Study On Some Problems In Fractal Geometry

Fractal geometry is a brand-new branch of mathematics,which was found by Mandelbrot in 1975.At present,Fractal geometry intersects with many branches of mathematics,such as Fourier analysis on fractals,Wave analysis on fractals,Brownian motion on fractals,Complex analysis on fractals,etc.One of the basic problems of Fourier analysis on fractals is the existence of orthogonal basis of exponential functions,that is,the spectral problem of fractal measure.In 1998,Jorgensen and Pederson discovered the first singular non-atomic spectral measure.This surprising discovery attracted widespread research interest.This cross-research has achieved rich results and becomes a popular research area.The cross-research of Fractal geometry and Complex analysis stems from the study of the analytical properties and geometric behavior of the Cauchy transform of selfsimilar measures by Lund ect.in 1998.Lund ect.studied the Cauchy transform of the normalized Hausdorff measure on the Sierpinski gasket,and put forward Cantor set conjecture(or Cantor boundary behavior).Their result expands the research content of Fractal geometry,and opens up a research direction of mathematics-Complex analysis on fractals.This thesis is mainly on Fourier analysis on fractals and Complex analysis on fractals.The first part is to study the spectrality of fractal measures,including the non-spectral property of m-Bernoulli convolution,the infinite orthogonality of twoelement Moran measures and the spectrality of three-element Moran measures;the second part is to consider the starlikeness and convexity of Cauchy transform.This thesis includes six chapters,the specific arrangement is as follows:In the first two chapters,we introduce the research questions and research background,main results and innovations of this thesis.In the third chapter,we mainly study of the non-spectral properties of m-Bernoulli convolution ??,m under the conditions that ?=±(q/p)1/r and ged(p,m)=1.We obtain that if gcd(q,m)=1,then there exist at most m orthogonal exponential functions in L2(??,m);if gcd(q,m)>1,then there are any number of orthogonal exponential functions in L2(??,m).In the fourth chapter,we mainly characterize the equivalent condition of the infinite orthogonality of the Moran measure generated by real contractions ?,?,?,?,…and the digit set {0,1}.We obtain that L2(?{?n},{0,1})admits infinite orthogonal exponential functions if and only if ??=(p/q)1/n,where r ? N+,pis odd,q is even and p<q.In the fifth chapter,we mainly consider the spectral and non-spectral properties of the Moran measure generated by integral contractions {kn} and digit sets {0,1,sn/tn},where kn ?3Z\{0} and {sn,tn}=?1,2}(mod 3).We obtain a sufficient condition for the Moran measure to become a spectral measure and give the form of its spectra.In the sixth chapter,we mainly study the analytic properties of the Cauchy transform.Let K be the square with vertex {1,i,-1,-i} and ? be the normalized twodimensional Lebesgue measure on K and F(z)=?Kd?(w)/(z-w)be the Cauchy transform of?.We obtain that F(z)is a starlike function on(C\K but not a convex function.Our result provides an idea for solving the conjecture of Cauchy transform on Sierpinski carpet.