通过轨道等价结合有理映射与Kleinian群

克莱因群与多项式配对映射的连续性

Continuity of matings of Kleinian groups and polynomials

本文在 Bullett-Harvey 手术构造基础上研究克莱因群与多项式的配对映射，提高映射次数，并证明其在参数空间内部解析、在拟共形刚性边界条件下连续。

The outside of the Teichmuller space of punctured tori in Maskit's embedding

We consider the following question: Which parameters in the extension of a rational pleating ray across the boundary of $\Cal M$, the Maskit embedding of the Teichmüller space of once punctured tori correspond to a Kleinian group? Using methods of Keen and Series and Wright we prove a local result, stating that on each rational ray there is a sequence of parameters in $\overline\Bbb H\setminus\Cal M$ accumulating at the boundary point of $\Cal M$ on the ray. These are the unique parameters on the extended $p/q$ ray for which the special word $W_{p/q}$ is a primitive elliptic Möbius transformat…

On density of horospheres in dynamical laminations

In 1985 D.Sullivan had introduced a dictionary between two domains of complex dynamics: iterations of rational functions on the Riemann sphere and Kleinian groups. The latters are discrete subgroups of the group of conformal automorphisms of the Riemann sphere. This dictionary motivated many remarkable results in both domains, starting from the famous Sullivan's no wandering domain theorem in the theory of iterations of rational functions. One of the principal objects used in the study of Kleinian groups is the hyperbolic 3- manifold associated to a Kleinian group, which is the quotient of its…

Dimension rigidity in conformal structures

Let $Λ$ be the limit set of a conformal dynamical system, i.e. a Kleinian group acting on either finite- or infinite-dimensional real Hilbert space, a conformal iterated function system, or a rational function. We give an easily expressible sufficient condition, requiring that the limit set is not too much bigger than the radial limit set, for the following dichotomy: $Λ$ is either a real-analytic manifold or a fractal in the sense of Mandelbrot (i.e. its Hausdorff dimension is strictly greater than its topological dimension). Our primary focus is on the infinite-dimensional case. An important…

Dynamical aspects of piecewise conformal maps

We study the dynamics of piecewise conformal maps in the Riemann sphere. The normality and chaotic regions are defined and we state several results and properties of these sets. We show that the stability of these piecewise maps is related to the Kleinian group generated by their transformations under certain hypotheses. The general motivation of the article is to compare the dynamics of piecewise conformal maps and those of the Kleinian groups and iterations of rational maps.

Trees, length spectra for rational maps via barycentric extensions and Berkovich spaces

In this paper, we study the dynamics of degenerating sequences of rational maps on Riemann sphere $\hat{\mathbb{C}}$ using $\mathbb{R}$-trees. Given a sequence of degenerating rational maps, we give two constructions for limiting dynamics on $\mathbb{R}$-trees: one geometric and one algebraic. The geometric construction uses the ultralimit of rescalings of barycentric extensions of rational maps, while the algebraic construction uses the Berkovich space of complexified Robinson's field. We show the two approaches are equivalent. The limiting dynamics on the $\mathbb{R}$-tree are analogues to i…

共形动力学的镜像：反有理映射、反射群、Schwarz反射与对应之间的相互作用

Mirrors of conformal dynamics: Interplay between anti-rational maps, reflection groups, Schwarz reflections, and correspondences

本文综述了反有理映射迭代、Kleinian反射群作用、正交域中Schwarz反射生成的动力学以及代数对应这四类共形动力学分支之间的密切联系，并通过实例和一般结果展示了它们之间的动力学关系及技术工具的应用。

Thurston equivalence to a rational map is decidable

We demonstrate that the question whether or not a given postcritically finite topological ramified covering map of the 2-sphere is Thurston equivalent to a rational map is algorithmically decidable.

Equivalence of Collet-Eckmann conditions for slowly recurrent rational maps

In this short note we observe that within the family of slowly recurrent rational maps on the Riemann sphere, the Collet-Eckmann, second Collet-Eckmann, and topological Collet-Eckmann conditions are equivalent and also invariant under topological conjugacy.

Coarse expanding conformal dynamics

Building on the dictionary between Kleinian groups and rational maps, we establish new connections between the theories of hyperbolic groups and certain iterated maps, regarded as dynamical systems. In order to make the exposition self-contained to researchers in many fields, we include detailed proofs and ample background.

Thurston equivalence for rational maps with clusters

We investigate rational maps with period one and two cluster cycles. Given the definition of a cluster, we show that, in the case where the degree is $d$ and the cluster is fixed, the Thurston class of a rational map is fixed by the combinatorial rotation number $ρ$ and the critical displacement $δ$ of the cluster cycle. The same result will also be proved in the case that the rational map is quadratic and has a period two cluster cycle, but that the statement is no longer true in the higher degree case.

垫片Julia集的一致化

Uniformization of gasket Julia sets

本文刻画了可由圆形垫片一致化的有理映射垫片Julia集，证明了胖垫片Julia集可拟共形一致化，拟圆盘Fatou分支的Julia集可通过David同胚一致化，且垫片Julia集与Kleinian群极限集可局部拟共形同胚。

配对、全纯对应与一个Bers切片

Matings, holomorphic correspondences, and a Bers slice

本文统一并推广了克莱因群与有理映射配对的两类框架，将其扩展到亏格为零且轨形点受限的双曲轨形，构造了与实对称双曲分支中多项式的对应配对，并建立了代数参数空间中的Bers切片类似物。

Bers Slices in Families of Univalent Maps

We construct embeddings of Bers slices of ideal polygon reflection groups into the classical family of univalent functions $Σ$. This embedding is such that the conformal mating of the reflection group with the anti-holomorphic polynomial $z\mapsto\overline{z}^d$ is the Schwarz reflection map arising from the corresponding map in $Σ$. We characterize the image of this embedding in $Σ$ as a family of univalent rational maps. Moreover, we show that the limit set of every Kleinian reflection group in the closure of the Bers slice is naturally homeomorphic to the Julia set of an anti-holomorphic po…

有理映射的复合与有限循环群的自由积之间的交配

Matings between compositions of rational maps and free products of finite cyclic groups

针对具有特定抛物不动点的有理映射对，构造双次数\((pq,pq)\)的代数对应\(F\)实现映射复合与循环群自由积表示的交配，表明\(F\)是特定有理映射删除覆盖对应之复合，推广方法构造多项式对复合与群表示的交配，给出不可逆例子。

二次有理映射的形变空间

On deformation spaces of quadratic rational maps

研究部分后临界有限分支自覆盖的自等价群，解答了关于有理映射形变空间可收缩性的问题。