  Mixing rates and decay of correlations for dynamics defined by potentials
with summable variations are well understood, but little is known for
non-summable variations. In this paper, we exhibit upper bounds for these
quantities in the case of dynamics defined by potentials with square summable
variations. We obtain these bounds as corollaries of a new block coupling
inequality between pair of dynamics starting with different histories. As
applications of our results, we prove a new weak invariance principle and a
Chernoff-type inequality.
