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Ergodic Averages for Weight Functions Moved by Non-Linear Transformations on ℝn

Published online by Cambridge University Press:  20 November 2018

David I. McIntosh
David I. McIntosh*
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6
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Abstract

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Let ℝ+ denote the non-negative half of the real line, and let λ denote Lebesgue measure on the Borel sets of ℝn. A function φ: ℝn → ℝ+ is called a weight function if ʃℝn φ dλ = 1. Let (X, ℱ, μ) be a non-atomic, finite measure space, let ƒ: X → ℝ+, and suppose { Tν}ν∊ℝn is an ergodic, aperiodic ℝn-flow on X. We consider the weighted ergodic averages where is a sequence of weight functions. Sufficient as well as necessary and sufficient conditions for the pointwise, almost-everywhere convergence of are developed for a particular class of weight functions φk. Specifically, let {τk: ℝn → ℝn} be a sequence of measurable, non-singular maps with measurable, non-singular inverses such that the Radon-Nikodym derivatives dλ oτk /dλ and dλ oτk-1 / dλ are L∞ (ℝn), and such that τk and τ-1 map bounded sets to bounded sets. We examine convergence for the sequence where θk is an a.e.-convergent sequence of weight functions which are dominated by a fixed L1(ℝn) function with bounded support.

Keywords

28D9960F99maximal ergodic theorems
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 4 , 01 August 1995 , pp. 852 - 876
Copyright
Copyright © Canadian Mathematical Society 1995

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