本文選題：加倍測度 + 廣義Cantor集�。� 參考：《華南理工大學(xué)》2016年博士論文

【摘要】：本論文主要研究了分形幾何中的三個方面問題。論文的第一部分,即第三章,研究廣義Cantor集關(guān)于加倍測度的胖性和瘦性。Buckley, Hanson和MacManus [8]研究了中間區(qū)間Cantor集關(guān)于加倍測度的胖性和瘦性,給出了判定中間區(qū)間Cantor集關(guān)于加倍測度是胖集和瘦集的充要條件。進(jìn)一步,Han, Wang(?)Wen [36], Peng和Wen[81]刻畫了齊次Cantor集關(guān)于加倍測度的胖性和瘦性。注意到,無論是中間區(qū)間Cantor集還是齊次Cantor集,它們的結(jié)構(gòu)都有很強(qiáng)的對稱性,其同階基本區(qū)間和間隔的長度相等。對于一般的Moran集,關(guān)于加倍測度的胖性和瘦性的刻畫是困難的。本文考慮的廣義Cantor集,其基本區(qū)間和間隔的長度可不同,我們引入了(αk)-正則,好的和相當(dāng)好的概念,分別在這些條件下,給出了廣義Cantor集關(guān)于加倍測度的胖性和瘦性刻畫。論文的第二部分,即第四章,研究了Borel測度的關(guān)聯(lián)維數(shù)。測度關(guān)聯(lián)維數(shù)的概念由Procaccia, Grassberger和Hentschel [82]于1982年引入。測度的關(guān)聯(lián)維數(shù)是通過能量表示,在隨機(jī)動力系統(tǒng)中應(yīng)用廣泛。本文考慮度量空間中Borel測度的關(guān)聯(lián)維數(shù)。首先,用積分形式和離散形式分別刻畫了測度的關(guān)聯(lián)維數(shù),其離散形式表明測度的關(guān)聯(lián)維數(shù)和測度的L2-譜是相等的；然后,研究了測度的關(guān)聯(lián)維數(shù)和測度的Hausdorff維數(shù)之間的關(guān)系,并給出了判定二者相等的一個充分條件；最后,指出測度的關(guān)聯(lián)維數(shù)是擬-Lipschitz不變量。論文的第三部分,即第五章,研究了Moran結(jié)構(gòu)集上測度的局部維數(shù)。對于自相似集上的自相似測度,Geronimo和Hardin [31]在強(qiáng)分離條件下證明了其局部維數(shù)幾乎處處為一個常數(shù)。Strichartz [93]則將這個結(jié)果推廣到開集條件。進(jìn)一步,Cawley和Mauldin [9]研究了一類特殊Moran集上的Moran測度,在這類Moran集在構(gòu)造中,逐階壓縮映射個數(shù)和壓縮比不變,他們在強(qiáng)分離條件下給出了這類Moran測度在幾乎處處意義下局部維數(shù)的公式。Lou和Wu[67]研究了一類更廣的Moran集上的Moran測度的局部維數(shù),在該類Moran集的構(gòu)造中,逐階壓縮映射個數(shù)和壓縮比均不同,她們在強(qiáng)分離條件下給出了這類Moran測度在幾乎處處意義下局部維數(shù)的公式。后來,Li和Wu[63]將這一結(jié)果推廣到開集條件。本文第五章考慮了更一般的Moran結(jié)構(gòu)集,在更弱的分離性條件下,我們給出了Moran結(jié)構(gòu)集上測度的下、上局部維數(shù)的刻畫。因此,本文研究的Moran結(jié)構(gòu)集上測度的局部維數(shù)結(jié)果無論是從研究對象上還是條件上都在一定程度上推廣了前述結(jié)果。
[Abstract]:This paper mainly studies three aspects of fractal geometry.In the first part of the paper, chapter 3, we study the fatness and thinness of generalized Cantor sets on doubling measures. Hanson and MacManus [8] study the fatness and thinness of intermediate interval Cantor sets on doubling measures.A necessary and sufficient condition for determining that the intermediate interval Cantor set is a fat set and a thin set is given.Further, Han, Wang(?)Wen [36], Peng and Wen [81] characterize the fatness and thinness of homogeneous Cantor sets with respect to doubling measures.It is noted that both the intermediate interval Cantor set and the homogeneous Cantor set have strong symmetries and the length of the same order basic interval and interval is equal.For a general Moran set, it is difficult to characterize the fatness and thinness of doubling measures.In the second part, chapter 4, we study the correlation dimension of Borel measure.The concept of measure correlation dimension was introduced by Procacia, Grassberger and Hentschel [82] in 1982.The correlation dimension of measure is widely used in stochastic dynamic systems by energy representation.In this paper, we consider the correlation dimension of Borel measure in metric space.A sufficient condition for determining the equality of the two is given, and finally, it is pointed out that the correlation dimension of the measure is quasi-Lipschitz invariant.In the third part, chapter 5, we study the local dimension of measure on Moran structure set.For the self-similar measure and Hardin [31] on the self-similar set, it is proved that the local dimension of the measure is almost everywhere a constant. Strichartz [93] is extended to the open set condition.Further, Cawley and Mauldin [9] study the Moran measure on a special Moran set. In the construction of this kind of Moran set, the number of contractive mappings and the ratio of contractions are invariant.Under the condition of strong separation, they gave the local dimension formula of the Moran measure in almost every sense. Lou and Wu [67] studied the local dimension of Moran measure on a class of Moran sets. In the construction of this kind of Moran set,The number of contractive mappings and the ratio of contractions are different from each other. Under the condition of strong separation, the formulas of the local dimension of this kind of Moran measure in almost everywhere sense are given.Later, Li and Wu [63] extended this result to the open set condition.In the fifth chapter, we consider the more general set of Moran structures. Under the condition of weaker separation, we give the characterization of the lower and upper local dimensions of the measure on the set of Moran structures.Therefore, the results of the local dimension of measures on the set of Moran structures studied in this paper generalize the above results to a certain extent, both on the object of study and on the condition.
【學(xué)位授予單位】：華南理工大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O174.12

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