几类不同分形集的维数和一类非对称Cantor集的上下密度
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摘要
本博士论文由四部分组成,第一部分引入一些基本概念、介绍我们所研究的问题背景以及前人的研究工作;第二部分研究一类由组频率诱导的莫朗(Moran)集子集的分形维数;第三部分考虑一类Cantor函数不可微点的维数问题;第四部分具体给出一类非对称Cantor集在每一点的上下密度并给出证明.
第二章我们研究了一类由组频率诱导的莫朗集子集的分形维数.一般情况下,为证明一给定集合的Hausdorff和Packing维数,需首先猜测其维数公式,这通常较为困难.但对这类由组频率诱导的特定子集,我们直接给出并证明其Hausdorff和Packing维数公式.结果表明,该类集合为正则集(即Hausdorff维数等于Packing维数),且其Hausdorff和Packing维数可套用公式计算而无需猜测.
第三章我们研究了一类Cantor函数不可微点的维数问题.目前所知结果均要求对任意i,p_i＞a_i(P_i为一给定概率向量的第i分量,a_i为产生Cantor集的迭代函数系统的第i个函数的压缩比).然而,若存在i,使得P_i＜a_i,已知文献中的办法将不再适用,这时猜测并证明该目标集的分形维数比较困难.我们在具体分析了Cantor函数不可微点的结构后,巧妙地联系起Olsen在文[43,45]中关于编码组频率发散点的结果,解决了该问题.
第四章我们研究了一类非对称Cantor集在每一点的上下密度.丰德军、华苏和文志英等在[16]中给出一类对称Cantor集的具体上下密度.而对非对称Cantor集,已知参考文献未有结果.
This dissertation consists of four parts.The first part introduces some basic concepts and the background of our research work as well as the work done by others.The second part is to study the dimensions of a class of subset of Moran fractals related to frequencies of their codings.The research of dimensions of non-differentiability points of a class of Cantor function is arranged at the third part.While the fourth part is devoted to the solution of the concrete upper and lower densities of a kind of non-symmetric Cantor set.
In chapter 2,we study the dimensions of a class of subset of Moran fractals induced by the mixed group frequencies in the codings.Generally,to prove the Hausdorff and Packing dimensions of a given set,we need to guess its dimension formula,which is usually hard to get.However,for some specific subsets of Moran fractals,we can treat their Hausdorff and Packing dimensions in a unified manner.The advantage of our approach is that the Hausdorff and Packing dimensions do not guess to be a priori.
In chapter 3,we study the dimensions of non-differentiability points of a class of Cantor function.As far as we know,the condition 'for all i,p_i > a_i' is required in all papers about the non-differentiability points of Cantor function,where P_i is the i-th component of a given probability vector and a_i is the contraction ratio of the i-th function of the Iterated Function System which generates the Cantor set.However, if there exists at least one i,such that p_i < a_i,how to guess and prove the fractal dimensions of non-differentiability points of a Cantor function is a difficult question.To crack this problem,we analyzed the structure of non-differentiability points of Cantor functions,and finally ingeniously used the conclusions of[43,45]to achieve our goal.
In chapter 4,we study the concrete upper and lower densities of a kind of non-symmetric Cantor set.Feng,Hua and Wen[16]got the upper and lower densities of Cantor sets based on the requirement that the target Cantor set is symmetric.However, we derived some concrete results about a kind of non-symmetrical Cantor set.
第二章我们研究了一类由组频率诱导的莫朗集子集的分形维数.一般情况下,为证明一给定集合的Hausdorff和Packing维数,需首先猜测其维数公式,这通常较为困难.但对这类由组频率诱导的特定子集,我们直接给出并证明其Hausdorff和Packing维数公式.结果表明,该类集合为正则集(即Hausdorff维数等于Packing维数),且其Hausdorff和Packing维数可套用公式计算而无需猜测.
第三章我们研究了一类Cantor函数不可微点的维数问题.目前所知结果均要求对任意i,p_i＞a_i(P_i为一给定概率向量的第i分量,a_i为产生Cantor集的迭代函数系统的第i个函数的压缩比).然而,若存在i,使得P_i＜a_i,已知文献中的办法将不再适用,这时猜测并证明该目标集的分形维数比较困难.我们在具体分析了Cantor函数不可微点的结构后,巧妙地联系起Olsen在文[43,45]中关于编码组频率发散点的结果,解决了该问题.
