On the interplay of two-scale convergence and translation

We study the effects of translation on two-scale convergence. Given a two-scale convergent sequence (u_ε(x))_ε with two-scale limit u(x,y), then in general the translated sequence (u_ε(x+t))_ε is no longer two-scale convergent, even though it remains two-scale convergent along suitable subsequences. We prove that any two-scale cluster point of the translated sequence is a translation of the original limit and has the form u(x+t,y+r) where the microscopic translation r belongs to a set that is determined solely by t and the vanishing sequence (ε). Finally, we apply this result to a novel homogenization problem that involves two different coordinate frames and yields a limiting behavior governed by emerging microscopic translations.