Convergence of Diagonal Ergodic Averages

Abstract

Tao has recently proved that if $T_1,...,T_l$ are commuting, invertible, measure-preserving transformations on a dynamical system then for any $L^{\infty }$ functions $f_1,...,f_l$, the average $\frac{1}{N}{\sum }_{n=0}^{N-1}{\prod }_{i\le l}{f}_i\circ T_i^n$ converges in the $L^2$ norm. Tao's proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence "backwards". In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao's argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties.

References fixed

References in corpus (2)

• Universal Characteristic Factors and Furstenberg Averages
• A General Correspondence between Averages and Integrals