Brushing the hairs of transcendental entire functions
| Autor: | Krzysztof Barański, Lasse Rempe, Xavier Jarque |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Mathematics::Dynamical Systems Transcendental entire maps media_common.quotation_subject Entire function Julia set Mathematics::General Topology Cantor bouquet Dynamical Systems (math.DS) 01 natural sciences 37F10 30D05 FOS: Mathematics Order (group theory) Compactification (mathematics) Mathematics - Dynamical Systems 0101 mathematics Complex Variables (math.CV) Mathematics - General Topology Mathematics media_common Mathematics - Complex Variables Plane (geometry) Mathematics::Complex Variables 010102 general mathematics Mathematical analysis General Topology (math.GN) Composition (combinatorics) Infinity 010101 applied mathematics Filled Julia set Straight brush Geometry and Topology |
| Zdroj: | Recercat: Dipósit de la Recerca de Catalunya Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) Dipòsit Digital de Documents de la UAB Universitat Autònoma de Barcelona Recercat. Dipósit de la Recerca de Catalunya instname |
| Popis: | Let f be a hyperbolic transcendental entire function of finite order in the Eremenko-Lyubich class (or a finite composition of such maps), and suppose that f has a unique Fatou component. We show that the Julia set of $f$ is a Cantor bouquet; i.e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic. We also show that if $f\in\B$ has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic, then the Julia set of f contains a Cantor bouquet. As part of our proof, we describe, for an arbitrary function $f\in\B$, a natural compactification of the dynamical plane by adding a "circle of addresses" at infinity. Comment: 19 pages. V2: Small number of minor corrections made from V1 |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|