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

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.