Asymptotics in a family of linked strip maps

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.