Spectral Dimension And Multifractal Decomposition Of Fractal Measure

This thesis consists of five chapters.We mainly study two topics:first,we study the spectral dimension of self-similar measures that are essentially of finite type on R;second,we study the Lq-spectrum of self-similar measures that are essentially of finite type on Rd(d>1).In this thesis,the essentially of finite type condition is firstly proposed,and Strichartz second-order identities are generalized.In Chapter 1,we introduce the backgrounds related to spectral dimension and Lq-spectrum,and state the main results of this thesis.In Chapter 2,we introduce some related definitions,such as neighbor,neighborhood,(neighborhood)type,(neighborhood)measure type,finite type condition,island,(island)type,(island)measure type,nonbasic island,cell,essentially of finite type(EFT),and so on.In Chapter 3,we introduce some examples satisfying(EFT)on R and R2.In Chapter 4,we study the spectral dimension of self-similar measures satisfying(EFT)on R.We derive a vector-valued renewal equation for eigen-value counting functions.Using(EFT)condition and properties of unitarily equivalent operators,and applying the vector-valued renewal theorem of Lau et al.[33],we prove that the solution of the renewal equation is bounded,and hence obtain the spectral dimension.In Chapter 5,we study the Lq-spectrum of self-similar measures satisfying(EFT)on Rd(d>1).We derive a vector-valued renewal equation for the integral defining the Lq-spectrum,and then apply the vector-valued renewal theorem again to prove that the solution of the renewal equation is bounded,which allows us to obtain the Lq-spectrum.