Singularity Of Certain Self-affine Measures And μ_γ Orthogonality

This thesis mainly investigates the singularity of self-affine measures which are produced by iterated function systems and the property of the Bernoulli measure μλ (λ∈(0,1)) which is a one parameter system of compact supported Broel probability measures.The main goal is to obtain a class of singular self-affine measure and a class of absolutely continuous self-affine measure for different A,D and P,and to study the maximal orthogonal exponentials in L2(μλ) for a given λ. The main results are as follows:(1)A sufficient condition on singularity of some self-affine measures is obtained by using the definition.A class of absolutely continuous self-affine measure is found by using Fourier transform.The condition under which the self-affine measure is singular is investigated by the estimation of the lower bound of the sequence and the Riemann-Lebesgue Lemma.When the weights of the iterated function system are extended from real numbers to complex,a type of singular measure is obtained.(2) Though E (Tk) may not be an ONB in L2(μλ),it may be the maximal system.By applying the property of Гk and the zero set of μλ,we show that E{Гk{3/8}} is the maximal orthogonal exponentials in L2(μ3/8). The special conclusion is extended to the general conclusion by the similar method and we show that E{Гk{p/2n}}(p is an odd) is the maximal orthogonal exponentials in L2(μp/2n). The results of this thesis improve the related conclusions finished by P.E.T.Jorgensen and J.-L.Li.It is important to deeply study the singularity of self-affine measure and the spectrality of the Bernoulli measure.