Stable components in the parameter plane of transcendental functions of finite type
| Autor: | Núria Fagella, Linda Keen |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Funcions de variables complexes
Shell (structure) Holomorphic function Boundary (topology) Dynamical Systems (math.DS) 01 natural sciences 37F10 Transcendental functions Functions of complex variables Parameter spaces 0103 physical sciences Simply connected space FOS: Mathematics Differentiable dynamical systems 0101 mathematics Mathematics - Dynamical Systems Mathematics Meromorphic function Plane (geometry) Transcendental function 010102 general mathematics Mathematical analysis Sistemes dinàmics diferenciables Exponential function Holomorphic dynamics 010307 mathematical physics Geometry and Topology |
| Zdroj: | Dipòsit Digital de la UB Universidad de Barcelona Recercat. Dipósit de la Recerca de Catalunya instname Dipòsit Digital de Documents de la UAB Universitat Autònoma de Barcelona Recercat: Dipósit de la Recerca de Catalunya Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Popis: | We study the parameter planes of certain one-dimensional, dynamically-defined slices of holomorphic families of entire and meromorphic transcendental maps of finite type. Our planes are defined by constraining the orbits of all but one of the singular values, and leaving free one asymptotic value. We study the structure of the regions of parameters, which we call {\em shell components}, for which the free asymptotic value tends to an attracting cycle of non-constant multiplier. The exponential and the tangent families are examples that have been studied in detail, and the hyperbolic components in those parameter planes are shell components. Our results apply to slices of both entire and meromorphic maps. We prove that shell components are simply connected, have a locally connected boundary and have no center, i.e., no parameter value for which the cycle is superattracting. Instead, there is a unique parameter in the boundary, the {\em virtual center}, which plays the same role. For entire slices, the virtual center is always at infinity, while for meromorphic ones it maybe finite or infinite. In the dynamical plane we prove, among other results, that the basins of attraction which contain only one asymptotic value and no critical points are simply connected. Our dynamical plane results apply without the restriction of finite type. Comment: 41 pages, 13 figures |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|