In this project we have given the prerequisites for, and developed the notion of, a conformal renormalisation scheme for compact sets whose corresponding Green's function has atleast one critical point. The conformal renormalisation scheme works by utilising a critical point of a Green's function, to construct a suitable quotient topology. We show that there exist an atlas of charts such that our quotient topology is a Riemann surface. By the uniformisation theorem, this Riemann surface is conformally equivalent to the Riemann sphere.
Given a non-polar and compact set in the complex plane, we can apply the conformal renormalisation scheme to obtain a conformal canonical map, between a subset of the compact set and its renormalised version. We then relate the Green's function for the compact set, to the Green's function for the renormalisation.
We apply the conformal renormalisation scheme to put forward a conjecture for the asymptotic behaviour of renormalisable component of disconnected sets. We give some motivation for why one would belive said conjecture, as well as an immediate consequence of the conjecture.
It is concluded that the conformal renormalisation scheme have prospects to help further our understanding of the asymptotic behaviour of logarithmic capacity of renormalisable components of disconnected compact sets.
|Uddannelser||Basis - Naturvidenskabelig Bacheloruddannelse, (Bachelor uddannelse) Bachelor|
|Udgivelsesdato||17 dec. 2019|
|Vejledere||Carsten Lunde Petersen|
- Complex Analysis
- Complex Dynamics
- Potential Theory