L~q-Spectrum Of A Class Of Asymmetric Self-similar Measures With Overlaps

In this thesis,we study the Lq-spectrum of a class of asymmetric self-similar measures ? with overlaps,where q>0 and ? does not satisfy the open set condition.We can use second order identities to calculate ?(T1TJT1A)and give a formula for the Lq-spectrum.For this purpose,we derive the renewal equation by using an equivalent definition of ?(q).The main difficulty of this paper is to estimate the error term of the renewal equation.This thesis consists of five chapters,arranged as followsIn Chapter 1,we give a brief summary of the research background related to spectral dimension and Lq-spectrum,and state some main results of this thesis.In Chapter 2,we introduce some related definitions,such as iterated function system,self-similar measure,Lq-spectrum,dimension of Lq-spectrum,the renewal equation and so on.We derive a family of the second-order identitiesIn Chapter 3,we calculate ?(T1TJT1A)by using the self-similar identity,and give the matrix form of cj;which is needed to estimate the error termIn Chapter 4,we obtain the renewal equation and a formula for ?(q)by using the equivalent definition of ?(q).we prove that ?(q)is equal to the solution of a particular equation,which is convenient for us to calculate ?(q)In Chapter 5,we estimate the error term of the renewal equation.