UoP 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.