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.Comment: PhD Thesis, Queen Mary University of London, 117 page
