Large deviations upper bound for the Teichmueller flow and applications

Based on the work of A. Eskin, M. Mirzakhani and A. Mohammadi on invariant measures of the $SL(2,R)$ action on translation surfaces, we provide an upper bound for deviations of Birkhoff integrals of the Teichmueller flow. It is a refinement of previous work of J. Chaika and A. Eskin. As a consequence, we obtain an upper bound for the Hausdorff dimension of angles in the periodic windtree model for which the associated flow is transient. This strengthen a previous result of A. Avila and P. Hubert who prove that this set has zero measure. Joint work with Artur Avila.