Extending local analytic conjugacies between parabolic fixed points

• Main Author: Morris, David
• Published: University of Warwick 2017
• Subjects: 510; QA Mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.742250

The focus of this thesis is a study of the extension properties of local analytic conjugacies between simple parabolic fixed points. Any given conjugacy itself will generally not have an extension to the immediate basin. However, we show that if both maps belong to a suitable class (which includes polynomial-like maps and rational maps with a simply connected parabolic basin) then for all n large enough g on o X does have an analytic extension to the immediate parabolic basin. We begin by studying qualitative models for the dynamics near a parabolic fixed point, leading us to the Parabolic Flower Theorem. We then construct Fatou coordinates, which conjugate f to the unit translation, and study extension and properties of these maps. By restricting ourselves to the case when the restriction of f to its parabolic basin is a proper map with finitely many critical points we are able to study covering properties of these extended Fatou coordinates. We also introduce the horn map and lifted horn maps and show that the former is a complete invariant of the local analytic conjugacy class. Working from the covering properties of the horn map, we develop an intuition for how critical orbits of two maps f and g with locally conjugate simple parabolic fixed points should be related. In our main theorem, Theorem 3.1.10, we show that if both maps have a proper parabolic basin and is a local analytic conjugacy from (f; z0) to (g; w0) then for all n large enough, the map g on o X has an analytic extension along any curve starting in a region near z0 contained in the basin of z0. Under the additional assumption that the immediate basin is simply connected we can then conclude that the map Xn := g on o X has an analytic extension to a semi-conjugacy between the immediate basins whenever n is large enough.