Spectral And Non-spectral Self-affine Measures In R~n

In this thesis, we will consider spectral and non-spectral problems of self-affine measures in Rn. This problems originate from the conjecture of Fuglede in 1974, and the discovery of Jorgensen and Pedersen that some fractal measures also admit exponential orthonormal bases, but some do not. It generates a lot of interest in understanding what kind of measures are spectral measure. For those measures failing to have exponential orthonormal bases, one of the main problems is to estimate the number of orthogonal exponentials in L2(μ).The self-affine measure μM,D corresponding to an expanding matrix M ∈ Mn(Z) and a finite subset D (?)Zn is supported on the attractor (or invariant set) of the iterated function system{φd(x)= M-1(x+d)}d∈D.It only depends upon an expanding matrix M and a finite digit set D. The spectral and non-spectral problems on μM,D, including the spectrum-tiling problem implied in them, have re-ceived much attention in recent years. The spectral problem of self-affine measures in Rn which originates from the research and spectral conjecture of Jorgensen and Pedersen. The non-spectral problem of self-affine measures in Rn which originates from the non-spectral conjecture of Li.This thesis considers spectrality and not-spectrality of self-affine measures μM,D with two-elements digit set, three-elements digit set, four-elements digit set and direct-sum-forms digit set. It should be noted that the currently available best results for spectrum and non-spectrum research dimension of the problem is less three-dimensional situation. The conclusions obtained relative to three dimen-sional case, we generalized the current best results, and answered the spectrum and non-spectrum conjectures in the part of the high dimensional open problem. Moreover, this paper gives some methods of solving such problems also have some reference.The structure of this paper is as follows.In Chapter 1, we firstly introduce the background and significance of spectral and non-spectral problems of self-affine measures, then list the definitions of self-affine measures, spectral measures, compatible pair and some known conclusions. Finally, we state the main results of this thesis.In Chapter 2, we study the spectral and non-spectral problems of self-affine measures μM,D with two-elements digit set in Rn. In Section 1, we consider general characterization of the zero set Z(μM,D), and give relations inside the zero set Z(μM,D).In Section 2, for the non-spectral conjecture, we estimate the number of orthogonal exponentials in L2(μM,D).In Section 3, for the spectral conjecture, we show that the L2(μM,D) possesses infinite families of orthogonal exponentials, and one of theses families forms an orthogonal exponential basis. That is, μM,D is a spectral measure. Such a spectral measure can be obtained without the condition of compatible pair. It should be pointed out that in Rn, the self-affine measure μM,D with two-element digit set is the extension of the Bernoulli convolution (or Bernoulli measure). The fourth section firstly cites some open problems, then we show that the main research methods of the spectral and non-spectral problem of self-affine measures in the chapter are used in the dimension n> 3.In Chapter 3, we study the spectral and non-spectral problems of self-affine measures μM,D with three-elements digit set in Rn. In Section 1, we consider characterization of the zero set Z(μM,D).In Section 2, we obtain some relations inside the zero set Z(μM,D) and some properties of the zero set Z(μM,D).In Section 3, for the non-spectral conjecture, we estimate the number of orthogonal exponentials in L2(μM,D).This provides some supportive evidences about the non-spectral conjecture. In Section 4, for the spectral conjecture, we show that the L2(μM,D) possesses an orthogonal exponential basis. That is, μM,D is a spectral measure. This provides some supportive evidences about the spectral conjecture. The five section firstly cites some open problems, then we show that the main research methods of the spectral and non-spectral problem of self-affine measures in the chapter are used in the dimension n> 3.In Chapter 4, we study the spectral and non-spectral problems of self-affine measures μM,D with direct-sum-forms digit set in Rn. Based on the study on spectrality and not-spectrality of self-affine measures in Chapter 2 and Chapter 3, for the non-spectral conjecture, we estimate the number of orthogonal exponentials in L2(μM,D).And for the spectral conjecture, we show that the L2(μM,D) possesses an orthogonal exponential basis. That is, μM,D is a spectral measure. The last section firstly cites some open problems, then we show that the main research methods of the spectral and non-spectral problem of self-affine measures in the chapter are used in the dimension n> 3.In Chapter 5, we study the spectral and non-spectral problems of self-affine measures μM,D with a finite digit set in Rn. In Section 1, we list all known con-clusions about the spectral and non-spectral problems of self-affine measures with four-elements digit set. In Section 2, we consider the spectrality and not-spectrality of a class of self-affine measures with a finite digit set. In Section 3, in order to study the relations inside the zero set Z(μM,D), we propose a method of classi-fication matrix. In Section 4, we give some sufficient conditions for finite and infinite families of orthogonal exponentials. In Section 5, for the non-spectral con-jecture, we estimate the number of orthogonal exponentials on the generalized three-dimensional Sierpinski gasket. For the spectral conjecture, we show that the L2(μM,D) possesses an orthogonal exponential basis. That is,μM,D is a spectral measure. This provides some supportive evidences about the spectral conjecture and non-spectral conjecture.In Chapter 6, we list some open problems.