摘要
本文主要讨论n维欧式空间Rn中的多模式分形集与方向向量各分量都为有理数的(n-m)-维子空间的截集问题.在一定的条件下,我们证明了截集的Hausdorff维数或者盒维数等于分形集的Hausdorff维数减m.我们还讨论了与之相关的问题.
具体的说,我们考虑的多模式分形是从一个单位立方体开始构造的,其构造方式如下:将单位立方体[0,1]n等分成若干全等的小立方体,然后按照某一模式去掉一些小立方体.模式事实上是包含一些小立方体位置信息的集合.对剩下的小立方体重复上面的过程.在同一步中,不同小立方体所用的模式是一样的.在不同步中,所用的模式可能不同.将这个过程无限次的进行下去,得到极限集E,称之为由模式生成的分形.若每一步所用的模式都是同一个,那么得到的极限集是一个自相似集.若各步中所用的模式所构成的序列不是最终周期的,那么得到的极限集具有Moran结构,我们称之为多模式分形.
在本文中,我们总假设在生成极限集中所用的模式与选取的子空间的方向向量之间满足一个特殊的“同余”条件,称之为(s-*)条件.
在第三章中,我们考虑了自相似集E与方向向量各分量为有理数且截距也为有理数的(n-1)-维超平面的截集.在(s-*)条件下,我们证明了截集的维数等于集合E的维数减1.
在第四章中,我们讨论了自相似集E与方向向量各分量为有理数但截距为无理数的(n-1)-维超平面的截集.我们证明了(s-*)条件是使得截集的典型Hausdorff维数取到Marstrand值(E的维数减1)的充分条件.
在第五章中,我们证明了第三、四章中结果的高维版本,并讨论了多模式分形与方向向量各分量为有理数的(n-m)-维子空间的截集的维数.我们证明了(s-*)条件是使得截集维数取到Marstrand值(E的维数减m)的充分条件.对自相似的极限集,在(s-*)条件下我们还证明了支撑在它上面的自相似测度的投影测度μV关于m维Lebesgue测度是绝对连续的.同时还讨论了当投影测度μV关于m维Lebesgue测度是绝对连续的情况下,投影测度的局部维数与截集的盒维数之间的关系.
In the dissertation, we concern to the intersection of a class of fractals in Rn, which isgenerated by multi-rules, with (n m)-dimensional subspace of integral direction vector.Under certain condition, we prove that the Haudorff dimension or box dimension of slicesequal to the Hausdorff dimension of the fractals minus m. Other related topics are alsodiscussed.
In detail, the fractals we discussed are generated by rules from initial cube. Partitionthe unit cube [0,1]ninto several congruent sub-cubes, and discard several of them accordingto a given rule which contains the information of positions of several sub-cubes that willbe discarded. Apply the same process, to each remaining sub-cubes. The rules we discardsub-cubes in each steps are given. And during a step, the rule we applied to each sub-cubesare the same. Repeat this operation ad infinitum, we get the limit set E, which we calledfractals generated by rules. If we use the same rule throughout the whole process, the limitset E is a self-similar set. If we use sequence of rules we used is not eventually periodic,the limit set E is a fractal of Moran structure, called fractals generated by multi-rules.
Throughout the dissertation, the rules we used and the orthogonal direction of the sub-space are chosen to satisfy a congruence condition which we called s-star condition.
In Chapter3, we discuss intersection of the self-similar fractal E with (n-1)-dimensional hyperplane of integer orthogonal direction and rational intercepts. Under s-starcondition we prove that the dimensions equal the dimension of E minus one.
In Chapter4, we discuss intersection of the self-similar fractal E with (n-1)-dimensional hyperplane of integer orthogonal direction and irrational intercepts. We provethat s-star condition is a sufficient condition to ensure that the typical Hausdorff dimensionof slices takes the value in Marstrand’s theorem, i.e., the dimension of the self-similar setminus one.
