First-return maps of Birkhoff sections of the geodesic flow

Closed geodesics in Alexandrov spaces of curvature bounded from above

In this paper, we show a local energy convexity of $W^{1,2}$ maps into $CAT(K)$ spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting.

Geodesic excursions into an embedded disc on a hyperbolic Riemann surface

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.

Teichmüller 测地流的例外方向与 Hausdorff 维数

Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension

本文研究Teichmüller测地流中Birkhoff与Oseledets定理偏离、平均发散及非弱混合IET等例外方向，通过维数估计证明其Hausdorff维数严格受限，并改进多项经典结果。

悬吊阿诺索夫流的几乎等价性

Almost equivalence of suspension Anosov flows

本文通过构造 genus 一的 Birkhoff 截面，证明了由正迹超几何矩阵生成的悬吊阿诺索夫流之间几乎等价，并给出了距离的显式界。

Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials

Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy-Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmueller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely the linear involutions. These maps are related to general measured foliations o…

Width and mean curvature flow

Given a Riemannian metric on a homotopy $n$-sphere, sweep it out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show: Each curve in the tightened sweepout whose length is close to the length of the longest curve in the sweepout must itself be close to a closed geodesic. In particular, there are curves in the sweepout that are close to closed geodesics. Finding closed geodesics on the 2-sphere by using sweepouts goes back to Birkhof…

无穷型曲面的Nielsen-Thurston分类：良态同胚映射

Towards Nielsen-Thurston classification for surfaces of infinite type: well-tempered homeomorphisms

该研究针对无穷型曲面，引入良态映射类并证明其满足Nielsen-Thurston型分类定理，给出曲面的典范不变子曲面分解，首次回归为周期或平移。