Aperiodic points for outer billiards

TL;DR

The paper resolves a longstanding question by proving that outer billiards about any regular $N$-gon with $N>4$, $N\neq 6$ possess aperiodic points. The authors reduce the 2D outer-billiard dynamics to bounded polygon exchange transformations and two complementary invariant frameworks: dynamic Hadwiger invariants in 2D and boundary Sah–Arnoux–Fathi invariants in 1D via boundary quasi-commutators. By constructing a nonzero invariant $Inv_L(g_X)$ on a bounded invariant domain $X$ and analyzing essential Julia sets, they demonstrate nonperiodicity, which yields aperiodic points; a second route uses induced maps and SAF theory in the twice-odd case. The results not only answer Schwartz’s ICM2022 question but also introduce robust dynamical-invariant tools for broad classes of piecewise isometries, with potential extensions to quasi-rational polygons and higher-dimensional PETs.

Abstract

Euclidean outer billiard on a regular polygon (that is not a triangle, square or a hexagon) has aperiodic points, i.e., points where all iterates of the outer billiard map are defined and yield pairwise distinct images. This result answers a question of R. Schwartz posed at ICM 2022.

Figures (5)

• Figure 1: Outer billiard map on a regular pentagon $\Pi$. Here, the vertices of $\Pi$ are the roots of unity $\zeta^k$, where $\zeta=\exp(2\pi \mathbf{i}/5)$. The domain $\mathrm{dom}(f)$ is the union of the 5 sectors $V_k$, and the map $f_\Pi$ acts on $V_k$ as the half-turn about $\zeta^k$.
• Figure 2: Left: the vassal polygon $\Pi^\dagger$, strip 0 and strip $-1$. Right: the polygon $\Pi$ and the piece $A_0$ for $N=8$.
• Figure 3: The action of the map $g=g_X$ on $X$. Left: components of $\mathrm{dom}(g)$. Right: their $g$-images.
• Figure 4: The domain of $f_\mathrm{ind}$. Left: $N=10$, middle: $N=14$, right: $N=18$ (the corresponding values of $m$ are $4$, $6$, and $8$, respectively). Dashed lines are the axes of the reflections $v_0$ and $v_k$ with $k=1$, $\dots$, $m+1$.
• Figure 5: Pieces $Q_k$ of $U$. Symmetry axes of $U$ and of all $Q_k$ are shown as dashed lines. This figure is simply a zoom-in of Fig. \ref{['fig:retV0']}.

Theorems & Definitions (127)

• Theorem 1.1
• Definition 2.1: Scissors congruence
• Remark 2.2: Groupoid structure
• Remark 2.3: Connection with outer billiards
• Definition 2.4: Pieces
• Definition 2.5: Boundary, periodic, aperiodic points
• Definition 2.6: Fatou and Julia sets
• Definition 2.7: Periodicity
• Lemma 2.8
• proof