In Chapter5, the Hausdorff dimension of the intersection of self-similar fractals in Eu-clidean space Rngenerated from initial cube pattern with (n-m)-dimensional hyperplaneV in a fixed direction is discussed. And we prove that s-star condition is sufficient to ensurethat the Hausdorff dimensions of the slices of the fractal sets generated by “multi-rules”take the valve in Marstrand’s theorem, i.e., the dimension of the self-similar sets minusm. For the self-similar fractals generated from with initial cube pattern, this sufficientcondition also ensures that the projection measure μVis absolutely continuous with respect to the Lebesgue measure Lm. When the projection measure is absolutely continuous withrespect to the m-dimensional Lebesgue measure, the connection of the local dimension ofμVand the box dimension of slices is also given.
引文
[1] Mandelbrot B B. The Fractal Geometry of Nature. New York: W. H. Freeman and Company,1982.
[2] Edgar G. Measure, Topology, And Fractal Geometry,2nd ed. New York: Springer,2008.
[3] Mandelbrot B B. Fractals. Encyclopedia of Physical Science and Technology,1987,5:579–593.
[4] Hart D. Multifractals: Theory and Applications. New York: Chapman&Hall/CRC Press,2001.
[5] Mandelbrot B B. Fractal: Form, Chance, and Dimension. New York: W. H. Freeman and Company,1977.
[6] Feder J. Fractals. New York: Plenum Press,1988.
[7] Fleischman M, Tildesley D J, Ball R C. Fractals in the Nature Science. Princeton: PrincetonUniversity Press,1990.
[8] David G, Semmes S. Fractured fractals and broken dreams: Self-similar geometry through metricand measure, Oxford Lecture Series in Mathematics and its Applications, vol.7. New York: TheClarendon Press Oxford University Press,1997.
[9] Falconer K. The geometry of fractal sets. Cambridge: Cambridge University Press,1985.
[10] Falconer K. Fractal geometry–Mathematical foundations and applications. John Wiely,1990.
[11] Falconer K. Techniques in Fractal Geometry. Chichester: John Wiley&Sons,1997.
[12] Federer H. Geometric Measure Theory. New York: Springer-Verlag,1969.
[13] Mattila P. Geometry of sets and measures in Euclidean spaces. Cambridge: Cambridge UniversityPress,1995.
[14] Peters E E. Fractal Market Analysis: Applying Chaos Theory to Investment and Economics. NewYork: John Wiely&Sons,1990.
[15] Mandelbrot B B. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. New York:Springer,1977.
[16] He M, Petoukhov S. Mathematics in Bioinformations: Theory, Practice, and Applycations. NewJersey: John Wiely&Sons,2011.
[17] Gouyet J F. Physics and Fractal Structures. Springer,1996.
[18] Fogg P. Substitutions in Dynamics, Arithmetics and Combinatorics, Vol.1794of Lecture Notes inMathematics. Springer-Verlag,2001.
[19] Allouche J P, Shallit J. Automatic sequences: Theroy, Applications, Generalizations. CambridgeUniversity Press,2003.
[20] Bugeaud Y. Approximation by algebraic numbers, Cambridge Tracts in Mathematics160. Cam-bridge University Press,2004.
[21] Nekrashevych V. Self-similar groups. American Mathematical Society,2005.
[22] Kigami J. Analysis on fractals. Cambridge: Cambridge University Press,2001.
[23] Marstrand J M. Some fundamental geometrical properties of plane sets of fractional dimension.Proc. London Mathematical Society,1954,4:257–302.
[24] Mattila P. Hausdorff dimension, orthogonal projections and intersections with planes. Ann. Acad.Sci. Fenn. Ser A I. Math,1975,1:227–244.
[25] Mattila P. Integralgeometric properties of capacities. Trans. Amer. Math. Soc,1981,266:539–554.
[26] Hawkes J. Some algebraic properties of small sets. Q. J. Math,1975,26:195–201.
[27] Kenyon R, Peres Y. Intersecting random translates of invariant Cantor sets. Invent. Math,1991,104:601–629.
[28] Benjamini I, Peres Y. On the Hausdorff dimension of fibres. Israel J. Math,1991,74:267–279.
[29]李军方. Sierpinski集截集的Hausdorff维数:[硕士学位论文].武汉:武汉大学,1999.
[30] Dekking F M. Recurrent sets. Advance in Mathematics,1982,44:78–104.
