COUNTING CLOSED GEODESICS IN ORBIT CLOSURES

The moduli space of Abelian differentials on Riemann surfaces admits a natural action by $\mathrm{SL}\left(2,\mathbb{R}\right)$.  This thesis is concerned with using the classification of invariant measures for this action due to Eskin and Mirzakhani, to study the growth of closed geodesics in the support of an invariant measure coming from the closure of an orbit for the $\mathrm{SL}\left(2,\mathbb{R}\right)$-action. These are always subvarieties of moduli space. For $0 \leq \theta \leq 1$, we obtain an exponential bound on the number of closed geodesics in the orbit closure, of length at most $R$, that have at least $\theta$-fraction of their length in a region with short saddle connections.