設T為一於二維環面T²上，特徵值為整數的線性自同態，而D為一T²的Markov分割。那麼每一個D上定義的符號空間有限型子轉移則對應一個T²的T -不變緊子集K。判斷K的 Hausdor和Minkowski維數何時相等是一有趣問題。Kenyon and Peres [15]說明了此問題與(K, T )的測度熵及拓撲熵關係密切。這篇論文將進一步說明兩種維數的相等與符號動力系統及矩陣乘積的漸近性態的密切關係。此外我們描述一種算法以判斷兩個譜半徑為1的本原矩陣的任意乘積的譜半徑何時維持1，以及此算法對於研究sofic自仿集K的應用。 / Let T be a linear endomorphism on the 2-torus T² with integer eigenvalues, and D be a natural Markov partition (c.f. Bowen [4]) of T² . Then a subshift of nite type over D corresponds to a T-invariant compact subset K of T². An interesting problem is to determine when the Hausdorff and Minkowski dimensions of K conincide. Kenyon and Peres [15] showed that this is closely related to the measure-theoretic and topological entropies of (K, T). In this thesis, we further show that the coincidence of dimensions has a deep connection to symbolic dynamics and the asymptotic behaviour of matrix products. Moreover, we develop an algorithm to determine when the spectral radii of arbitrary products of two primitive matrices, with spectral radius 1, are preserved, and apply this algorithm to some sofic self-affine sets considered above. / Detailed summary in vernacular field only. / Lo, Chiu Hong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 55-56). / Abstracts also in Chinese. / Chapter 1 --- Introduction and Main Results --- p.6 / Chapter 2 --- Preliminaries --- p.12 / Chapter 2.1 --- Basic symbolic dynamics --- p.12 / Chapter 2.2 --- The symbolic representations --- p.13 / Chapter 2.3 --- An adapted covering of the invariant set KT (A) --- p.17 / Chapter 2.4 --- Some basic lemmas and theorems --- p.18 / Chapter 3 --- Proofs of Proposition 1.2 and Theorem 1.3 --- p.22 / Chapter 3.1 --- Proof of Proposition 1.2 --- p.22 / Chapter 3.2 --- Proof of Theorem 1.3 --- p.24 / Chapter 4 --- Projection of measure of maximal entropy for sub-shifts of finite type --- p.26 / Chapter 4.1 --- Projection of the Parry measure via a general factor map --- p.26 / Chapter 4.2 --- Proof of Theorem 1.4 --- p.36 / Chapter 5 --- Spectral radii of products of primitive matrices --- p.38 / Chapter 5.1 --- The algorithm --- p.40 / Chapter 5.2 --- Some applications and examples --- p.51 / Bibliography --- p.55
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328617 |
| Date | January 2012 |
| Contributors | Lo, Chiu Hong., 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, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (56 leaves) : 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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