Some pinching deformations of the Fatou function
Tom 228 / 2015
Fundamenta Mathematicae 228 (2015), 1-15
MSC: Primary 37F10; Secondary 37D05.
DOI: 10.4064/fm228-1-1
Streszczenie
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.
