In [DS1,2,3], Denker and Sato initiated a new point of view to study the problem. They identified the Sierpinski gasket as a Martin boundary of some canonical Markov chain and used the associated theory to consider the problem. In this thesis, we will extend their result so as to be applicable to all single-point connected monocyclic post critically finite (m.p.c.f.) self-similar sets. / In the first chapter, we review some basic facts of the self-similar sets and the Martin boundaries, and we prove that every m.p.c.f. self-similar set K is homeomorphic to the quotient space of the symbolic space associated with K, moreover, the homeomorphism is a Lipschitz equivalence for some special m.p.c.f. self-similar sets. / In the second chapter, we first prove that the quotient space of the symbolic space associated with K is homeomorphic to the Martin boundary with respect to the state space associated with K if K is a single-point connected m.p.c.f. self-similar set. Combining this result and the result in the first chapter, we conclude that every single-point connected m.p.c.f. self-similar set can be identified with the Martin boundary of some canonical Markov chain. Then for the 3-level Sierpinski gasket, we prove that there exists a one to one relation between the strongly P-harmonic functions on the 3 state space and K-harmonic functions constructed by Kigami. / In the third chapter, we define a new Markov chain on the pentagasket K which is a single-point connected m.p.c.f. self-similar also. Under the new Markov chain, we prove that K can be identified with the Martin boundary of the new Markov chain and that there exists a one to one relation between the strongly P-harmonic functions and the K-harmonic functions. / One of the fundamental problems in fractal analysis is to construct a Laplacian on fractals. Since fractals, like the Sierpinski gasket and the pentagasket, do not have any smooth structures, it is not possible to construct it from the classical point of view. Hence, until now there is no systematic way to define such a notion on the general class of fractals. / There are two approaches for the problem which have achieved some success in certain special situations. The first one is a probabilistic approach via constructing Brownian motions on self-similar sets. The second approach is an analytical one proposed by Kigami. He approximated the underlying self-similar set K by an increasing sequence of finite sets equipped with the discrete Laplacians Hm in a consistent way. He showed that if K is strongly symmetric, then Hm converge to a Laplacian on K. / by Ju, Hongbing. / "March 2008." / Adviser: Lau Ka Sing. / Source: Dissertation Abstracts International, Volume: 70-03, Section: B, page: 1702. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 91-94). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_344170 |
| Date | January 2008 |
| Contributors | Ju, Hongbing., Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, theses |
| Format | electronic resource, microform, microfiche, 1 online resource (94 p. : ill.) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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