Near-integrable behaviour in a family of discretised rotations

Abstract

We consider a one-parameter family of invertible maps of a two-dimensional lattice, obtained by applying round-off to planar rotations. All orbits of these maps are conjectured to be periodic. We let the angle of rotation approach pi/2, and show that the limit of vanishing discretisation is described by an integrable piecewise-affine 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. Furthermore, the flow is nonlinear (unlike the original rotation), and a suitably chosen Poincare return map satisfies a twist condition. The round-off perturbation introduces KAM-type phenomena: we identify the unperturbed curves which survive the perturbation, and show that they form a set of positive density in the phase space. We prove this considering symmetric orbits, under a condition that allows us to obtain explicit values for densities. Finally, we show that the motion at infinity is a dichotomy: there is one regime in which the nonlinearity tends to zero, leaving only the perturbation, and a second where the nonlinearity dominates. In the domains where the nonlinearity remains, numerical evidence suggests that the distribution of the periods of orbits is consistent with that of random dynamics, whereas in the absence of nonlinearity, the fluctuations result in intricate discrete resonant structures.