# Effective Equidistribution and Hecke Operators in Dynamical Systems

# Effective Equidistribution and Hecke Operators in Dynamical Systems

Hecke Operators are ubiquitous in the theory of automorphic forms. We present a simple construction of averaging operators on state spaces of measurable dynamical systems. This is a common generalization of Hecke operators from number theory and Markov shifts. It is also closely related to the Laplacian on a Riemannian manifold. Our first objective is showing an effective ergodic theorem with an exponential rate — large deviations — using a norm gap for the averaging operator. I will present a general criterion for states spaces of dynamical systems which implies a relatively sharp large deviations result. A large class of such systems arises from S-arithmetic quotients of reductive groups. This part builds upon the work of Kahale and Ellenberg, Michel and Venkatesh. Last I will discuss how these methods imply an effective version of uniqueness of the measure of maximal entropy and non-escape of mass for sequences of measures with high entropy for the dynamical systems in question. The relation between large deviations and equidistribution was pioneered by Linnik and later studied by Ellenberg, Michel and Venkatesh.