March 5, 2005 (Saturday)
Room
517, Meng Wah Complex, HKU
Professor Mitsuhiro Shishikura
Kyoto University, Japan
Renormalization
in Complex Dynamics and Teichmüller spaces
Abstract
The idea of renormalization was imported from statistical
physics to the theory of dynamical systems, and supplied important techniques
in studying the systems which involve quasi-periodic motions or phase
transitions. Given certain classes of dynamical systems, a typical way of
defining renormalization is to find a suitable subdomain
in the phase space, take the “first return map” to this domain and scale the
new system so that it belongs to the original class of maps. The
renormalization can be regarded as a dynamical system defined in a space of
dynamical systems, and key questions are existence of a fixed point and the hyperbolicity.
In the theory of holomorphic
dynamical systems, this approach was particularly successful in studying
quadratic-like maps or Feigenbaum-type
renormalization (Sullivan, McMullen, Lyubich and
others) and maps which are “rotation-like” (Yoccoz,
de Faria-deMelo, Epstein-Yampolsky).
In this talk, I will review the theory of renormalization for some classes, and
introduce a new class of renormalization which is defined for germs of holomorphic functions with parabolic or near-parabolic
fixed points. The hyperbolicity of this
renormalization will be shown by connecting the space of maps to a certain Teichmüller space, where holomorphic
self-maps do not expand its metric and with some more condition, they are
actually contracting.