Large deviations and applications in homogeneous dynamics

The Birkhoff Ergodic Theorem can be viewed as a dynamical version of the law of large numbers. A natural question that arises is how rapidly the measure of the exceptional set decays, leading to the study of large deviations. In this talk, we present large deviation estimates for diagonalizable actions on homogeneous spaces and discuss applications to the effective uniqueness of the measure of maximal entropy and quantitative pointwise equidistribution along expanding horospheres. This is joint work with Ron Mor.