本文選題：Space-filling曲線 + 自相似集��； 參考：《華中師范大學(xué)》2016年博士論文

【摘要】：在本學(xué)位論文中,我們主要研究自相似集的最佳參數(shù)化,所謂最佳參數(shù)化,也就是幾乎處處一對(duì)一,在一定意義下保測(cè)和1/3-Holder連續(xù),這里S是自相似集的豪斯道夫維數(shù).特別地,在本文中,我們構(gòu)造出一套系統(tǒng)且嚴(yán)格的方法,能夠?qū)臻g填充曲線有更深入的了解.通過這套方法,我們?cè)O(shè)計(jì)出了一套程序,只要輸入相應(yīng)的數(shù)據(jù),就能得到對(duì)應(yīng)的最佳參數(shù)化.首先,我們引進(jìn)一個(gè)概念Hnear GIFS(帶有序結(jié)構(gòu)的圖遞歸函數(shù)迭代系統(tǒng)).我們證明了，，當(dāng)一個(gè)linear GIFS的不變集滿足開集條件時(shí),它們有最佳參數(shù)化!因此,構(gòu)造不變集的空間填充曲線,也就是需要造出一個(gè)恰當(dāng)?shù)膌inear GIFS.其次,為了研究自相似集的linear GIFS結(jié)構(gòu),我們引進(jìn)不變集的skeleton這個(gè)新的定義.一個(gè)給定的有限skeleton決定一個(gè)完全有向圖Go.將G0代入函數(shù)迭代系中迭代,我們得到一個(gè)由G0的自仿像構(gòu)成的有向圖.通過研究G0和這個(gè)新的有向圖的關(guān)系,我們可以得到不同的歐拉代換規(guī)則,因此可以誘導(dǎo)不同的linear GIFS.這里驗(yàn)證開基條件的問題可以由分形幾何的理論解決.這樣我們就證明了,在一定的假設(shè)條件下,滿足開集條件和有有限skeleton的自相似集,有最佳參數(shù)化.同時(shí),本文還對(duì)經(jīng)典的空間填充曲線進(jìn)行了比較詳細(xì)的研究,并且給出了構(gòu)造經(jīng)典的空間填充曲線的三種方法.在本文的最后一部分,我們會(huì)證明前面給出的一些假設(shè)條件是多余的,并證明了在滿足開集條件和有限skeleton條件的下,不變集有最佳參數(shù)化.這也說明了我們的理論囊括了自相似集中的很大一類,其中也包括滿足有限型條件的自相似集,并且我們的結(jié)果幾乎包含了之前在這方面的所有結(jié)論.
[Abstract]:In this dissertation, we mainly study the optimal parameterization of self-similar sets, the so-called optimal parameterization, that is, almost everywhere one-to-one, in a certain sense, preserving and 1 / 3-Holder continuity, where S is the Hausdorf dimension of the self-similar set. In particular, in this paper, we construct a systematic and strict method to understand the space filling curve more deeply. Through this method, we design a program, which can get the best parameterization by inputting the corresponding data. First, we introduce the concept Hnear GIFS (Graph Recursive function iterative system with ordered structure). We prove that when an invariant set of linear GIFS satisfies the open set condition, they have the best parameterization! Therefore, it is necessary to construct an appropriate linear GIFSs to construct the space filling curve of the invariant set. Secondly, in order to study the linear GIFS structure of self-similar sets, we introduce a new definition of skeleton of invariant sets. A given finite skeleton determines a completely directed graph Go. By substituting G0 into the iterative system of functions, we obtain a directed graph composed of the self-imitating image of G0. By studying the relation between G0 and this new digraph, we can obtain different Euler substitution rules, so we can induce different linear GIFSs. The problem of verifying the open basis condition can be solved by the theory of fractal geometry. In this way, we prove that under certain assumptions, the open set condition and the self-similar set with finite skeleton have the best parameterization. At the same time, the classical space filling curve is studied in detail, and three methods of constructing the classical space filling curve are given. In the last part of this paper, we prove that some of the assumptions given above are superfluous, and prove that the invariant set has the best parameterization under the open set condition and the finite skeleton condition. This also shows that our theory contains a large class of self-similar sets, including self-similar sets satisfying finite type conditions, and our results contain almost all previous results in this respect.
【學(xué)位授予單位】：華中師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O189

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