This is a survey talk on the ICMS conference "Resonances of complex dynamics", 9-13 July 2018. Three examples which show how dynamical considerations lead to new results in complex function theory are discussed. These examples are: a) a refinement of of the Wiman-Valiron method, b) the proof of Hayman's conjecture on the value distribution of derivatives of meromorphic functions, and c) a new simplified proof of the result of Lin and Wang on the number of critical points of the Green function on a torus.
In this note we review a selection of contemporary research themes in holomorphic dynamics. The main topics that will be discussed are: geometric (laminar and woven) currents and their applications, bifurcation theory in one and several variables, and the problem of wandering Fatou components.
In [M3], Milnor found Tricorn-like sets in the parameter space of real cubic polynomials. We give a rigorous definition of these Tricorn-like sets as suitable renormalization loci, and show that the dynamically natural straightening map from such a Tricorn-like set to the original Tricorn is discontinuous. We also prove some rigidity theorems for polynomial parabolic germs, which state that one can recover unicritical holomorphic and anti-holomorphic polynomials from their parabolic germs.
内容：1.拟共形手术和变形：Ben Bielefeld，Quasiconformal Surgery; Curt McMullen，Rational maps and Teichmüller space; John Milnor，Thurston's algorithm without critical finiteness; Mary Rees，A possible approach to a complex renormalization problem。2. Julia集的几何：Lennart Carleson，Julia集的几何; John Milnor，局部连通性问题。3.可测量的动态：Mikhail Lyubich，Measure and Dimension of Julia Sets; Feliks Przytycki，On invariant measures for iterations of holomorphic maps. 4.整个函数的迭代：Robert Devaney，非理性复杂动力学中的开放问题; Alexandre Eremenko和Mikhail Lyubich，全纯映射的游荡域。5.斯科特·萨瑟兰（Scott Sutherland），《牛顿法的坏多项式》（Bad Polynomials for Newton's Method）作者
This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue of the Poincaré recurrence theorem for meromorphic correspondences with respect to certain dynamically interesting measures associated with them. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result concerning invariance prop…
The Kobayashi distance in holomorphic dynamics and operator theory
这些是我在里尔2015年5月的“Aspects metriques et dynamiques en analyze complete”学校上的一个短期课程的笔记。本笔记的目的是描述如何使用几何结构（即小林距离）来探索和编码定义在复流形上的全纯函数和映射的解析性质。我们将首先描述Kobayashi距离的主要性质，然后我们将介绍应用于紧流形、强伪凸区域和凸区域中的全纯动力学，以及Bergman空间中的算子理论（Carleson测度和Toeplitz算子）。
In dimensions greater than or equal to 3, the local structure of a singular holomorphic foliation conceals a globally defined foliation on the projective space of dimension one less. In this paper, we will discuss how the global dynamics of the latter foliation exerts influence on several problems that apparently have a purely local nature. In the course of the discussion, a few recent results and open problems in the area will be reviewed as well.
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 $\mathbb{C}^2$.
It is well known that baby Mandelbrot sets are homeomorphic to the original one. We study baby Tricorns appearing in the Tricorn, which is the connectedness locus of quadratic anti-holomorphic polynomials, and show that the dynamically natural straightening map from a baby Tricorn to the original Tricorn is discontinuous at infinitely many explicit parameters. This is the first known example of discontinuity of straightening maps on a real two-dimensional slice of an analytic family of holomorphic polynomials. The proof of discontinuity is carried out by showing that all non-real umbilical cor…
We study the backward dynamics of one-parameter semigroups of holomorphic self-maps of the unit disk. More specifically, we introduce the speeds of convergence for petals of the semigroup, namely the total, orthogonal, and tangential speeds. These are analogous to speeds of convergence introduced by Bracci, yet profoundly different due to the nature of backward dynamics. Results are extracted on the asymptotic behavior of speeds of petals, depending on the type of the petal. We further discuss the asymptotic behavior of the hyperbolic distance along non-regular backward orbits.
Lucjan Emil Böttcher (1872-1937)-- the Polish pioneer of holomorphic dynamics
本文部分基于与斯坦尼斯瓦夫·多莫拉兹基的联合工作，涉及Lucjan Emil Böttcher（1872年至1937年）-一位活跃在Lwów的鲜为人知的波兰数学家。我概述了Böttcher的学术道路，重点是他在莱比锡的研究与Sophus Lie，他后来在Lwów的学术生涯和他在其进步所面临的障碍。我也非常简要地介绍了他的数学成就。鉴于后来的事态发展全纯动力学，我认为，尽管缺陷，他的工作，Böttcher应被视为不仅是一个贡献者的领域，但实际上作为其创始人之一。
