Contemporary Mathematics

Volume 444, 2007

Topics in Ergodic Theory and Harmonic Analysis: An

Overview

Ahmed I. Zayed

ABSTRACT. The aim of this introductory chapter is to give an overview of the

monograph and provide a synopsis of each chapter.

There are strong connections between harmonic analysis and ergodic theory

(the study of the long term behavior of dynamical systems). In particular, the

question of convergence is central to both areas. Indeed, using the Calderon transfer

principle, it is often possible to directly translate results from one of these areas to

the other. For example, because of the Calderon transfer principle, the local ergodic

theorem is a consequence of (and implies) the Lebesgue differentiation theorem.

Maximal functions and covering lemmas play a critical role in both areas, and tools

developed in the context of one of the areas will often prove useful to solve problems

in the other.

To illustrate this, observe that techniques developed by Nagel and Stein to gen-

eralize non-tangential convergence of harmonic functions have been used to study

moving ergodic averages. Furthermore, Fourier transform techniques developed to

study singular integrals have been used to study convergence of some non-standard

ergodic averages. In the other direction, techniques developed to study variation

of ergodic averages have been used to obtain similar results about singular integral

operators in harmonic analysis. Thus, each of the two areas enriches the other in

terms of both results and techniques.

A more recent example of the interaction between harmonic analysis and er-

godic theory is the proof of the spectacular result by Terence Tao and Ben Green

that the set of prime numbers contains arbitrarily long arithmetic progressions. The

breakthrough achieved by Tao and Green is attributed to their use and application

of techniques from ergodic theory and harmonic analysis to number theory.

This monograph is based on talks delivered by plenary speakers at a conference

on Harmonic Analysis and Ergodic Theory that took place at DePaul University,

Chicago, December 2-4, 2005. The conference was in honor of two members of

1991

Mathematics Subject Classification.

Primary 42-06, 37-06; Secondary 42-02, 37-02.

Key words and phrases.

Borel foliations, Central Limit Theorem for Random Walks, Ergodic

Averages, Function Interpolation, Low Pass Filters, Nehari's Theorem for Polydisk, Restricted

Weak-type Operators, s-functions, Uniqueness Theorems for Multi-dimensional Trigonometric

Series, Wavelets .

@2007 American Mathematical Society

http://dx.doi.org/10.1090/conm/444/08572