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POINTWISE CONVERGENCE OF AVERAGES ALONG CUBES II Abstract. Let (X, B, , T) be a measure preserving system. We prove the pointwise
 

Summary: POINTWISE CONVERGENCE OF AVERAGES ALONG CUBES II
I. ASSANI
Abstract. Let (X, B, , T) be a measure preserving system. We prove the pointwise
convergence of averages along cubes of 2k
- 1 bounded and measurable functions for all k.
1. Introduction
Let (X, B, , T) be a dynamical system where T is a measure preserving transformation on
the measure space (X, B, , T). In [1] we proved the pointwise convergence of the averages
1
N2
N-1
n,m=0
f1(Tn
x)f2(Tm
x)f3(Tn+m
x)
and of similar averages with seven bounded functions fi. We also showed that if T is weakly
mixing then similar averages for 2k -1 bounded functions converge a.e to the product of the
integrals of the functions fi. The averages of three functions were used in [3] to generalize
Khintchine recurrence result [5]. In [2] B.Host and B.Kra proved that the averages of 2k -1

  

Source: Assani, Idris - Department of Mathematics, University of North Carolina at Chapel Hill

 

Collections: Mathematics