Statistical properties of Hecke correspondences

Abstract

This paper studies the statistical properties of the dynamical system generated by a Hecke correspondence on the modular curve over $\mathbb{C}$ and, for every prime number $p$, over $\mathbb{C}_p$. Over $\mathbb{C}$, it proves that the equidistribution to the hyperbolic measure established by Clozel and Otal occurs at an exponential rate. Moreover, it determines the sharp rate, assuming an affirmative solution to the Ramanujan-Petersson conjecture. Over $\mathbb{C}_p$, two distinct types of behavior arise. In the first, every orbit converges towards the Gauss point in the Berkovich affine line, and this paper establishes the sharp exponential convergence rate. In the second, a form of unique ergodicity holds on each orbit closure, and this paper establishes a spectral gap property and, as a consequence, a central limit theorem. This complements the central limit theorem proved by Cantat and Le Borgne over $\mathbb{C}$. Finally, the paper extends and strengthens the results of Goren and Kassaei on the associated random walks.