[31]赵燕芬. Sierpinski集截集的维数:[博士学位论文].武汉:武汉大学,2006.
[32] Rao H, Wen Z Y. Some studies of a class of self-similar fractals with overlap structure. Adv. Appl.Math,1998,20:50–72.
[33] Liu Q H, Xi L F, Zhao Y F. Dimensions of inersections of the Sierpinski carpet with lines of rationalslopes. Proc. Edinb. Math. Soc,2007,50:411–428.
[34] Manning A, Simon K. Dimension of slices through the Sierpinski carpet. Trans. Amer. Math. Soc,2013,365:213–250.
[35] Ba′ra′ny B, Ferguson A, Simon K. Slicing the Sierpinski gasket. Nonlinearity,2012,25:1753–1770.
[36] Wen Z Y, Xi L F. On the dimensions of sections for the graph-directed sets. Annales AcademiScientiarum Fennic. Math,2010,35:515–535.
[37] Furstenberg H. Ergodic fractal measures and dimension conservation. Ergodic Theory Dynam.Systems,2008,28No.2:405–422.
[38] Morgan F. Geometric Measure Theory: A Beginner’s Guide,4th ed. Singapore: Elsevier (Singa-pore) Pte Ltd.,2009.
[39] Edgar G. Classics on Fractals. Boulder: Westview Press,1992.
[40] Edgar G. Integral Probability, and Fractal Measures. New York: Springer,1998.
[41] Lapidus M L, Frankenjuijsen M V. Fractal Geometry, Complex Dimensions and Zeta Functions.New York: Springer,2006.
[42]文志英.分形几何的数学基础.上海:上海科技教育出版社,2000.
[43] Halmos P R. Measure Theory. New York: Springer-Verlag,1974.
[44] Doob D L. Measure Theory. New York: Springer,1980.
[45] Einsiedler M, Ward T. Ergodic Theory: with a view towards Number Theory. London: Springer-Verlag,2011.
[46] Rudin W. Principles of Mathematical Analysis,3rd. McGraw Hill Higher Education,2006.
[47] Royden H L, Fitzpatrick P M. Real Analysis,4th ed. Prentice Hall,2010.
[48] Folloand G B. Real Analysis,2nd. John Wiley&Sons: A Wiley-Interscience Publication,1999.
[49] Roger C A. Hausdorff Measures. Cambridge: Cambridge University Press,1970.
[50] Barnsley M F. Fractals Everywhere,2nd ed. Boston: Academic Press,1993.
[51] Hurewicz W, Wallman H. Dimension Theory. Princeton: Princeton University Press,1941.
[52] Engelking R. Dimension Theory. Amsterdam: North-Holland,1978.
[53] Nagata J. Modern Dimension Theory. Groningen: Noordhoff,1965.
[54] Nagami K. Dimension Theory. Boston: Academic Press,1970.
[55] Pears A. Dimension Theory of General Spaces. Cambridge: Cambridge University Press,1975.
[56] Moran P A P. Additive functions of intervals and Hausdorff measure. Proc. Camb Philo. Soc.,1946,42:15–23.
[57] Hutchinson J E. Fractals and self similarity. Indiana Univ. Math. J,1981,30:713–747.
[58] Schief A. Separation properties for self-similar sets. Proc. Amer. Math. Soc.,1994,122:111–115.
[59] Zerner M. Weak separation properties for self-similar sets. Proc. Amer. Math. Soc.,1996,121:3529–3539.
[60] Falconer K J. Dimension and measure of quasi-self-similar sets. Proc. Amer. Math. Soc.,1989,106:543–554.
[61] Anderson L M. Recussive construction of fractals. Ann. Acad. Sci. Fenn. Ser. A I Math. Disserta-tions.,1992,86.
[62] Bandt C. Self-similar sets1. Topological Markov chains and mixed self-similar sets. Math. Nachr.,1989,142:107–123.
[63] Bandt C. Self-similar sets3. Constructions with sofic systems. Monatsh. Math.,1989,108:89–12.
[64] Bandt C. Self-similar sets5. Integer matrices and tilings of Rn. Proc. Amer. Math. Soc.,1991,122:549–562.
