  We consider a prototypical two-parameter family of invertible maps of
$\mathbb{Z}^2$, representing rotations with decreasing rotation number. These
maps describe the dynamics inside the island chains of a piecewise affine
discrete twist map of the torus, in the limit of fine discretisation. We prove
that there is a set of full density of points which, depending of the parameter
values, are either periodic or escape to infinity. The proof is based on the
analysis of an interval-exchange map over the integers, with infinitely many
intervals.
