Renormalization Map’s Properties

Abstract:In the1970s, in order to explain some physical phenomena, physi-cists wanted to define the "Laplace" operator on a fractal set. Up to now, mathematicians have developed two methods to construct energy forms(ie,the Laplacian) on the finitely ramified self-similar sets. One comparatively direct approach is inner graph approximation:constructing a self-similar compatible resistance networks sequence on graph sequence. however, to get this com-patible sequence, we need to find an energy E and weight r such that the renormalization map Δr meets Δr(E)=E.This paper mainly discusses some properties for more general renormaliza-tion mapping Mn(E). First of all, we introduce, some basic concepts and the energy form on fractals taking Sierpinski gasget, Vicsek snowflake for example. Then we introduce some properties of renormalization mapping Mn(E):posi-tively Homogeneous. monotonicity,etc. Finally, with help of Hilbert Projection metric, a property on Mn(E)(see Proposition3.2.1) is obtained.