Spectral Property Of Self-affine Measures

Fractal geometry has penetrated into all major branches of mathematics,pro-moted cross-fusion of the subject directions,and achieved rich research results.For example:On the cross-research of fractal geometry and harmonic analysis,Jor-gensen and Pedersen first discovered a singular,non-atomic fractal measure(one-fourth Cantor measure)? admits an exponential orthonormal basis EA={e2?i<?,x>:??A} for the space L2(?).This surprising discovery quickly made the Fourier analysis on fractal sets to be a hot topic in mathematics.The measure ? with the above properties is called a spectral measure and A is called a spectrum for ?.This thesis consists of two parts.In the first part,we consider the spectral property of fractal measures.In the second part,we study the Loewner differential equation in the upper half-plane H.The study of the spectrality of fractal measure ? is to investigate whether the function in L2(?)has a Fourier expansion,which is a very significant thing.To determine the spectral property of ?,we need to figure out the distribution of ze-ros for the Fourier transform ? of ?.At present,little is known about this,but there are various expectations.The main contribution of this thesis is to construct a class of self-affine measures,which are generalization of the well-known Bernoulli convolution in high dimensional space.These measures provide various examples such that the distribution of zeros for the Fourier transform ? exactly match peo-ple's expectations.We investigate these spectral properties from the knowledge and skills of analysis and algebra:If ? is a spectral measure,we find the spectrum A;otherwise,we determine the maximum number of orthogonal exponential function-s EA.Some special case of these results contains some of the recent well-known conclusions[9,22,55]etc..The second task of our research on spectrality of fractal measures is to partially prove the non-spectral conjecture[54,55]:we prove that if the zero set of the mask polynomial of digit set D on(0,1]n is a finite and rational set,then the non-spectral conjecture is true.The third task is to give some sufficient conditions for certain types of Sierpiriski-Moran measures to become spectral measures,and some spectra A are found.The second part of this thesis is to discuss the Loewner differential equation in the upper half plane H:(?),where ?(t)is a continuous real-valued function,called the driving function;and b(t)? Cl is the half-plane capacity.If the driving function is random,the equation becomes the famous stochastic Loewner evolutions(SLE),which belongs to the cross-domain of stochastic analysis,complex analysis,and fractal geometry.SLE is the mainstream direction of mathematics,and it also has important significance in statistical.physics,seepage and other disciplines.We study the trace Kt in this equation as the circular arc slit (?).The exact expressions of the driving function ?(t)and the half-plane capacity b(t)are given,and thus we get the exact growth order of the driving function ?(t)near t = 0· These two quantities reflect the geometric and analytical properties of mapping gt(z),which occupy a very important position in the theory of SLE.