2013/08/15 by Luka Boc-Thaler, Boc-Thaler, Luka, Han Peters +3
Mathematics · #32E20 #32E30 #32H02 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:32E20 #msc:32E30 #msc:32H02
paper · pdf · doi:10.48550/arxiv.1308.3407
14 pages
arxiv created 2013/08/15 · arxiv updated 2013/08/16
We study invariant Fatou components for holomorphic endomorphisms in ℙ2. In the recurrent case these components were classified by Sibony and the second author in 1995. In 2008 Ueda completed this classification by proving that it is not possible for the limit set to be a punctured disk. Recently Lyubich and the third author classified non-recurrent invariant Fatou components, under the additional hypothesis that the limit set is unique. Again all possibilities in this classification were known to occur, except for the punctured disk. Here we show that the punctured disk can indeed occur as the limit set of a non-recurrent Fatou component. We provide many additional examples of holomorphic and polynomial endomorphisms of ℂ2 with non-recurrent Fatou components on which the orbits converge to the regular part of arbitrary analytic sets.