第四章我们研究了一类非对称Cantor集在每一点的上下密度.丰德军、华苏和文志英等在[16]中给出一类对称Cantor集的具体上下密度.而对非对称Cantor集,已知参考文献未有结果.
This dissertation consists of four parts.The first part introduces some basic concepts and the background of our research work as well as the work done by others.The second part is to study the dimensions of a class of subset of Moran fractals related to frequencies of their codings.The research of dimensions of non-differentiability points of a class of Cantor function is arranged at the third part.While the fourth part is devoted to the solution of the concrete upper and lower densities of a kind of non-symmetric Cantor set.
In chapter 2,we study the dimensions of a class of subset of Moran fractals induced by the mixed group frequencies in the codings.Generally,to prove the Hausdorff and Packing dimensions of a given set,we need to guess its dimension formula,which is usually hard to get.However,for some specific subsets of Moran fractals,we can treat their Hausdorff and Packing dimensions in a unified manner.The advantage of our approach is that the Hausdorff and Packing dimensions do not guess to be a priori.
In chapter 3,we study the dimensions of non-differentiability points of a class of Cantor function.As far as we know,the condition 'for all i,p_i > a_i' is required in all papers about the non-differentiability points of Cantor function,where P_i is the i-th component of a given probability vector and a_i is the contraction ratio of the i-th function of the Iterated Function System which generates the Cantor set.However, if there exists at least one i,such that p_i < a_i,how to guess and prove the fractal dimensions of non-differentiability points of a Cantor function is a difficult question.To crack this problem,we analyzed the structure of non-differentiability points of Cantor functions,and finally ingeniously used the conclusions of[43,45]to achieve our goal.
In chapter 4,we study the concrete upper and lower densities of a kind of non-symmetric Cantor set.Feng,Hua and Wen[16]got the upper and lower densities of Cantor sets based on the requirement that the target Cantor set is symmetric.However, we derived some concrete results about a kind of non-symmetrical Cantor set.
引文
[1]文志英.分形几何的数学基础.上海科技教育出版社,2000.
[2]L.Olsen ans S.Winter.Multifractal analysis of divergence points of deformed measure theoretical birkhoff averages,ⅱ:Non-linearity,divergence points and banach space valued spectra.Bulletin des Sciences Math(?)matiques,131:518-558,2007.
[3]A.Salli.Upper density properties of hausdorff measures on fractals.Ann.Acad.Sci.Fenn.Ser.A I Math.Dissertationes,55:49pp,1985.
[4]L.Barreira.Variational properties of multifractal spectra.Nonlinearity,14:259 - 274,2001.
[5]C.Tricot.Two definitions of fractional dimension.Math.Proc.Cambridge Philos.Soc.,91:57-74,1982.
[6]R.Darst.The hausdorff dimension of the nondifferentiability set of the cantor function is [log2/log3]~2 Proc.Arner.Math.Soc.,119:105-108,1993.
[7]R.Darst.Hausdorff dimension of sets of non-differentiability points of cantor function.Math.Proc.Camb.Phil.Soc.,117:185-191,1995.
[8]F.M.Dekking and WX Li.How smooth is a devil's staircase? Fractals,11:101-107,2003.
[9]D.W.Spear.Measure and self-similarity.Adv.Math.,91:143-157,1992.
[10]E.Ayer and B.S.Strichartz.Exact hausdorff measure and intervals of maximum density for cantor sets.Trans.Amer.Math.Soc.,35 1:3725-3741,1999.
[11]K.J.Falconer.Fractal Geometry:Mathematical Foundations and Applications.John Wiley &Sons,Chichester,1990.
[12]K.J.Falconer.sub-self-similar sets.Trans.Amer.Math.Soc.,347(8):3121-3128,1995.
[13]K.J.Falconer.Techniques in Fractal Geometry.John Wiley & Sons,1997.
[14]K.J.Falconer.One-sided multifractal analysis and points of non-differentiability points of devil's staircase.Math.Proc.Camb,Phil.Soc.,136:167-174,2004.
[15]DJ Feng.Exact packing measure of liner cantor sets.Math.Nachr.,248-249:102-109,2003.
