Near-integrable behaviour in a family of discretised rotations
  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. 
Heather Reeve-Black and Franco Vivaldi
math.DS