Linear Flows on Translation Prisms

TL;DR

The paper develops a rigorous bridge between translation surfaces and linear flows on translation prisms to study billiards in prisms. It uses Furstenberg’s and Veech’s criteria, along with Rauzy induction and zippered-rectangle decompositions, to characterize when prism flows are uniquely ergodic and when they exhibit non-ergodic behavior via explicit Pisot eigenfunctions. It proves that for Pisot expansion factors of degree d, the unstable direction flow has a rank-d eigenvalue group and admits a semi-conjugacy to a hyperbolic automorphism on T^d, yielding torus-filling dynamics in suitable cases. The work also constructs explicit non-ergodic prism flows in right prisms over regular polygons (notably the double heptagon and genus-3 Arnoux–Yoccoz-type examples) and discusses implications for billiards in regular prisms, including the existence of minimal yet non-uniquely ergodic directions. Overall, the results illuminate how spectral data of translation surfaces governs prism-flow ergodicity and provides concrete, computable examples of both ergodic and non-ergodic prism dynamics with connections to toral dynamics.

Abstract

Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat $3$-manifolds which we call translation prisms. Using ideas of Furstenberg and Veech, we connect results about weak mixing properties of flows on translation surfaces to ergodic properties of linear flows on translation prisms, and use this to obtain several results about unique ergodicity of these prism flows and related billiard flows. Furthermore, we construct explicit eigenfunctions for translation flows in pseudo-Anosov directions with Pisot expansion factors, and use this construction to build explicit examples of non-ergodic prism flows, and non-ergodic billiard flows in a right prism over a regular $n$-gons for $n=7, 9, 14, 16, 18, 20, 24, 30$.

Figures (6)

• Figure 1: Surface of the double heptagon
• Figure 2: First return of the flow of slope $-y = 1+a-a^2$ to a transverse (indeed, orthogonal) interval on the staircase surface associated to the double heptagon.
• Figure 3: An eigenfunction for the straight-line flow in the direction of slope $6 \, \sin\left(\tfrac{3}{7} \, \pi\right) - 2 \, \sin\left(\tfrac{2}{7} \, \pi\right) - 6 \, \sin\left(\tfrac{1}{7} \, \pi\right) \approx 1.683$on the double heptagon. For more detail see Remark \ref{['rem:heptagon figure']}.
• Figure 4: Top depicts the level set $L=\Psi_c^{-1}(\{0\})$, drawn in color. The bottom depicts $U$, the union of five saddle connections obtained by straightening $L$. Colors of $U$ chosen so that hue remains constant across the saddle connection and lightness varies linearly (in the HSL color space), and these colors were pushed forward to $L$ via the conjugacy of the return maps. The black lines are initial segments of separatrices in direction $\pm u$ until the first return to $L$ or $U$.
• Figure 5: The IET $\tau$ carries the top arrangement to the bottom arrangement by piecewise translation. Colors come from $U$ via $h:U \to [0, 1)$ which conjugates $r:U \to U$ to $\tau$.

Theorems & Definitions (56)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Corollary 1.9
• Corollary 1.10