The Properties Of Compatible Pair And The Orthogonal Exponentials Of Bernoulli Convolution Measure

In this paper, we study and investigate the properties of the compatible pair and the orthogonal exponentials for Bernoulli iterated function systems. Let M be an expanding integer matrix, D be a finite sets, μM,D is a self-affine measure associates with iterated function system and A(M, S) is defined by The key points and the main contents of this paper are as follow:In Chapter one, we mainly introduce some elementary concepts and corollaries, accordingly, some present research and results are described.In Chapter two, the properties of the compatible pair are discussed. Firstly, we study the relations of the compatible pair, the integral self-affine tile and the compatible pair, the integral self-affine tile and the spectral pair, respectively, explore the properties of the compatible pair and the integral self-affine tile.Furthermore,the relations between the compatible pair and the spectral pair are discussed. Finally, we obtain a necessary and sufficient condition for (μM,D, A(M, S)) to be a spectral pair. Condition of a spectral pair is given and some existing results are given.In Chapter three, we investigate the orthogonal exponentials for Bernoulli iterated function systems when λ=(1/2,1)∩Q. Especially, the question whether some of these Ak sets can be combined to form larger collections of orthogonal exponentials is discussed.