齐次完全集的拟对称极小性
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摘要
作为Cantor型集的推广,文志英和吴军引入了齐次完全集的概念,并基于齐次完全集的基本区间的长度以及基本区间之间的间隔的长度,得到了齐次完全集的Hausdorff维数.本文研究齐次完全集的拟对称极小性,证明在某些条件下Hausdorff维数为1的齐次完全集是1维拟对称极小的.
Wen and Wu introduced the notion of homogeneous perfect sets as a generalization of Cantor type sets and determined their exact Hausdorff dimension based on the length of their basic intervals and the gaps between them.In this paper,we considered the quasisymmetrically minimality of the homogeneous perfect sets,proved the homogeneous perfect sets with Hausdorff dimension 1 are 1-dimensional quasisymmetrically minimal under some conditions.
Wen and Wu introduced the notion of homogeneous perfect sets as a generalization of Cantor type sets and determined their exact Hausdorff dimension based on the length of their basic intervals and the gaps between them.In this paper,we considered the quasisymmetrically minimality of the homogeneous perfect sets,proved the homogeneous perfect sets with Hausdorff dimension 1 are 1-dimensional quasisymmetrically minimal under some conditions.
引文
[1]Dai Y.X.,Wen Z.X.,Xi L.F.,et al., Quasisymmetrically minimal Moran sets and Hausdorff dimension,Ann.Acad.Sci.Fenn.Math, 2011,36:139-152.
[2]Falconer K.J.,Fractal Geometry:Mathematical Foundations and Applications,John Wiley,New Jersey,1990.
[3]Gehring F.W.,Vaisala J.,Hausdorff dimension and quasiconformal mappings,J.London Math.Soc.,1973,6:504-512.
[4]Hakobyan H.A.,Cantor sets which are minimal for quasi-symmetric mappings,J.Contemp.Math.Anal.,2006,41(2):13-21.
[5]Hu M.D.,Wen S.Y.,Quasisymmetrically minimal uniform Cantor sets,Topol.Appl., 2008,155:515-521.
[6]Mattila P.,Fourier Analysis and Hausdorff Dimension,Cambridge University Press,Cambridge,2015.
[7]Staples S.,Ward L.,Quasisymmetrically thick sets,Ann.Acad.Sci.Fenn.Math.,1998,23:151-168.
[8]Tukia P.,Hausdorff dimension and quasisymmetrical mappings,Math.Scand., 1989,65:152-160.
[9]Wen Z.Y.,Wu J.,Hausdorff dimension of homogeneous perfect sets,Acta.Math.Hunger, 2005,107:35-44.
[10]Wu J.M.,Null sets for doubling and dyadic doubling measures,Ann.Acad.Sci.Fenn.Math., 1993,18:77-91.
[11]Wang W.,Wen S.Y.,On quasisymmetric minimality of Cantor sets,Topol.Appl., 2014,178:300-314.
[12]Xiao Y.Q.,Quasisymmetrically minimal homogeneous perfect sets,Acta Math.Sinica,Chin.Ser., 2013,56:527-536.
[13]Yang J.J.,Wu M.,Li Y.Z.,On quasisymmetric minimality of homogeneous perfect sets,Fractals.,2018,26:1850010(10 pages).
[2]Falconer K.J.,Fractal Geometry:Mathematical Foundations and Applications,John Wiley,New Jersey,1990.
[3]Gehring F.W.,Vaisala J.,Hausdorff dimension and quasiconformal mappings,J.London Math.Soc.,1973,6:504-512.
[4]Hakobyan H.A.,Cantor sets which are minimal for quasi-symmetric mappings,J.Contemp.Math.Anal.,2006,41(2):13-21.
[5]Hu M.D.,Wen S.Y.,Quasisymmetrically minimal uniform Cantor sets,Topol.Appl., 2008,155:515-521.
[6]Mattila P.,Fourier Analysis and Hausdorff Dimension,Cambridge University Press,Cambridge,2015.
[7]Staples S.,Ward L.,Quasisymmetrically thick sets,Ann.Acad.Sci.Fenn.Math.,1998,23:151-168.
[8]Tukia P.,Hausdorff dimension and quasisymmetrical mappings,Math.Scand., 1989,65:152-160.
[9]Wen Z.Y.,Wu J.,Hausdorff dimension of homogeneous perfect sets,Acta.Math.Hunger, 2005,107:35-44.
[10]Wu J.M.,Null sets for doubling and dyadic doubling measures,Ann.Acad.Sci.Fenn.Math., 1993,18:77-91.
[11]Wang W.,Wen S.Y.,On quasisymmetric minimality of Cantor sets,Topol.Appl., 2014,178:300-314.
[12]Xiao Y.Q.,Quasisymmetrically minimal homogeneous perfect sets,Acta Math.Sinica,Chin.Ser., 2013,56:527-536.
[13]Yang J.J.,Wu M.,Li Y.Z.,On quasisymmetric minimality of homogeneous perfect sets,Fractals.,2018,26:1850010(10 pages).