  We consider a one-parameter family of invertible maps of a two-dimensional
lattice, obtained by discretising the space of planar rotations. We let the
angle of rotation approach $\pi/2$, and show that the limit of vanishing
discretisation is described by an integrable piecewise-smooth Hamiltonian flow,
whereby the plane foliates into families of invariant polygons with an
increasing number of sides. Considered as perturbations of the flow, the
lattice maps assume a different character, described in terms of strip maps, a
variant of those found in outer billiards of polygons. The perturbation
introduces phenomena reminiscent of the Kolmogorov-Arnold-Moser scenario: a
positive fraction of the unperturbed curves survives. We prove this for
symmetric orbits, under a condition that allows us to obtain explicit values
for their density, the latter being a rational number typically less than 1.
This result allows us to conclude that the infimum of the density of all
surviving curves (symmetric or not) is bounded away from zero.

Dynamical Systems