Extremal parameters and their duals for Fuchsian boundary maps

带尖点的富克斯群的约化理论

Reduction theory for Fuchsian groups with cusps

研究与带尖点的第一类有限生成富克斯群相关的类似鲍文 - 系列映射，通过构造特殊基本多边形，证明其二维自然扩张有双射定义域和有限矩形结构全局吸引子，证实唐·扎吉尔猜想。

超越对应：当富克斯群接管整函数的盆域时

Transcendental correspondences: when Fuchsian groups take over basins of entire maps

本文系统研究超越整函数与富克斯群共形组合产生的\((\infty:\infty)\)全纯对应，结合整函数盆域外动力学与群作用，证明其为特定复合，当整函数奇异值有限时，相关亚纯函数奇异值也有限且线复形可明确描述。

某些连分数映射的首返回系统

First return systems for some continued fraction maps

该研究证明了Calta等人关于区间映射族T_{3,n,α}的猜想，通过先证Hecke三角形群相关映射族的对应结果并比较平面区域完成证明，还给出了相关熵函数的性质及精确公式。

Extremal maps of the universal hyperbolic solenoid

We show that the set of points in the Teichmuller space of the universal hyperbolic solenoid which do not have a Teichmuller extremal representative is generic (that is, its complement is the set of the first kind in the sense of Baire). This is in sharp contrast with the Teichmuller space of a Riemann surface where at least an open, dense subset has Teichmuller extremal representatives. In addition, we provide a sufficient criteria for the existence of Teichmuller extremal representatives in the given homotopy class. These results indicate that there is an interesting theory of extremal (and …

通过轨道等价结合有理映射与Kleinian群

Combining Rational maps and Kleinian groups via orbit equivalence

本研究提出复多项式与Kleinian曲面群全纯配对的轨道等价框架，定义高阶Bowen-Series映射，明确配对Teichmüller空间分支，证明圆周Fuchsian群无轨道等价刚性，并分类可配对的Kleinian Bers边界群。

渐近弗uchs面在封闭双曲3流形中的极限

Limits of asymptotically Fuchsian surfaces in a closed hyperbolic 3-manifold

研究了封闭双曲3流形中渐近弗uchsian面的极限性质，证明其测度极限由双曲测度与完全测地面测度的凸组合构成。

Tessellation and Lyubich-Minsky laminations associated with quadratic maps II: Topological structures of 3-laminations

According to an analogy to quasi-Fuchsian groups, we investigate topological and combinatorial structures of Lyubich and Minsky's affine and hyperbolic 3-laminations associated with the hyperbolic and parabolic quadratic maps. We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same 3-laminations. This gives a good reason to regard the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasi-Fuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolic-to…

Openness of momentum maps and persistence of extremal relative equilibria

We prove that for every proper Hamiltonian action of a Lie group G in finite dimensions the momentum map is locally G-open relative to its image (i.e. images of G-invariant open sets are open). As an application we deduce that in a Hamiltonian system with continuous Hamiltonian symmetries, extremal relative equilibria persist for every perturbation of the value of the momentum map, provided the isotropy subgroup of this value is compact. We also demonstrate how this persistence result applies to an example of ellipsoidal figures of rotating fluid, and provide an example with plane point vortic…

离散双曲与抛物动力学中的极值收敛速率

Extremal rate of convergence in discrete hyperbolic and parabolic dynamics

本文研究上半平面全纯自映射的动力学行为，利用 Herglotz 表示、Koenigs 函数共形性及双曲距离等工具，刻画了轨道以最慢速率逼近 Denjoy--Wolff 点的双曲与抛物自映射的极值收敛速率。

Adler and Flatto revisited: cross-sections for geodesic flow on compact surfaces of constant negative curvature

We describe a family of arithmetic cross-sections for geodesic flow on compact surfaces of constant negative curvature based on the study of generalized Bowen-Series boundary maps associated to cocompact torsion-free Fuchsian groups and their natural extensions, introduced in Katok-Ugarcovici's 2017 paper. If the boundary map satisfies the short cycle property, i.e., the forward orbits at each discontinuity point coincide after one step, the natural extension map has a global attractor with finite rectangular structure, and the associated arithmetic cross-section is parametrized by the attract…

连续动力学中的极值收敛速率

Extremal rate of convergence in continuous dynamics

本文刻画上半平面全纯自映射半群以最慢速率收敛至Denjoy--Wolff点的条件，重点解决抛物情形，并利用Herglotz表示与Koenigs函数共形性提供多种等价描述。

Continuity of entropy for all $α$-deformations of an infinite class of continued fraction transformations

We extend the results of our 2020 paper in the Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. There, we associated to each of an infinite family of triangle Fuchsian groups a one-parameter family of continued fraction maps and showed that the matching (or, synchronization) intervals are of full measure. Here, we find planar extensions of each of the maps, and prove the continuity of the entropy function associated to each one-parameter family. We also introduce a notion of "first pointwise expansive power" of an eventually expansive interval map. We prove that for every map …

施瓦茨反射、B-对合与代数对应的一般动力学理论

A general dynamical theory of Schwarz reflections, B-involutions, and algebraic correspondences

本文建立了施瓦茨反射、B-对合与代数对应的一般动力学理论，证明通用（反）多项式与Fuchsian反射群圆周映射的配对存在性，并构造代数对应为Fatou统一理论提供新证据。