无共轭点的紧致高亏格曲面的测地流具有扩张因子

Geodesic flows modeled by expansive flows

Given a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves time-parametrization. It is concluded that the geodesic flow has a unique measure of maximal entropy.

The structure of pointwise recurrent expansive homeomorphisms

Let $X$ be a compact metric space and let $f:X\rightarrow X$ be a homeomorphism on $X$. We show that if $f$ is both pointwise recurrent and expansive, then the dynamical system $(X, f)$ is topologically conjugate to a subshift of some symbolic system. Moreover, if $f$ is pointwise positively recurrent, then the subshift is semisimple; a counterexample is given to show the necessity of positive recurrence to ensure the semisimilicity.

无共轭点粗双曲流形上测地流最大熵测度的唯一性

Uniqueness of the measure of maximal entropy for geodesic flows on coarse hyperbolic manifolds without conjugate points

本文在无共轭点且基本群为Gromov双曲、剩余有限的闭黎曼流形上，通过对扩张集施加适当条件，证明了测地流最大熵测度的唯一性，从而推广了此前仅在负截面曲率背景度量下成立的相关结果。

Local product structure for expansive homeomorphisms

Let $f\colon M\to M$ be an expansive homeomorphism with dense topologically hyperbolic periodic points, $M$ a compact manifold. Then there is a local product structure in an open and dense subset of $M$. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is uniform. In particular, we conclude that the homeomorphism is conjugated to a linear Anosov diffeomorphism of a torus.

Unique ergodicity of the horocycle flow of a higher genus compact surface with no conjugate points and continuous Green bundles

We show that the horocyclic flow of an orientable compact higher genus surface without conjugate points and with continuous Green bundles is uniquely ergodic. The result applies to nonflat nonpositively curved surfaces and generalizes a classical result of Furstenberg and Marcus in negative curvature. The proof relies on the definition of a uniformly expanding parametrization on the quotient by the strips of the surface.

Closed geodesics on surfaces without conjugate points

We obtain Margulis-type asymptotic estimates for the number of free homotopy classes of closed geodesics on certain manifolds without conjugate points. Our results cover all compact surfaces of genus at least 2 without conjugate points.

无共轭点的紧致高亏格曲面测地流的扩张密度

Density of expansivity for geodesic flows of compact higher genus surfaces without conjugate points

本文证明了在亏格大于1且无共轭点的紧致连通光滑曲面中，无带状区域的测地集合在单位切丛中构成稠密轨道集，推广了无焦点条件下的已知结论。

Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points

We prove that for closed surfaces $M$ with Riemannian metrics without conjugate points and genus $\geq 2$ the geodesic flow on the unit tangent bundle $T^1M$ has a unique measure of maximal entropy. Furthermore, this measure is fully supported on $T^1M$ and the flow is mixing with respect to this measure. We formulate conditions under which this result extends to higher dimensions.

A priori bounds for GIETs, affine shadows and rigidity of foliations in genus 2

We prove a rigidity result for foliations on surfaces of genus two, which can be seen as a generalization to higher genus of Herman's theorem on circle diffeomorphisms and, correspondingly, flows on the torus. We prove in particular that, if a smooth, orientable foliation with non-degenerate (Morse) singularities on a closed surface of genus two is minimal, then, under a full measure condition for the rotation number, it is differentiably conjugate to a linear foliation. The corresponding result at the level of Poincaré sections is that, for a full measure set of interval exchange transformati…

无共轭点的磁流平均洛伦兹力的上界——以水平圆的测地曲率表示

An upper bound for the mean Lorentz force of magnetic flows without conjugate points in terms of the geodesic curvature of horocycles

本文研究亏格至少二的闭黎曼曲面上无共轭点的磁流，证明在 Mañé 临界值之上，洛伦兹力积分受水平圆测地曲率约束，并由此推出基于面积和高斯曲率的显式上界。

无共轭点的均匀可见流形上测地流的遍历性质

On ergodic properties of geodesic flows on uniform visibility manifolds without conjugate points

本文研究无共轭点均匀可见流形上测地流的遍历性质，证明了闭曲面情形下测地流的遍历性，构造了唯一极大熵测度并证明其 Bernoulli 性质，在压力间隙条件下得到平衡态唯一性，并应用于计数闭测地线及体积渐近，最后建立了 Hopf-Tsuji-Sullivan 二分法。

亏格2拟Fuchs群与余紧Kleinian群的子群

Subgroups of Genus-2 Quasi-Fuchsian groups and Cocompact Kleinian Groups

本文研究余紧Kleinian群中曲面子群的几何控制，证明对任意亏格2拟Fuchs群和余紧Kleinian群，存在曲面子群与有限指标子群拟共形共轭。

仿射圆柱与环面上测地流的共轭性

Conjugacies of geodesic flows in affine cylinders and tori

本文研究仿射圆柱与仿射环面上测地流的共轭性，给出了在拓扑与全纯范畴下共轭的显式充要条件。

Linearization of topologically Anosov homeomorphisms of non compact surfaces

We study the dynamics of Topologically Anosov homeomorphisms of non compact surfaces. In the case of surfaces of genus zero and finite type, we classify them. We prove that if $f:S \to S$, is a Topologically Anosov homeomorphism where $S$ is a non-compact surface of genus zero and finite type, then $S= \R ^ 2$ and $f$ is conjugate to a homothety or reverse homothety (depending on wether $f$ preserves or reverses orientation). A weaker version of this result was conjectured in a previous work.

高亏格Cherry流与全族GIET

Higher genus Cherry flows and full families of GIETs

该研究在紧致曲面上发展Cherry流的高亏格对应物，构造满足Keane性质的$C^\fty$ Cherry流族，实现带平坦片的GIET并支撑对应数量的遍历不变测度，为相关动力系统研究提供了最优参数化方案。

闭曲面上Anosov度量的路径连通性

Path Connectivity of Anosov Metrics on Surfaces

研究了闭曲面上Anosov度量的路径连通性，通过共形变形保持Anosov性质并连接正负曲率度量。

曲面自同胚的关键因素

Essential factors for surface homeomorphisms

针对亏格至少为1的闭可定向曲面上同痕于伪阿诺索夫自同胚的自同胚，通过坍缩万有覆盖中迭代直径一致有界的极大连续统构造单调因子，该因子具强混沌特征且可保面积，其极大熵测度为全支撑伯努利测度。