Rational maps without Herman rings

An Experimental View of Herman Rings for Dianalytic Maps of $\mathbb{RP}^2$

We provide an experimental study of the existence of Herman rings in a parametrized family of rational maps preserving antipodal points, and a discussion of their properties on $\mathbb{RP}^2$. We study analytic maps of the sphere that project to dianalytic maps on the nonorientable surface, $\mathbb{RP}^2$. They have a known form and we focus on a subset of degree $3$ dianalytic maps and explore their dynamical properties. In particular, we focus on maps for which the Fatou set has a Herman ring. We appeal to dynamical properties of particular maps to justify our assertion that these Fatou co…

A decomposition theorem for Herman maps

In 1980s, Thurston established a topological characterization theorem for postcritically finite rational maps. In this paper, a decomposition theorem for a class of postcritically infinite branched covering termed `Herman map' is developed. It's shown that every Herman map can be decomposed along a stable multicurve into finitely many Siegel maps and Thurston maps, such that the combinations and rational realizations of these resulting maps essentially dominate the original one. This result gives an answer to a problem of McMullen in a sense and enables us to prove a Thurston-type theorem for …

具有光滑退化 Herman 环的有理映射

Rational maps with smooth degenerate Herman rings

本文利用 Avila–Buff–Chéritat 的光滑 Siegel 盘构造及连续的 Siegel 到 Herman 拟共形手术，证明存在带光滑退化 Herman 环的有理映射，并得到正面积无处稠密 Julia 集的不可重整化例子。

Herman环的先验界与有界类型旋转数的退化

A priori bounds and degeneration of Herman rings with bounded type rotation number

本文通过构造具有任意度数和组合学的有理映射，证明了具有有界类型旋转数的Herman环退化为Herman曲线，并展示了Herman曲线的重正化理论的存在。

二次Hénon映射的Exotic rotation domains和Herman环的存在性

Existence of Exotic rotation domains and Herman rings for quadratic Hénon maps

本文证明了二次Hénon映射中“Exotic rotation domains”和耗散情形下吸引性Herman环的存在性，并提供了数值生成这些Herman环的理论框架。

Rational maps with bad reduction and domains of quasiperiodicity

Consider a rational map $R$ of degree $d\geq 2$ with coefficients over the non-archimedean field $\mathbb{C}_p$, with $p$ a fixed prime number. If $R$ has a cycle of Siegel disks and has good reduction, then it was shown by Rivera-Letelier in his PhD dissertation that a new rational map $Q$ can be constructed from $R$, in such a way that $Q$ will exhibit a cycle of $m$-Herman rings. In this paper, we address the case of rational maps with bad reduction and provide an extension of Rivera-Letelier's result for these class of maps.

Sur la structure des ensembles de Fatou p-adiques

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).

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。

J-旋转有理映射的刚性及临界准圆映射

Rigidity of J-rotational rational maps and critical quasicircle maps

研究J-旋转有理映射和临界准圆映射的刚性性质，证明其在几何有限条件下的Julia集无线场，并展示对Herman环退化问题的组合刚性结果。

Rational group ring elements with kernels having irrational dimension

We prove that there are examples of finitely generated groups G together with group ring elements Q \in \bbQ G for which the von Neumann dimension \dim_{LG}\ker Q is irrational, so (in conjunction with other known results) answering a question of Atiyah.