Homogenization at different linear scales, bounded martingales and the Two-Scale Shuffle limit

TL;DR

This paper studies the behavior of two-scale homogenization limits with increasing periods, showing they form a bounded martingale that converges strongly and almost everywhere to a comprehensive limit called the Two-Scale Shuffle limit.

Contribution

It introduces the Two-Scale Shuffle limit concept, demonstrating convergence properties and the martingale structure of homogenization limits with increasing periods.

Findings

Two-scale limits form a bounded martingale after rearrangement.

The Two-Scale Shuffle limit converges strongly in L^2 and almost everywhere.

The limit encapsulates all information from the sequence of two-scale limits.

Abstract

In this paper, we consider two-scale limits obtained with increasing homogenization periods, each period being an entire multiple of the previous one. We establish that, up to a measure preserving rearrangement, these two-scale limits form a martingale which is bounded: the rearranged two-scale limits themselves converge both strongly in $\mathrm{L}^2$ and almost everywhere when the period tends to $+\infty$. This limit, called the Two-Scale Shuffle limit, contains all the information present in all the two-scale limits in the sequence.