关于测度的重分形分析
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摘要
测度的重分形分析是分形几何的一个重要研究方向,它广泛应用于动力系统、湍流、降雨量模型、地震和金融时间序列模型.发展重分形测度的数学理论和方法至关重要.文中简要阐述测度的重分形分析的基本思想和方法,并介绍笔者及其课题组在该领域取得的主要研究成果.
Multifractal analysis of measures is known as an important research direction of fractal geometry.It has been widely used in dynamical systems,turbulence analysis,rainfall modeling,earthquake analysis,and financial time series modeling.Developing the mathematical theory and methods of multifractal measures is of utmost importance.This paper briefly explains the basic ideas and methods of the multifractal analysis of measures and describes the author's major findings and achievements in this field.
Multifractal analysis of measures is known as an important research direction of fractal geometry.It has been widely used in dynamical systems,turbulence analysis,rainfall modeling,earthquake analysis,and financial time series modeling.Developing the mathematical theory and methods of multifractal measures is of utmost importance.This paper briefly explains the basic ideas and methods of the multifractal analysis of measures and describes the author's major findings and achievements in this field.
引文
[1]Mandelbrot B B.The fractal geometry of nature[M].New York:W.H.Freeman and Co.,1982.
[2]Kolmogorov A N.The local structure of turbulence in in-compressible viscous fluid for very large Reynolds num-bers[J].Comptes Rendus(Doklady)de I'Aacdemie desSciences de I'URSS30,1941:301-305.
[3]Halsey T C,Jensen M H,Kadanoff L P,et al.Fractalmeasures and their singularities:the characterization ofstrange sets[J].Phys Rev A,1986,33:1141-1151.
[4]Falconer K J.Techniques in fractal geometry[M].Eng-land:John Wiley and Sons,Ltd Chichester,1997.
[5]Falconer K J.Fractal geometry-mathematical foundationsand applications.[S.l.]:John Wiley,1990.
[6]Stein E M.Harmonic analysis:real-variable methods,or-thogonality,and oscillatory integrals,with the assistance ofTimothy S Murphy,Volume 43 of Princeton MathematicalSeries,Monographs in Harmonic AnalysiscmⅢ[M].Princeton:Princeton University Press,1993.
[7]Arbeiter M,Patzschke N.Random self-similar multifrac-tals[J].Math Nachr,1996,181:5-42.
[8]Olsen L.Bounds for the Lq-spectra of a self-similar multi-fractals not satisfying the open set condition[J].J MathAnal Appl,2009,355:12-21.
[9]Li Jin-jun,Olsen L.Wu min.Bounds for the Lq-spectra ofa self-similar multifractals without any separation condi-tions[J].J Math Anal Appl,2012,387:77-89.
[10]Olsen L.Empirical multifractal moment measures andmoment scaling functions of self-similar multifractals[J].Math Proc Camb Phil Soc,2002,33:459-485.
[11]Xiao Jia-qing,Wu Min,Olsen L.The exact rate of con-vergence of the Lq-spectra of self-similar measures forq<0[J].J Math Anal Appl,2008,338(1):726-741.
[12]Xiao Jiaqing,Wu Min.Empirical multifractal momentmeasures of self-similar measures for q<0[J].MonatshMath,2009,156:175-185.
[13]Barreira L,Schmeling J.Sets of“non-typical”pointshave full topological and full Hausdorff dimensions[J].Israel J Math,2000,116:29-70.
[14]Xiao Jia-qing,Wu Min.Divergence points of self-similarmeasures satisfying the OSC[J].J Math Anal Appl,2011,379:834-841.
[15]Li Jinjun,Wu min,Xiong Yin.Hausdorff dimensions ofthe divergence points of self-similar measures with theopen set condition[J].Nonlinearity,2012,25:93-105.
[16]Olsen L,Winter S.Normal and non-normal points of self-similar sets and divergence points of self-similar mea-sures[J].J Lordon Math Soc,2003,67:103-122.
[17]Geronino J S,Hardin D P.An exact formula for themeasure dimensions associated with a class of piecewiselinear maps[J].Constr Approx,1989,5:89-98.
[18]Strichartz R S.Self-similar measures and their Fouriertransforms[J].Iniana Univ Math J,1990,39:797-817.
[19]Cawley R,Mauldin R D.Multifractal decompositions ofMoran fractals[J].Adv Math,1992,92:196-236.
[20]Lou Manli,Wu Min.The pointwise dimensions of Moranmeasures[J].Sci China Math,2010,53(5):1283-1292.
[21]Li Jin-jun,Wu Min.Pointwise dimensions of general Mo-ran measures with open set condition[J].Sci ChinaMath,2011,54(4):699-710.
[22]Wu Min.The singularity spectrum F(α)of some Moranfractal[J].Monatshefte fǔr mathematik,2005,144:141-155.
[23]Wu Min.The multifractal spectrum of some Moran mea-sures[J].Sci China Math,2005,48:1097-1112.
[24]Wu Min,Xiao Jia-qing.The singularity spectrum of somenon-regularity Moran fractals[J].Chaos Solitons Frac-tals,2011,44(7):548-557.
[25]Xiao Jiaqing,Wu Min.The multifractal dimension func-tions of homogeneous Moran measure[J].Fractals,2008,16(2):175-185.
