A divergent Teichmuller geodesic with uniquely ergodic vertical foliation

Divergent on average directions of Teichmuller geodesic flow

The set of directions from a quadratic differential that diverge on average under Teichmuller geodesic flow has Hausdorff dimension exactly equal to one-half.

Limit sets of Teichmüller geodesics with minimal non-uniquely ergodic vertical foliation

We describe a method for constructing Teichmüller geodesics where the vertical measured foliation $ν$ is minimal but is not uniquely ergodic and where we have a good understanding of the behavior of the Teichmüller geodesic. The construction depends on various parameters, and we show that one can adjust the parameters to ensure that the set of accumulation points of such a geodesic in the Thurston boundary is exactly the set of all possible measured foliations in the homotopy class of $ν$. With further adjustment of the parameters, one can even take $ν$ to be an ergodic measure on a non-unique…

Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, II

Given a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of which projectively converges to one of the ergodic measures on the ending lamination. The conditions are sufficiently robust, allowing us to construct sequences on a closed surface of genus $g$ for which the space of measures has the maximal dimension $3g-3$, for example. We also s…

Limits in PMF of Teichmuller geodesics

We consider the limit set in Thurston's compactification PMF of Teichmueller space of some Teichmueller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that a) there are quadratic differentials so that the limit set of the geodesic is a unique point, b) there are quadratic differentials so that the limit set is a line segment, c) there are quadratic differentials so that the vertical foliation is ergodic and there is a line segment as limit set, and d) there are quadratic differentials so that the vertical foliation is ergodic and…

关于 $\mathbb{CP}^2$ 上具有唯一奇点的叶状结构的注记

Remarks on foliations on $\mathbb{CP}^2$ with a unique singular point

本文研究 $\mathbb{CP}^2$ 上具有唯一奇点的叶状结构，给出次数 7 的非临界情形下具有发散分隔线的显式例子，并探讨低次数下的不同族。

Slow Divergence and Unique Ergodicity

Masur showed that a Teichmuller geodesic that is recurrent in the moduli space of closed Riemann surfaces is necessarily determined by a quadratic differential with a uniquely ergodic vertical foliation. In this paper, we show that a divergent Teichmuller geodesic satisfying a certain slow rate of divergence is also necessarily determined by a quadratic differential with unique ergodic vertical foliation. As an application, we sketch a proof of a complete characterization of the set of nonergodic directions in any double cover of the flat torus branched over two points.