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

Homogenization at different linear scales, bounded martingales and the two-scale shuffle limit