[Paper Review] Teichmueller geodesics with d-dimensional limit sets

Anna Lenzhen, Babak Modami+1 more|arXiv (Cornell University)|Aug 29, 2016

Morphological variations and asymmetry17 references3 citations

TL;DR

This paper constructs a Teichmüller geodesic ray in the Teichmüller space of a genus $d+1$ surface whose limit set in the Thurston boundary is a $d$-dimensional simplex of projective measured foliations. Using a construction of $d+1$ rotated square tori with irrational slope foliations, glued along vertical slits, the authors show that under specific continued fraction growth conditions on the slopes, the geodesic ray's limit set spans a $d$-dimensional simplex, answering Masur's question about higher-dimensional limit sets.

ABSTRACT

We construct an example of a Teichmueller geodesic ray whose limit set in Thurston boundary of Teichmueller space is a d-dimensional simplex.

Motivation & Objective

• To resolve a long-standing question posed by Masur regarding whether Teichmüller geodesic rays can have limit sets of dimension greater than one.
• To construct explicit examples of Teichmüller geodesic rays whose limit sets in the Thurston boundary are $d$-dimensional simplices for any $d \geq 2$.
• To demonstrate that such higher-dimensional limit sets arise from a specific geometric construction involving $d+1$ rotated square tori with irrational foliations glued along slits.
• To establish that the limit set is the full $d$-dimensional simplex spanned by ergodic measured foliations supported on individual tori.

Proposed method

• Construct a translation surface $X_0$ of genus $d+1$ by gluing $d+1$ rotated square tori $T^i$, each with an irrational slope foliation $\theta^i$, along vertical slits of fixed size $s_0$.
• Define a Teichmüller geodesic ray $\mathbf{r}$ starting at $X_0$ in the direction of the holomorphic quadratic differential $\phi_0$ with a single zero of order $2d$.
• Use continued fraction expansions of the slopes $\theta^i$ with specific growth conditions on the coefficients to control the asymptotic behavior of the geodesic.
• Estimate the hyperbolic length and twist of simple closed curves along the ray using the extremal length and width estimates in the thick and thin parts of the surface.
• Apply coarse estimates involving $\sim$, $\asymp$, and $\stackrel{\ast}{\asymp}$ to compare lengths of curves corresponding to convergents of the continued fractions at different times along the ray.
• Show that the limit set contains the $d+1$ ergodic measured foliations $\nu^i$ and that their convex hull forms a $d$-dimensional simplex in $\mathcal{PMF}(S)$.

Experimental results

Research questions

• RQ1 Can Teichmüller geodesic rays have limit sets of dimension greater than one in the Thurston boundary of Teichmüller space?
• RQ2 What geometric and dynamical conditions on the initial surface and geodesic direction lead to a $d$-dimensional limit set?
• RQ3 How do continued fraction growth conditions on the slopes of foliations on individual tori affect the asymptotic behavior of the geodesic?
• RQ4 Is it possible for the limit set to be the entire $d$-dimensional simplex spanned by $d+1$ ergodic measured foliations?

Key findings

• For any $d \geq 2$, there exists a Teichmüller geodesic ray whose limit set in $\mathcal{PMF}(S)$ is a $d$-dimensional simplex.
• The limit set is the convex hull of $d+1$ ergodic measured foliations $\nu^0, \nu^1, \dots, \nu^d$, each supported on a separate torus in the construction.
• The construction relies on continued fraction expansions of the slopes $\theta^i$ with coefficients satisfying specific growth conditions to ensure the limit set is full-dimensional.
• The asymptotic length estimates of curves along the ray show that the limit set contains all measures in the simplex, confirmed via coarse estimates of hyperbolic length and twist.
• The limit set is not a point or a one-dimensional interval, but a full $d$-dimensional simplex, resolving Masur’s question affirmatively.