Spectral Dimension Of A Class Of Laplace Operators Defined By Asymmetric Fractal Measures With Overlaps

Spectral dimension is one of the most basic quantities in the study of Laplace operators that arise in fractal analysis.Spectral dimension can be used to study heat kernel estimates,which can in turn be used to study other problems such as wave propagation speed.In addition,spectral dimension also arises in the study of fractal Schrodinger operators.Let(?)(m?2)be an iterated function system(IFS),S:Rn?Rn be contractive similitudes,and pi,(?),pi>0 be probability weights.Then,there exists a unique probability measure ? that satisfies the identity#12? is called a self-similar measure.Ngai studied the spectral dimension of a class of one-dimensional self-similar measures that satisfy second-order identities but do not satisfy the open set condition or the post-critically finite condition.He computed the spectral dimension of the Laplacian where ? is the self-similar measure defined by the IFS (?)(1)with probability weight pi=p2=1/2.In this thesis,we focus on the spectral dimension of Laplacian which defined by self-similar measures ? with general probability weight(i.e.p1=p,p2=1-p,0<p<1/2).The thesis is divided into three chapters and arranged as follows:In Chapter 1,we introduce the research background of spectral dimension and the main results of this thesis.In Chapter 2,we introduce basic notions and results related to spectral dimension,including eigenvalue counting function,second-order identity,uni-tarily equivalent operator and vector-valued renewal equations.Then,we use properties of unitarily equivalent operators to derive a system of functional equations for the eigenvalue counting function.In Chapter 3,we calculate the spectral dimension of Laplacian defined by asymmetric infinite Bernoulli convolutions of(1).The proofs of some important lemmas,error estimates,and the main theorem are given in this chapter.