  We apply round-off to planar rotations, obtaining a one-parameter family of
invertible maps of a two-dimensional lattice. As the angle of rotation
approaches pi/2, the fourth iterate of the map produces piecewise-rectilinear
motion, which develops along the sides of convex polygons.
  We characterise the dynamics ---which resembles outer billiards of
polygons---as the concatenation of so-called strip maps, each providing an
elementary perturbation of an underlying integrable system. Significantly,
there are orbits which are subject to an arbitrarily large number of these
perturbations during a single revolution, resulting in the appearance of a
novel discrete-space version of near-integrable Hamiltonian dynamics.
  We study the asymptotic regime of the limiting integrable system
analytically, and numerically some features of its very rich near-integrable
dynamics. We unveil a dichotomy: there is one regime in which the nonlinearity
tends to zero, and a second where it doesn't. In the latter case, numerical
experiments suggest that the distribution of the periods of orbits is
consistent with that of random dynamics; in the former case the fluctuations
result in an intricate structure of resonances.