[16]DJ Feng,S Hua,and ZY Wen.The pointwise densities of the cantor measure.J.Math.Anal.Appl.,250:692-705,2000.
[17]DJ Feng,ZY Wen,and J Wu.Dimension of the homogeneous moran sets.Sci.China(Ser.A),40:475-482,1997.
[18]H.G.Eggleston.The fractional dimension of a set defined by decimal properties.Quart.Journ.Math.,20(March):31-36,1949.
[19]S Hua.The dimensions of generalized self-similar sets.Acta Math.Appl.Sinica,17:551-558,1994.
[20]S Hua and WX Li.Packing dimension of generalized moran sets.Prog.Natural Sci.,2:148-152,1996.
[21]J.E.Hutchinson.Fractal and self-similarity.Indiana Univ.Math.J,30:713-747,1981.
[22]I.S.Baek,L.Olsen,and N.Snigireva.Divergence points of self-similar measures and packing dimension.Adv.Math.,214:267-287,2007.
[23]L.Barreira and B.Saussol.Variational principles and mixed multifractal spectra.Trans.Amer.Math.Soc.,353:3919-3944,2001.
[24]L.Barreira,B.Saussol,and J.Schmeling.Distribution of frequencies of digits via multifractal analysis.J.Number Theory,97:410-438,2002.
[25]L.Barreira,B.Saussol,and J.Schmeling.Higher dimensional multifratal analysis.J.Math.Pures Appl.,81:67-91,2002.
[26]L.Barreira and J.Schmeling.Sets of 'non-typical' points have full topological entropy and full hausdorff dimension.Israel J.Math.,116:29-70,2000.
[27]WX Li.An equivalent definition of packing dimension and its application,to appear in Nonlinear Analysis:Real World Applications.
[28]WX Li.Non-differentiability points of cantor functions.Math.Nachr.
[29]WX Li.Points of infinite derivative of cantor functions.Real Analysis Exchange,32(1):87 - 96,2006.
[30]WX Li and F.M.Dekking.The dimension of subsets of moran sets determined by the success run behaviour of their codings.Monatsh.Math.,131:309-320,2000.
[31]WX Li and F.M.Dekking.Hausdorff dimension of subsets of moran fractals with prescribed group frequency of thier codings.Nonlinearity,16:187-199,2003.
[32]WX Li,L Olsen,and ZY Wen.Hausdorff and packing dimensions of subsets of moran fractals with prescribed mixed group frequency of their codings,to appear in Aequationes Mathematicae.
[33]WX Li and DM Xiao.A note on generalized moran set.Acta Mathematica Scientia(suppl.),18:88-93,1998.
[34]WX Li,DM Xiao,and F.M.Dekking.Non-differentiability of devil's staircases and dimensions of subsets of moran fractals.Math.Proc.Camb.Phil.Soc,133:345-355,2002.
[35]P.Marttila.Geometry of sets and measures in Euclidean space.Cambridge University Press,1995.
[36]R.D.Mauldin and S.C.Williams.Hausdorff dimension in graph directed construction.Trans.Amer.Math.Soc,309:811-829,1988.
[37]J.Morris.The hausdorff dimension of the nondifferentiability set of a nonsymmetric cantor function.Rocky mountain Journal of Mathematics,32:357-370,2002.
[38]L.Olsen.Distribution of digits in integers:Besicovitch-eggleston subsets of N.J.London Math.Soc,67(3):561-579,2003.
[39]L.Olsen.Multifractal analysis of divergence points of deformed measure theoretical birkhoff averages.J.Math.PuresAppl,82:591-1649,2003.
[40]L.Olsen.Applications of multifractal divergence points to sets of d-tuples of numbers defined by their N-adic expansion.Bulletin des Sciences Mathematiques,128:265-289,2004.
[41]L.Olsen.On the hausdorff dimension of generalized besicovitch-eggleston sets of d-tuples of numbers.Proc.Edinb.Math.Soc,15(4):535-547,2004.
[42]L.Olsen.Mixed generalized dimensions of self-similar measures.J.Math.Anal.Appl,306:516-539,2005.
[43]L.Olsen.A generalizations of a result by w.li and f.dekking on the hausdorff dimension of subsets of self-similar sets with presribed group frequency of meir coding.Aequationes Math,72:10-26,2006.
