自相似测度L-%27q--谱的收敛速率与Moran测度的重分形谱.pdf

中文摘要中文摘要本文由两个部分组成. 第一部分(即第2,3章)自相似测度的Lq-谱的收敛速率以及重分形矩测度的弱收敛. 当q≥0且开集条件满足时,文【1]得到了自相似测度的Lq一谱的收敛速率, <0时,文【l】,当q<0时Lq-谱对测度肛的微小变化相当敏感,因此对其的分析_般被认为相当困难,此时文f1】,我们首先证明了自相似测度是Ahlfors正则的,并以此为基础,得到了当q<0且开集条件成立时一大类自相似测度的Lq一谱的收敛速率,在第三章,我们利用第二章的结果,证明了口<, 我们对猜想给出了肯定的回答. 本文第二部分(即第4章)一类Moran测度的重分形谱. 文献f4】讨论了—i序列相关的齐次Moran测度的重分形谱,文【5】-谱,重分形维数函数以及重分形谱之间的关系,利用了文【5l的思想,给出驴一谱与重分形维数函数的勒让德变换等于重分形谱的一个充分条件,并以此为基础,得出了一类比(4,5],本章讨论了一类Lq一谱不存在的齐次Moran测度,我们分别得到了其上(下)伊一谱和Hausdorff(Packing) 重分形维数. 关键词;自相似集,Lq一谱,重分形机理,Moran测度,重分形豪斯道夫测度, 重分形填充测度,重分形矩测度,重分形维数函数湖北大学博上论文 Abstract ThisP印er contains two parts, Inthe firstpart ofthispaper,we study the exact rate ofconvergence of themultifractal Lq-spectra ofself-similarmeasures and weak convergence ofthe multifractalmoment measures ofself-similarmeasures. Forq≥0,if theopen setcondition satisfiedOlsen[1lhasattained theexact rate ofconvergence oftheLq——spectra of aself-similarmeasure and showed that themultifractal moment measures converges weakly tothenormalized multifractal q< hasgiven two problem of analysing L叮一spectra forq<0 isgenerally considered significantly more difficult since theL4一spectra are extremely sensitive tosmall variations ofp forq<0. Here themethod infl】is disabled forg< thechapter 2,we obtained the Ahlfors regular condition forself-similarmeasures,and attained the exact rate of convergence oftheLq-spectra of alarge setofself-similarmeasures forq<0 the open setcondition 3,we showed themultifractal moment measures converges weakly tothenormalized multifractal measures forq<0 using theresults inthechapter wegave affirmative answer totheconjectures。 Inthesecond part of thispaper,we study the singularity spectra ofsome Moran measures. The paper f4】discussed themultifractal spectrum ofsome homogeneous Moran measures assciated iseries,and thepa