On some rational piecewise linear rotations

TL;DR

This paper analyzes the dynamics of rational piecewise linear rotations in the complex plane, revealing that the regular set consists of open, bounded, periodic components with convex polygonal boundaries, and all points in this set are periodic.

Contribution

It characterizes the structure and dynamics of the regular set for these rotations, showing periodicity and polygonal boundaries, which is a novel detailed geometric and dynamical analysis.

Findings

Connected components of the regular set are open, bounded, and periodic.

Each component's boundary is a convex polygon with a maximum number of sides.

All points in the regular set are periodic under the rotation.

Abstract

We study the dynamics of the piecewise planar rotations $F_{\lambda}(z)=\lambda (z-H(z)), $ with $z\in\C$, $H(z)=1$ if $\mathrm{Im}(z)\ge0,$ $H(z)=-1$ if $\mathrm{Im}(z)<0,$ and $\lambda=\mathrm{e}^{i \alpha} \in\C$, being $\alpha$ a rational multiple of $\pi$. Our main results establish the dynamics in the so called regular set, which is the complementary of the closure of the set formed by the preimages of the discontinuity line. We prove that any connected component of this set is open, bounded and periodic under the action of $F_\lambda$, with a period $\ell,$ that depends on the connected component. Furthermore, $F_\lambda^\ell $ restricted to each component acts as a rotation with a period which also depends on the connected component. As a consequence, any point in the regular set is periodic. Among other results, we also prove that for any connected component of the regular set, its boundary is a convex polygon with certain maximum number of sides.