科研之友
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摘要
We consider a one-parameter family of invertible maps of a two-dimensional lattice, obtained by discretizing the space of planar rotations. We let the angle of rotation approach pi/2, and show that the limit of vanishing discretization 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.
- 出版日期2013-5