[65] Bandt C, Graf S. Self-similar sets7. A characterization of self-similar fractals with positive Haus-dorff measure. Proc. Amer. Math. Soc.,1992,114:995–1001.
[66] Bandt C, Kuschel. Self-similar sets8. Average interior distance in some fractals. Rend. Circ. Mat.Palermo(2). Suppl.,1992,28:307–317.
[67] Falconer K J. Sub-self-similar sets. Trans. Amer. Math. Soc.,1995,347:3121–3129.
[68] Hata M. On the structure of self-similar sets. Japan J. Appl. Math.,1985,2:381–414.
[69] Hata M. Fractals-On self-similar sets. Sugaku Expositions.,1993,6:107–123.
[70] Lalley S P. The packing and covering function of some self-similar fractals. Indiana Univ. Math. J.,1988,37:3:699–709.
[71] Mattila P. On the structure of self-similar fractals. Ann. Acad. Sci. Fenn. Ser. A I. Math.,1982,7:189–195.
[72] Reyes M. An analytic study on the self-similar fractals: Differention of integrals. Collect. Math.,1989,40:159–167.
[73] Reyes M. An analytic study of the functions defined on self-similar fractals. Real Anal. Exchange.,1991-92,16:197–214.
[74] Spear D W. Measure and self similarity. Adv. Math.,1992,91:143–157.
[75] Barnsley M F, Hutchinson J E, StenfloO¨. V-variable fractals: Fractals with partial self similarity.Advance in Mathematics,2008,218:2051–2088.
[76]严加安.测度论讲义(第二版).北京:科学出版社,2004.
[77] Katznelson Y. An introduction to Harmornic analysis,3rd. Cambridge University Press,2004.
[78] Stein E M, Shakarchi R. Fourier Analysis: an introduction. Princeton and Oxford: PrincetonUniversity Press,2003.
[79] Bedford T. Dimension and dynamics of fractal recurrent sets. J. London Math. Soc.,1986,(2)33:89–100.
[80] Mauldin R D, Williams S C. Hausdorff dimension in graph directed constructions. Trans. Amer.Math. Soc,1988,309(1-2):811–839.
[81] Diestel R. Graph Theory,3rd ed. Berlin: Springer-Verlag,2006.
[82] Walters P. An introduction to ergodic theory. New York: Springer-Verlag,1982.
[83] Billingsley P. Ergodic Theory and Information. New York: Wiley,1965.
[84] Bowen R E. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphism. Lecture Notesin Math.470.: Springer,1975.
[85] Friedman N A. Introduction to Ergodic Theory. Van Nostrand,1970.
[86] Halmos P. Lectures on Ergodic Theory. New York: Chelsea,1953.
[87] Sinai Y G. Introduction to Ergodic Theory. Princeton: Princeton University Press,1970.
[88] Gantmacher F R. The theory of matrices vol.2. Chelsea: Chelsea Publishing Company,1960.
[89] Mattila P, Mauldin R D. Measure and dimension functions: measurability and densities. Math.Proc. Cambridge Philos. Soc,1997,121:1:81–100.
[90] Feng D J, Hu H. Dimension theory of iterated function systems. Communications on Pure andApplied Mathematics,2009,62:1435–1500.
[91] Young L S. Dimension entropy and Lyapunov exponents. Ergodic Theory and Dynamical Systems,1982,2No.1:109–124.
[92] Niu M, Xi L F. Singularity of a class of self-similar measures. Chaos, Solitons and Fractals,2007,34:376–382.
[93] Wen Z Y. Moran sets and Moran classes. Chinese Sci. Bull,2001,46(22):1849–1856.
[94] Peres Y, Schlag W, Solomyak B. Sixty years of Bernoulli convolutions In Fractal geometry andstochastics, II (Greifsward/Koserow,1998),. Progr. Probab,2000,46, Birkhuser, Basel:39–65.
[95] Protasov V. Refinement equations with nonnegative coefficients. J. Fourier Anal Appl,2000,6(1):55–77.
[96] Zygmmund A. Trigonometric series, vol1,3rd,. Cambridge: Cambridge University Press,2002.