  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.
