$≤R in a connected component $${{\mathcal {C}}}$$C of a stratum of quadratic differentials. We prove that, for any $$0\le \theta \le 1$$0≤θ≤1, the number of closed geodesics $$\gamma $$γ of length at most R such that $$\gamma $$γ spends at least $$\theta $$θ-fraction of its time outside of a compact subset of $${{\mathcal {C}}}$$C is exponentially smaller than $$N({{\mathcal {C}}}, R)$$N(C,R). The theorem follows from a lattice counting statement. For points x, y in the moduli space $${{{\mathcal {M}}}(S)}$$M(S) of Riemann surfaces, and for $$0 \le \theta \le 1$$0≤θ≤1 we find an upper-bound for the number of geodesic paths of length $$\le R$$≤R in $${{\mathcal {C}}}$$C which connect a point near x to a point near y and spend at least a $$\theta $$θ-fraction of the time outside of a compact subset of $${{\mathcal {C}}}$$C.