Linear Sequences and Weighted Ergodic Theorems

Tanja Eisner

doi:10.1155/2013/815726

Abstract

We present a simple way to produce good weights for several types of ergodic theorem including the Wiener-Wintner type multiple return time theorem and the multiple polynomial ergodic theorem. These weights are deterministic and come from orbits of certain bounded linear operators on Banach spaces. This extends the known results for nilsequences and return time sequences of the form for a measure preserving system and , avoiding in the latter case the problem of finding the full measure set of appropriate points .

Highlights

The classical mean and pointwise ergodic theorems due to von Neumann and Birkhoff, respectively, take their origin in questions from statistical physics and found applications in quite different areas of mathematics such as number theory, stochastics, and harmonic analysis
We show that one can take linear sequences as weights in the multiple Wiener-Wintner type generalisation of the return time theorem due to Zorin-Kranich [16] and Assani et al [25] discussed in the introduction
Every nilsequence is a good weight for the multiple polynomial ergodic theorem

Summary

Introduction

The classical mean and pointwise ergodic theorems due to von Neumann and Birkhoff, respectively, take their origin in questions from statistical physics and found applications in quite different areas of mathematics such as number theory, stochastics, and harmonic analysis. Assani and Presser [14] and Zorin-Kranich [16], gave a generalisation of the return time theorem and showed that (in the previous notation) the sequence (g(Sny)) is for almost every y a universally good weight for multiple ergodic averages; see Definition 4 later. The most general class of systems for which the convergence in the multiple return time theorem is known to hold everywhere, leading to good weights which are easy to construct, are nilsystems, that is, systems of the form Y = G/Γ for a nilpotent Lie group G, a discrete cocompact subgroup Γ, the Haar measure μ on G/Γ, and the rotation S by some element of G For such system (Y, μ, S), g ∈ C(Y) and y ∈ Y, the sequence (g(Sny)) is called a basic nilsequence. Multiple polynomial correlation sequences provide another class of deterministic examples of good weights for the Wiener-Wintner type multiple return time theorem and the multiple polynomial ergodic theorem discussed in Sections 3 and 4

Linear Sequences and Their Structure

A Wiener-Wintner Type Result for the Multiple Return Time Theorem

Weighted Multiple Polynomial Ergodic Theorem

A Counter Example