Renormalization for Siegel discs

Student thesis: Doctoral Thesis


This thesis is concerned with a domain of linearizability, otherwise known as a Siegel disc, around an irrationally indi erent fixed point of a complex analytic map. In particular, we investigate the existence of Siegel discs and examine the properties of their boundary curves for golden mean rotation number. The key tool used is the idea of a renormalization operator acting on a space of functions. Firstly, a computer-assisted proof is discussed and verified, which establishes the existence of a fixed point of the relevant renormalization operator. In particular, the proof yields a ball of functions around an approximate fixed point that is guaranteed to contain the true fixed point. The rigorous computational techniques which allow computers to be used for this purpose are then discussed. Given the existence of the renormalization fixed point, we verify certain topological conditions, known as the necklace hypotheses, on the action of the maps making up the fixed point. This proves the existence of a Siegel disc having a Holder continuous (invariant) boundary curve for all maps attracted to the fixed point. Further, it is shown that the motion on the boundary is conjugate to a pure rotation, that the boundary curve passes through a critical point of the map, and that the conjugator is not differentiable on a dense set of points. Finally, by viewing the invariant curve as the limit set of an iterated function system (IFS), a further investigation is made to get rigorous bounds on the fractal dimension
of the Siegel disc boundary. This involves calculating bounds on the contractivities and coercivities of the maps of the IFS and solving corresponding partition equations. In
particular, a rigorous upper bound on the dimension of 1.08523 is obtained.
Date of Award1997
Original languageEnglish
Awarding Institution
  • Loughborough University
SponsorsEngineering and Physical Sciences Research Council

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