[44]L.Olsen.Multifractal analysis of divergence points of deformed measure theoretical birkhoff averages,Ⅲ.Aequationes Mathematicae,71:29-53,2006.
[45]L.Olsen.Multifractal analysis of divergence points of deformed measure theoretical birkhoff averages,iv:Divergence points and packing dimension.Bulletin des Sciences Mathematiques,132(8):650-678,2008.
[46]Y.Pesin.Dimension theory in dynamical systems:Contemporary and applications.The University of Chicago Press,1997.
[47]R.Cawley and R.D.Mauldin.Multifractal decomposition of moran fractals.Adv.Math.,92:196-236,1992.
[48]P.Walters.An Introduction to Ergodic Theory.Springer-Verlag,New York,1982.
[49]ZL Zhou.The hausdorff measure of the self-similar set-koch curve.Sci.China Ser.A,28:103-107,1998.
[50]Z.Y.Wen.Moran sets and moran classes.Chinese Sci.Bull.,46(22):1849-1856,2001.
[2]L.Olsen ans S.Winter.Multifractal analysis of divergence points of deformed measure theoretical birkhoff averages,ⅱ:Non-linearity,divergence points and banach space valued spectra.Bulletin des Sciences Math(?)matiques,131:518-558,2007.
[3]A.Salli.Upper density properties of hausdorff measures on fractals.Ann.Acad.Sci.Fenn.Ser.A I Math.Dissertationes,55:49pp,1985.
[4]L.Barreira.Variational properties of multifractal spectra.Nonlinearity,14:259 - 274,2001.
[5]C.Tricot.Two definitions of fractional dimension.Math.Proc.Cambridge Philos.Soc.,91:57-74,1982.
[6]R.Darst.The hausdorff dimension of the nondifferentiability set of the cantor function is [log2/log3]~2 Proc.Arner.Math.Soc.,119:105-108,1993.
[7]R.Darst.Hausdorff dimension of sets of non-differentiability points of cantor function.Math.Proc.Camb.Phil.Soc.,117:185-191,1995.
[8]F.M.Dekking and WX Li.How smooth is a devil's staircase? Fractals,11:101-107,2003.
[9]D.W.Spear.Measure and self-similarity.Adv.Math.,91:143-157,1992.
[10]E.Ayer and B.S.Strichartz.Exact hausdorff measure and intervals of maximum density for cantor sets.Trans.Amer.Math.Soc.,35 1:3725-3741,1999.
[11]K.J.Falconer.Fractal Geometry:Mathematical Foundations and Applications.John Wiley &Sons,Chichester,1990.
[12]K.J.Falconer.sub-self-similar sets.Trans.Amer.Math.Soc.,347(8):3121-3128,1995.
[13]K.J.Falconer.Techniques in Fractal Geometry.John Wiley & Sons,1997.
[14]K.J.Falconer.One-sided multifractal analysis and points of non-differentiability points of devil's staircase.Math.Proc.Camb,Phil.Soc.,136:167-174,2004.
[15]DJ Feng.Exact packing measure of liner cantor sets.Math.Nachr.,248-249:102-109,2003.
[16]DJ Feng,S Hua,and ZY Wen.The pointwise densities of the cantor measure.J.Math.Anal.Appl.,250:692-705,2000.
[17]DJ Feng,ZY Wen,and J Wu.Dimension of the homogeneous moran sets.Sci.China(Ser.A),40:475-482,1997.
[18]H.G.Eggleston.The fractional dimension of a set defined by decimal properties.Quart.Journ.Math.,20(March):31-36,1949.
[19]S Hua.The dimensions of generalized self-similar sets.Acta Math.Appl.Sinica,17:551-558,1994.
[20]S Hua and WX Li.Packing dimension of generalized moran sets.Prog.Natural Sci.,2:148-152,1996.
[21]J.E.Hutchinson.Fractal and self-similarity.Indiana Univ.Math.J,30:713-747,1981.
[22]I.S.Baek,L.Olsen,and N.Snigireva.Divergence points of self-similar measures and packing dimension.Adv.Math.,214:267-287,2007.
