pretty thingUoP maths seminar (2013-14)

Renormalization for Siegel Discs.

Speaker: Andrew Burbanks

Venue: Wednesday 19th March 2014, 2pm, LG2.04a (refreshments provided)

Abstract: This talk will discuss the domain of linearizability - otherwise known as a "Siegel disc" - around an irrationally indifferent fixed point of a complex map.

The question of the existence of a Siegel disc leads to a "small divisor problem"; it is heavily dependent on the number-theoretic properties of the rotation number of the map. In 1942, Siegel gave the first proof of convergence of a small divisor series by showing that a domain of linearizability exists provided that the rotation number satisfies a Diophantine condition. This proof was a precursor of the celebrated "KAM theorem" of dynamical systems theory.

Attention has been focused on universal scaling properties that are observed on the boundary of the Siegel disc. This talk will show how these properties are related to the existence of fixed points of a renormalization operator. (Analogous scenarios occur in the study of the dynamics of circle maps and twist maps.)

By comparison with the simpler renormalization scenario constructed for period-doubling in 1-dimensional unimodal maps of the interval, we demonstrate how the existence of such renormalization fixed points may be established. This is done by means of a computer-assisted proof, in which rigorous error-bounds are maintained during all calculations.