Teichmüller disks with small limit sets in PMF

TL;DR

The article investigates how Teichmüller disks embedded in Teichmüller space can accumulate on the Thurston boundary, focusing on the smallest possible limit sets. It introduces the canonical circle $C(X,q)$ of vertical foliations and proves that disks with $\Lambda(X,q)$ equal to a circle are extremely rare: for fixed genus $g\ge 2$ there are only finitely many such disks up to the mapping class group, and in general $\Lambda(X,q)$ properly contains $C(X,q)$. For Veech surfaces, the limit set is contained in $M(X,q)=C(X,q)\cup\bigcup_\theta \Delta^{\circ}(F_\theta,\mu_\theta)$, with precise restrictions when accumulation occurs in non-uniquely ergodic directions. The paper further shows that disks with circle limit sets must be square-tiled with balanced cylinder heights, leading to finiteness results for such disks, and it provides explicit square-tiled examples where the limit set coincides with $C(X,q)$. Overall, the work clarifies the rigidity of boundary behavior for Teichmüller disks and connects circle-limit phenomena to square-tiled and Veech structures, informing both structural and classification questions in Teichmüller dynamics.

Abstract

We study limit sets of Teichmüller disks in the Thurston boundary of Teichmüller space of a closed surface S of genus at least 2. It is well known that almost every Teichmüller geodesic ray converges to a point on the boundary. We show that unlike rays, Teichmüller disks with smallest possible limit sets are extremely rare.

Theorems & Definitions (39)

• Lemma 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: Collar Lemmabuser:GSC
• Theorem 2.2: maskit:HE
• Theorem 2.3
• Theorem 2.4: kerckhoff:EBF
• Theorem 2.5: veech:TC
• Theorem 2.6: smillie:SW
• Lemma 3.1