Weak disks of Denjoy minimal sets

Angular and unrestricted limits of one-parameter semigroups in the unit disk

We study local boundary behaviour of one-parameter semigroups of holomorphic functions in the unit disk. Earlier under some addition condition (the position of the Denjoy - Wolff point) it was shown in [M.D.Contreras, S.Diaz-Madrigal and Ch.Pommerenke, Ann. Acad. Sci. Fenn. Math. 29(2004), No.2, 471-488] that elements of one-parameter semigroups have angular limits everywhere on the unit circle and unrestricted limits at all boundary fixed points. We prove stronger versions of these statements with no assumption on the position of the Denjoy - Wolff point. In contrast to many other problems, i…

On the convergence of Denjoy-Wolff points

If $φ$ is an analytic function from the unit disk $\mathbb{D}$ to itself, and $φ$ is not a conformal automorphism, we denote by $λ_φ$ its Denjoy-Wolff point, that is, the limit of the iterates $φ(φ(\cdotsφ(0)\cdots))$. A result of Heins shows that, given a sequence $(φ_{n})_{n\in\mathbb{N}}$ of such analytic functions that convergence pointwise to $φ$, it follows that $\lim_{n\to\infty}λ_{φ_{n}}=λ_φ$. This allows us to improve results about the contnuous extensions of the subordination functions that arise in the study of free convolutions. We also offer an alternate proof of the result of Hei…

极小康托系统的同源映射：从斯特姆系统到德若瓦系统与区间交换变换

Isogenies of minimal Cantor systems: from Sturmian to Denjoy and interval exchanges

本文以实数连分数展开为背景，引入零维系统同源映射概念，完成斯特姆、德若瓦等系统的同源映射分类，刻画区间交换变换的流等价性，证明相关猜想。

Holomorphic self-maps of the disk intertwining two linear fractional maps

We characterize (in almost all cases) the holomorphic self-maps of the unit disk that intertwine two given linear fractional self-maps of the disk. The proofs are based on iteration and a careful analysis of the Denjoy-Wolff points. In particular, we characterize the maps that commute with a given linear fractional map (in the cases that are not already known) and, as an application, determine all "roots" of such maps in the sense of iteration (if any). This yields as a byproduct a short proof of a recent theorem on the embedding of a linear fractional transformation into a semigroup of holomo…

On the abundance of $k$-fold semi-monotone minimal sets in bimodal circle maps

Inspired by a twist maps theorem of Mather we study recurrent invariant sets that are ordered like rigid rotation under the action of the lift of a bimodal circle map $g$ to the $k$-fold cover. For each irrational in the interior of the rotation set the collection of the $k$-fold ordered semi-Denjoy minimal sets with that rotation number contains a $(k-1)$-dimensional ball in the weak topology on their unique invariant measures. We also describe completely their periodic orbit analogs for rational rotation numbers. The main tool is a generalization of a construction of Hedlund and Morse which …

解析辛映射在球面、圆柱体和圆盘上显示最小的遍历性

Analytic symplectomorphisms displaying minimal ergodicity on the sphere, cylinder and disk

本文提出了一种在不同几何体上构造具有最小遍历性的解析辛映射的方法，通过推广Berger的原则实现仅3个遍历测度的构造。

A Classification of Minimal Sets of Torus Homeomorphisms

We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Periodic bounded disks can only occur in type 3. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. I…