From limit theorems to mixing limit theorems

TL;DR

This paper develops a general method to upgrade limit theorems to mixing limit theorems for dynamical systems, with applications to the Kontsevich-Zorich cocycle and geodesic complexity.

Contribution

It introduces a new method for deriving mixing limit theorems from existing limit theorems under mild assumptions, extending previous work.

Findings

Established a general method for mixing limit theorems

Applied the method to the Kontsevich-Zorich cocycle

Set the stage for future work on geodesic complexity

Abstract

Motivated by work of Dolgopyat and N\'andori, we establish a general method for upgrading limit theorems for Birkhoff sums and cocycles over dynamical systems to mixing limit theorems under mild ergodicity and hyperbolicity assumptions. Building on previous work of Al-Saqban and Forni, we apply this method to obtain mixing limit theorems for particular subbundles of the Kontsevich-Zorich cocycle. In forthcoming work of Arana-Herrera and Honaryar these results are applied to study the arithmetic/homological complexity of long simple closed geodesics on negatively curved surfaces.