On some rational piecewise linear rotations

TL;DR

This work analyzes the dynamics of the piecewise planar rotation $F_{\lambda}(z)=\lambda\,(z-H(z))$ with $H(z)$ partitioning the plane by the real axis and $\lambda=e^{i\alpha}$ for rational multiples of $\pi$. By focusing on the regular set $\mathcal{U}=\mathbb{C}\setminus\overline{\mathcal{F}}$, the authors prove that every connected component is open, bounded and periodic under $F_{\lambda}$, with an associated $\\ ext{ℓ}$-cycle of components and a rotation of order $k$ on each, implying all points in $\mathcal{U}$ are periodic. They show the boundary of each tile is a convex polygon with a number of sides bounded by $q$ (or $2q$ for odd $q$) when $\alpha=2\pi p/q$, and provide precise conditions under which the boundary is a regular polygon; they also demonstrate non-regular polygon boundaries for certain angles. The paper includes detailed proofs, geometric analysis of the critical set, and numerical/geometric evidence of fractal-like behavior and unbounded periods in non-regular cases, including explicit constructions for $\alpha=11\pi/6$ and $\alpha=8\pi/5$. These results illuminate the structure of the tessellation induced by the critical set and yield a comprehensive picture of the regular dynamics in rational-angle cases, with implications for polygonal dual billiards and related systems.

Abstract

We study the dynamics of the piecewise planar rotations $F_λ(z)=λ(z-H(z)), $ with $z\in\C$, $H(z)=1$ if $\mathrm{Im}(z)\ge0,$ $H(z)=-1$ if $\mathrm{Im}(z)<0,$ and $λ=\mathrm{e}^{i α} \in\C$, being $α$ a rational multiple of $π$. 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 $F_λ$, with a period $\ell,$ that depends on the connected component. Furthermore, $F_λ^\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.

Figures (6)

• Figure 1: Necklaces for $\alpha={8\pi}/{5}$. The critical set, formed by the union of a numerable sets of lines, in grey, in the left picture. In the right picture two invariant necklaces.
• Figure 2: The 20-periodic irregular hexagons associate to the hexagon $H$, for $\alpha=11\pi/6$.
• Figure 3: Apparent fractalization of $\mathcal{F}$ and $\mathcal{U}$ when $\alpha=2\pi\frac{p}{q}\notin\mathcal{R}$, for the cases $\alpha=\frac{11\pi}{6}$ and $\alpha=\frac{8\pi}{5}$, respectively.
• Figure 4: Sequence of nested triangles defined by the critical curves for $\alpha=\frac{8\pi}{5}$. In blue, $\triangle QRS$, $\triangle QR_1S_1$ and $\triangle QR_2S_2$. In magenta, the periodic points $P_0,P_1$ and $P_2$.
• Figure 5: The $7$-periodic orbit associate with $P_1=r(P_0)$ in Magenta. All the points in the corresponding pentagons are $35$-periodic. In blue a $35$-periodic orbit.

Theorems & Definitions (22)

• Theorem A
• Theorem B
• Theorem 1: Goetz & Quas, GQ
• Lemma 2
• proof
• Proposition 3
• proof
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['t:teoa']}