非奇异$α$ -刚性映射：简短证明

解析临界圆映射的刚性问题

The rigidity problem for analytic critical circle maps

证明了若$f$和$g$是任意两个解析临界圆映射，且无理旋转数相同，则$f$的临界点映射到$g$的临界点的共轭在临界点处具有正则性$C^{1+α}$，其普适值为$α>0$ .由此，给出了临界圆映射的完全重整化马蹄的双曲性的一个新证明。

分段线性圆周映射及其与刚性有理旋转的共轭性

Piecewise linear circle maps and conjugation to rigid rational rotations

本文研究分段线性圆周同胚与有理旋转数刚性旋转的共轭准则，通过两类映射验证结论，揭示其与无理旋转数情形的对应性及自然映射族中无模锁定的线性标度特性。

Non-Rigid Rank-One Infinite Measures on the Circle

For a class of irrational numbers, depending on their Diophantine properties, we construct explicit rank-one transformations that are totally ergodic and not weakly mixing. We classify when the measure is finite or infinite. In the finite case they are isomorphic to irrational rotations. We also obtain rank-one nonrigid infinite invariant measures for irrational rotations, and, for each Krieger type, nonsingular measures on irrational rotations. In the third version, in the infinite case we use the constructions to provide examples of non-weakly mixing infinite measure-preserving ergodic trans…

Rigidity and flexibility of entropies of boundary maps associated to Fuchsian groups

Given a closed, orientable surface of constant negative curvature and genus $g \ge 2$, we study the topological entropy and measure-theoretic entropy (with respect to a smooth invariant measure) of generalized Bowen--Series boundary maps. Each such map is defined for a particular fundamental polygon for the surface and a particular multi-parameter. We present and sketch the proofs of two strikingly different results: topological entropy is constant in this entire family ("rigidity"), while measure-theoretic entropy varies within Teichmüller space, taking all values ("flexibility") between zero…

A rigidity theorem for Hénon maps

The purpose of this note is two fold. First, we study the relation between a pair of Hénon maps that share the same forward and backward non-escaping sets. Second, it is shown that there exists a continuum of $Short-\mathbb{C}^2$'s that are biholomorphically inequivalent and finally, we provide examples of $Short-\mathbb{C}^2$'s that are neither Reinhardt nor biholomorphic to Reinhardt domains.

多项式Thurston刚性的代数证明

An Algebraic Proof of Thurston's Rigidity for a Polynomial

研究了临界点都具有有限前向轨道的$\mathbb{P}^{1}$的有理自映射. Thurston刚性定理指出，在一个已知的映射族之外，在$\mathbb{C}$上有许多这样的映射具有固定的度数和临界轨道长度。对于至少有一个临界点是周期的tamely分歧映射，我们给出了这一事实的代数证明。我们还产生了广泛分歧的反例。

亚纯映射和Ahlfors岛映射的刚性和不存在线场

Rigidity and absence of line fields for meromorphic and Ahlfors islands maps

本文给出了一元解析函数的锥朱利亚集上不存在不变线场的初等证明。这个证明不仅适用于有理函数和超越亚纯函数（在那里它是以前已知的），而且甚至适用于亚当·爱泼斯坦所定义的阿尔福斯岛映射的极其一般的设定。事实上，我们证明了一个更一般的结果不变_微分_的情况下，可测量的关于一个共形措施，支持（无分支）锥朱利亚集。这包括$\log|f'|$的上同调方程的研究，这些方程与一些著名的刚性问题有关。

映射刚性扰动下不动点的持久性

Persistence of fixed points under rigid perturbations of maps

设$f:S^1\times [0,1]\to S^1\times [0,1]$是一个与恒等映射同伦且保持面积形式的实解析环同态。假设对于某个提升$\tilde {f}:\mathbb{R}\times [0,1]\rightarrow \mathbb{R}\times [0,1]$，我们有${\rm Fix}(\tilde{f})=\mathbb{R}\times \{0\}$，并且$\tilde{f}$正向平移$\mathbb{R}\times \{1\}$中的点。令$\tilde{f}_ε$为刚性水平平移$(x,y)\mapsto (x+ε,y)$对$\tilde{f}$的扰动。我们表明，对于所有的$ε>0$足够小，我们有${\rm Fix} (\tilde{f}_ε)=\emptyset $。证明如下从Kerékjártó的建设布劳威尔线的方向保持同胚的平面没有不动点。这个结果在正则性假设方面是尖锐的：存在一个满足上述所有性质的自同构$f$，除了$f$不是实解析的，而只是光滑的，所以上面的结论是假的。这种映射是通过生成函数构造的。

复动力系统中的同宿轨道、乘子谱与刚性定理

Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics

本文利用复动力学与非阿基米德动力学工具，给出 McMullen 乘子谱刚性定理的新证明，并将其推广到长度谱情形，同时解决 Milnor 关于例外映射的两个猜想。