 
Summary: POINTWISE CONVERGENCE OF ERGODIC AVERAGES ALONG
CUBES
I. ASSANI
Abstract. Let (X, B, µ, T) be a measure preserving system. We prove the pointwise
convergence of ergodic averages along cubes of 2k
 1 bounded and measurable functions
for all k. We show that this result can be derived from estimates about bounded sequences
of real numbers. We apply these estimates to establish the pointwise convergence of some
weighted ergodic averages and ergodic averages along cubes for not necessarily commuting
measure preserving transformations.
1. Introduction
Let (X, B, µ, T) be a dynamical system on a finite measure space, where T : X X is
a measure preserving transformation i.e. µ(T1A) = µ(A) for all measurable subsets of B.
We will assume that T is invertible. A factor of the system (X, B, µ, T) is a sub algebra
invariant under T. For convenience we shall denote by the same letter a factor Z and the
L2 space built on this invariant sub algebra.
We will assume in some of the statements that T is ergodic. This means that the only
invariant functions for T are the constant functions. As we look for pointwise results we
Department of Mathematics, UNC Chapel Hill, NC 27599, assani@math.unc.edu.
Keywords: characteristic factors, ergodic averages along the cubes, Wiener Wintner averages.