[26]Wu J M.Hausdorff dimension and doubing measures onmetric spaces[J].Proc Amer Math Soc,1998,126(5):1453-1459.
[27]Lou Manli,Wen Shengyou,Wu Min.Two examples on a-tomic doubling measures[J].J Math Anal Appl,2007,333:1111-1118.
[28]Kaufman R,Wu J M.Two problems on doubing measures[J].Rev Mat Iberoamericana,1995,11(3):527-545.
[29]Lou Man-li,Wu Min.Doubling measures with doublingcontinuous part[J].Proc Amer Math Soc,2010,138(10):3585-3589.
[2]Kolmogorov A N.The local structure of turbulence in in-compressible viscous fluid for very large Reynolds num-bers[J].Comptes Rendus(Doklady)de I'Aacdemie desSciences de I'URSS30,1941:301-305.
[3]Halsey T C,Jensen M H,Kadanoff L P,et al.Fractalmeasures and their singularities:the characterization ofstrange sets[J].Phys Rev A,1986,33:1141-1151.
[4]Falconer K J.Techniques in fractal geometry[M].Eng-land:John Wiley and Sons,Ltd Chichester,1997.
[5]Falconer K J.Fractal geometry-mathematical foundationsand applications.[S.l.]:John Wiley,1990.
[6]Stein E M.Harmonic analysis:real-variable methods,or-thogonality,and oscillatory integrals,with the assistance ofTimothy S Murphy,Volume 43 of Princeton MathematicalSeries,Monographs in Harmonic AnalysiscmⅢ[M].Princeton:Princeton University Press,1993.
[7]Arbeiter M,Patzschke N.Random self-similar multifrac-tals[J].Math Nachr,1996,181:5-42.
[8]Olsen L.Bounds for the Lq-spectra of a self-similar multi-fractals not satisfying the open set condition[J].J MathAnal Appl,2009,355:12-21.
[9]Li Jin-jun,Olsen L.Wu min.Bounds for the Lq-spectra ofa self-similar multifractals without any separation condi-tions[J].J Math Anal Appl,2012,387:77-89.
[10]Olsen L.Empirical multifractal moment measures andmoment scaling functions of self-similar multifractals[J].Math Proc Camb Phil Soc,2002,33:459-485.
[11]Xiao Jia-qing,Wu Min,Olsen L.The exact rate of con-vergence of the Lq-spectra of self-similar measures forq<0[J].J Math Anal Appl,2008,338(1):726-741.
[12]Xiao Jiaqing,Wu Min.Empirical multifractal momentmeasures of self-similar measures for q<0[J].MonatshMath,2009,156:175-185.
[13]Barreira L,Schmeling J.Sets of“non-typical”pointshave full topological and full Hausdorff dimensions[J].Israel J Math,2000,116:29-70.
[14]Xiao Jia-qing,Wu Min.Divergence points of self-similarmeasures satisfying the OSC[J].J Math Anal Appl,2011,379:834-841.
[15]Li Jinjun,Wu min,Xiong Yin.Hausdorff dimensions ofthe divergence points of self-similar measures with theopen set condition[J].Nonlinearity,2012,25:93-105.
[16]Olsen L,Winter S.Normal and non-normal points of self-similar sets and divergence points of self-similar mea-sures[J].J Lordon Math Soc,2003,67:103-122.
[17]Geronino J S,Hardin D P.An exact formula for themeasure dimensions associated with a class of piecewiselinear maps[J].Constr Approx,1989,5:89-98.
[18]Strichartz R S.Self-similar measures and their Fouriertransforms[J].Iniana Univ Math J,1990,39:797-817.
[19]Cawley R,Mauldin R D.Multifractal decompositions ofMoran fractals[J].Adv Math,1992,92:196-236.
[20]Lou Manli,Wu Min.The pointwise dimensions of Moranmeasures[J].Sci China Math,2010,53(5):1283-1292.
[21]Li Jin-jun,Wu Min.Pointwise dimensions of general Mo-ran measures with open set condition[J].Sci ChinaMath,2011,54(4):699-710.
[22]Wu Min.The singularity spectrum F(α)of some Moranfractal[J].Monatshefte fǔr mathematik,2005,144:141-155.
[23]Wu Min.The multifractal spectrum of some Moran mea-sures[J].Sci China Math,2005,48:1097-1112.
[24]Wu Min,Xiao Jia-qing.The singularity spectrum of somenon-regularity Moran fractals[J].Chaos Solitons Frac-tals,2011,44(7):548-557.
[25]Xiao Jiaqing,Wu Min.The multifractal dimension func-tions of homogeneous Moran measure[J].Fractals,2008,16(2):175-185.
[26]Wu J M.Hausdorff dimension and doubing measures onmetric spaces[J].Proc Amer Math Soc,1998,126(5):1453-1459.
[27]Lou Manli,Wen Shengyou,Wu Min.Two examples on a-tomic doubling measures[J].J Math Anal Appl,2007,333:1111-1118.
[28]Kaufman R,Wu J M.Two problems on doubing measures[J].Rev Mat Iberoamericana,1995,11(3):527-545.
[29]Lou Man-li,Wu Min.Doubling measures with doublingcontinuous part[J].Proc Amer Math Soc,2010,138(10):3585-3589.
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