Juan Rivera-Letelier
A classification of the periodic components of the Fatou set of $p$-adic rational maps. Each such periodic component is either an immediate attracting basin or an open affinoid, where the dynamics is quasi-periodic (the $p$-adic analogues of Siegel discs and Herman rings.) The main tool is a fixed point theorem for connected subsets of Berkovich's projective line (or p-adic hyperbolic space).
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This paper is an introduction to the dynamics of rational fractions over a p-adic field. It follows the lectures that Jean-Christophe Yoccoz gave during the États de la recherche in May 2006, and develops certain points. Most of the results to be explained here are due to Benedetto, Bézivin, Hsia, Lubin, Morton, Rivera-Letelier and Silverman. ----- Ce texte est une introduction à la dynamique des fractions rationnelles sur un corps $p$-adique. Il reprend le cours que Jean-Christophe Yoccoz avait fait lors des \emph{États de la recherche} de mai 2006 et en développe certains points. La plupart …
This text deals with birationnal diffeomorphisms of real algebraic surfaces which have simple real dynamics and rich complex dynamics. We give an example of such a transformation on P^1xP^1, then we show that this situation is exceptional and implies strong conditions on both the topology of the real locus and the real dynamics.
In this paper, we study hyperbolic rational maps with finitely connected Fatou sets. We construct models of post-critically finite hyperbolic tree mapping schemes for such maps, generalizing post-critically finite rational maps in the case of connected Julia set. We show they are general limits of rational maps as we quasiconformally stretch the dynamics. Conversely, we use quasiconformal surgery to show any post-critically finite hyperbolic tree mapping scheme arises as such a limit. We construct abundant examples thanks to the flexibilities of the models, and use them to construct a sequence…
On Possible Limit Functions on a Fatou Component in non-Autonomous Iteration
研究非自治迭代中Fatou分支上的极限函数可能性，证明其行为范围远超单函数迭代，通过拟共形手术等方法构造二次多项式序列，使其Fatou分支能得到Schlicht族所有成员作为极限函数。
This survey is an introduction to the classification of Fatou components in holomorphic dynamics. We start with the description of the Fatou and Julia sets for rational maps of the Riemann sphere, and finish with an account of the recent results on Fatou components for polynomial skew-products in complex dimension two, where we focus on the key steps in the construction giving the existence of a wandering domain for a polynomial endomorphism of $\mathbb{C}^2$.
We consider rational surface automorphisms with positive entropy. A Fatou component is said to be a rotation domain if the automorphism induces a torus action on it. Here we construct a rational surface automorphism with positive entropy with the following property: it has a rotation domain which contains both a curve of fixed points and isolated fixed points. This Fatou component cannot be imbedded into complex euclidean space, so we introduce a global linear model space and show that it can be globally linearized in this model.
Dynamics on projective K3 surfaces: Herman rings
本文构造了射影K3曲面上具有正拓扑熵和非空Fatou集的自同构，其不变域为Herman环的二维类比，解答了McMullen和Cantat的开放问题。
A Survey of Baker Wandering Domains
本文综述了Baker环状域的当前研究现状，回顾了首个例子及其他实例，探讨了其对函数奇点值和动力学的影响，并讨论了无此类域的函数类别，提出若干问题。
This thesis deals with automorphisms of real algebraic surfaces, which are polynomial transformations with a polynomial inverse. The main concern is whether their restriction to the real locus reflects all the richness of the complex dynamics. This question is declined in two directions: the topological entropy and the Fatou set. For the first one, we introduce a purely geometric quantity depending only on the surface, and we call it concordance. Then we show that the ratio of real and complex entropies is linked to this quantity. The concordance is explicitely computed for many examples of su…
Transcendental Julia Sets of Minimal Hausdorff Dimension
本文证明存在超越整函数，其朱利亚集豪斯多夫维数为1且每个法图分支无限连通，并存在单点补分支，解答了Rippon和Stallard的问题，方法基于拟共形手术。
Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlevé 6
本文研究仿射三次曲面自同构群的动力学，定义并分析其Julia与Fatou集，证明Julia集中存在稠密轨道，并揭示Fatou/Julia二分法与局部离散性二分法的关联，为Painlevé 6方程及特征簇研究提供新视角。
We prove the existence of a locally dense set of real polynomial automorphisms of C 2 displaying a wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These Fatou components have non-empty real trace and their statistical behavior is historical with high emergence. The proof is based on a geometric model for parameter families of surface real mappings. At a dense set of parameters, we show that the dynamics of the model displays a historical, high emergent, stable domain. We show that this model can be embedded into fami…
We study pairs $(f, Γ)$ consisting of a non-Archimedean rational function $f$ and a finite set of vertices $Γ$ in the Berkovich projective line, under a certain stability hypothesis. We prove that stability can always be attained by enlarging the vertex set $Γ$. As a byproduct, we deduce that meromorphic maps preserving the fibers of a rationally-fibered complex surface are algebraically stable after a proper modification. The first article in this series examined the limit of the equilibrium measures for a degenerating 1-parameter family of rational functions on the Riemann sphere. Here we co…
Schwarz-Pick and Denjoy-Wolff for uniformly quasiregular maps
本文证明单位球上一致拟正则自映射的Denjoy-Wolff定理，利用Schwarz-Pick引理和链度量，并应用于稳定Fatou分量的分类，构造了非单射例子。
We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if $A$ and $B$ are two connected components of the Julia set, then one of $A$ and $B$ surrounds the other. From this, it is shown that each connected component of the Fatou set is either simply or doubly connected. Moreover, we show that the Julia set of such a semigroup is uniformly perfect. An upper estimate of the…
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