Some pinching deformations of the Fatou function

Patricia Domínguez ¹   ; Guillermo Sienra ²

¹ Facultad de Ciencias Físico Matemáticas Benemérita Universidad Autónoma de Puebla Puebla, Mexico
² Departamento de Matemáticas Facultad de Ciencias UNAM México, D.F., Mexico

Fundamenta Mathematicae, Tome 228 (2015) no. 1, pp. 1-15

Résumé

We are interested in deformations of Baker domains by a pinching process in curves. In this paper we deform the Fatou function $F(z)=z+1+e^{-z}$, depending on the curves selected, to any map of the form $F_{p/q} (z)=z+e^{-z}+2{\pi }ip/q$, $p/q$ a rational number. This process deforms a function with a doubly parabolic Baker domain into a function with an infinite number of doubly parabolic periodic Baker domains if $p=0$, otherwise to a function with wandering domains. Finally, we show that certain attracting domains can be deformed by a pinching process into doubly parabolic Baker domains.

Détail

DOI : 10.4064/fm228-1-1

Keywords: interested deformations baker domains pinching process curves paper deform fatou function z depending curves selected map form z rational number process deforms function doubly parabolic baker domain function infinite number doubly parabolic periodic baker domains otherwise function wandering domains finally certain attracting domains deformed pinching process doubly parabolic baker domains