[23]L.Barreira and B.Saussol.Variational principles and mixed multifractal spectra.Trans.Amer.Math.Soc.,353:3919-3944,2001.
[24]L.Barreira,B.Saussol,and J.Schmeling.Distribution of frequencies of digits via multifractal analysis.J.Number Theory,97:410-438,2002.
[25]L.Barreira,B.Saussol,and J.Schmeling.Higher dimensional multifratal analysis.J.Math.Pures Appl.,81:67-91,2002.
[26]L.Barreira and J.Schmeling.Sets of 'non-typical' points have full topological entropy and full hausdorff dimension.Israel J.Math.,116:29-70,2000.
[27]WX Li.An equivalent definition of packing dimension and its application,to appear in Nonlinear Analysis:Real World Applications.
[28]WX Li.Non-differentiability points of cantor functions.Math.Nachr.
[29]WX Li.Points of infinite derivative of cantor functions.Real Analysis Exchange,32(1):87 - 96,2006.
[30]WX Li and F.M.Dekking.The dimension of subsets of moran sets determined by the success run behaviour of their codings.Monatsh.Math.,131:309-320,2000.
[31]WX Li and F.M.Dekking.Hausdorff dimension of subsets of moran fractals with prescribed group frequency of thier codings.Nonlinearity,16:187-199,2003.
[32]WX Li,L Olsen,and ZY Wen.Hausdorff and packing dimensions of subsets of moran fractals with prescribed mixed group frequency of their codings,to appear in Aequationes Mathematicae.
[33]WX Li and DM Xiao.A note on generalized moran set.Acta Mathematica Scientia(suppl.),18:88-93,1998.
[34]WX Li,DM Xiao,and F.M.Dekking.Non-differentiability of devil's staircases and dimensions of subsets of moran fractals.Math.Proc.Camb.Phil.Soc,133:345-355,2002.
[35]P.Marttila.Geometry of sets and measures in Euclidean space.Cambridge University Press,1995.
[36]R.D.Mauldin and S.C.Williams.Hausdorff dimension in graph directed construction.Trans.Amer.Math.Soc,309:811-829,1988.
[37]J.Morris.The hausdorff dimension of the nondifferentiability set of a nonsymmetric cantor function.Rocky mountain Journal of Mathematics,32:357-370,2002.
[38]L.Olsen.Distribution of digits in integers:Besicovitch-eggleston subsets of N.J.London Math.Soc,67(3):561-579,2003.
[39]L.Olsen.Multifractal analysis of divergence points of deformed measure theoretical birkhoff averages.J.Math.PuresAppl,82:591-1649,2003.
[40]L.Olsen.Applications of multifractal divergence points to sets of d-tuples of numbers defined by their N-adic expansion.Bulletin des Sciences Mathematiques,128:265-289,2004.
[41]L.Olsen.On the hausdorff dimension of generalized besicovitch-eggleston sets of d-tuples of numbers.Proc.Edinb.Math.Soc,15(4):535-547,2004.
[42]L.Olsen.Mixed generalized dimensions of self-similar measures.J.Math.Anal.Appl,306:516-539,2005.
[43]L.Olsen.A generalizations of a result by w.li and f.dekking on the hausdorff dimension of subsets of self-similar sets with presribed group frequency of meir coding.Aequationes Math,72:10-26,2006.
[44]L.Olsen.Multifractal analysis of divergence points of deformed measure theoretical birkhoff averages,Ⅲ.Aequationes Mathematicae,71:29-53,2006.
[45]L.Olsen.Multifractal analysis of divergence points of deformed measure theoretical birkhoff averages,iv:Divergence points and packing dimension.Bulletin des Sciences Mathematiques,132(8):650-678,2008.
[46]Y.Pesin.Dimension theory in dynamical systems:Contemporary and applications.The University of Chicago Press,1997.
[47]R.Cawley and R.D.Mauldin.Multifractal decomposition of moran fractals.Adv.Math.,92:196-236,1992.
[48]P.Walters.An Introduction to Ergodic Theory.Springer-Verlag,New York,1982.
[49]ZL Zhou.The hausdorff measure of the self-similar set-koch curve.Sci.China Ser.A,28:103-107,1998.
[50]Z.Y.Wen.Moran sets and moran classes.Chinese Sci.Bull.,46(22):1849-1856,2001.