Spectrality Of Self-affine Measures

Fractal geometry is a very active theory in today's world.Its appear-ance makes people describe the world from a new angle.With the intersection and the integration of disciplines,fractal geometry has close relationship with major branches of mathematics.For example:On the cross-research of frac-tal geometry and harmonic analysis,Jorgensen and Pedersen first discovered that the singular,non-atomic measure(one-fourth Cantor measure ?1/4)was spectral measure.This discovery opens up a new research direction for the intersection of fractal geometry and fourier analysis(Fourier analysis on frac-tals).This direction has quickly become a research hotspot in mathematics.Let ?be a compactly supported probability measure on Rn and ?(?)Rn be a discrete set.We call ? a spectral measure if E(?)={e2?i?x:???}forms an orthonormal basis for L2(?)In this case,we call ? a spectrum of ?.Let M ? Mn(R)be an n x n expanding real matrix,D(?)Rn be a finite digit set,and let the iterated function system {?d(x)}d?D defined by{?d(x)=M-1(x+d)}d?D determine the unique attractor.From[37],we can see that the iterated func-tion system {?d(x)}d?D generates a unique self-affine measure ?:=?M,D,which satisfies?=1/|D|(?)?(?)??-1.In this thesis,we mainly consider the spectrality of self-affine measures?=?M,D,i.e.,investigate whether the function in L2(?)has a Fourier expan-sion.Since the spectral property of the measure ?M,D has close relationship with its Fourier transformation ?M,D,we can establish the necessary and suf-ficient conditions for some kind of self-affine measures are spectral measures or nonspectral measures,by the distribution of zeros for ? and the knowledge and skills of analysis,number theory,algebra and so on.This thesis consists of five chapters.In Chapter 1,the background,motivation and current situation of self-affine measures are introduced.At the same time,the main results of this thesis are given.[11]gives the necessary and sufficient conditions for the existence of an infinite orthogonal set of exponential functions for the self-similar measures generated by M=1/??R and D={0,1,…,m-1}(?)Z,where 0<|?|<1 and m? 2 is a prime number.In Chapter 2,we extend this important result to the case of arbitrary integer.If A is a spectrum of self-similar measure ?,then there has an interesting question that under which assumptions,K? is also a spectrum of self-similar measure ?.This problem is called the problem of scaling spectra.In Chapter 3,we discuss the scaling spectra of self-similar measures ?M,D generated by M=bn and D={0,1,…,b-1} C N,where b ? N+.We give sufficient conditions for the scaling spectra of ?M,D.In Chapter 4,we consider the spectrality of self-affine measures generated by the digit set and arbitrary integer matrices.We divide all integer matrices into four disjoint classes.Then we prove that the matrices have the same spectrality in each class.In Chapter 5,we study the digit set D={(0,0)t,(?1,?2)t,(?3,?4)t}(?)k(?1?4-?2?3}D(?)Z2 with decomposable form,where D is a digit set of finite integers and k,?1 ?i?4)are all integers.Let M ? M2(Z)be an expanding matrix satisfying gcd(det(M),3)=1.Then the maximal number of the orthogonal exponential functions in L2(?M,D)is 9?+8,where ?=max{r:3r|(?1?4-?2?3)}.Research significance:(1)For the self-similar measures ? with con-tinuous digit sets in R,we give the necessary and sufficient conditions for the existence of an infinite orthogonal set of exponential functions in L2(?).(2)We prove that under which conditions a class of self-similar measures have scaling spectra.(3)For a class of self-affine measures generated by the three integer digit sets,we skillfully use conjugate transformation to clarify the spectral properties of these self-affine measures.This provides a theoretical basis for using Fourier expansion theory to study L2(?).(4)For the self-affine measures with decomposable digit sets,we solve the problem that under which condi-tions these measures contain finite orthogonal sets of exponential functions.These enrich the research content of fractal analysis.