Ergodic Theory and Dynamical Systems



Convergence of Conze–Lesigne averages


BERNARD HOST a1 and BRYNA KRA a1
a1 Equipe d'analyse et de mathématiques appliquées, Université de Marne la Vallée, 77454 Marne la Vallée Cedex, France (e-mail: {host,kra}@math.univ-mlv.fr)

Abstract

We study the convergence of N^{-1} \sum f_1(T^{a_1n}x)f_2(T^{a_2n}x)f_3(T^{a_3n}x), for a measure-preserving system (X, \mathcal{B}, \mu, T) and f_{1}, f_{2}, f_{3} \in L^{\infty}(\mu). This generalizes the theorem of Conze and Lesigne on such expressions and simplifies the proof. We also obtain a description of the limit.

(Received May 20 1999)
(Revised October 20 1999)