Near-integrable behaviour in a family of discretised rotations

TL;DR

This paper studies a family of discretised planar rotations, showing that as the discretisation vanishes, the system approaches an integrable flow with invariant polygons, and demonstrates the persistence of some invariant curves under perturbation.

Contribution

It introduces a novel analysis of discretised rotations approaching a critical angle, revealing the persistence of invariant structures akin to KAM theory in a piecewise-smooth setting.

Findings

Limit of discretised maps approaches integrable polygonal flow.

A positive fraction of invariant curves persists under perturbation.

Explicit density values for symmetric orbits are obtained, bounded away from zero.

Abstract

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.