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| 研究生： |
蔡承育 Tsai, Cheng-Yu |
|---|---|
| 論文名稱： |
擴張樹與非擴張樹子平移熵與混合性質之研究 Entropy and Mixing Properties of Shifts on Expandable and Non-expandable Trees |
| 指導教授： |
班榮超
Ban, Jung-Chao |
| 口試委員: |
班榮超
Ban, Jung-Chao 曾睿彬 Tseng, Jui-Pin 許正雄 Hsu, Cheng-Hsiung 吳恭儉 Wu, Kung-Chien 陳國璋 Chen, Kuo-Chang |
| 學位類別： |
博士
Doctor |
| 系所名稱： |
理學院 - 應用數學系 Department of Mathematical Sciences |
| 論文出版年： | 2026 |
| 畢業學年度： | 114 |
| 語文別： | 英文 |
| 論文頁數： | 89 |
| 中文關鍵詞： | 拓樸熵 、條型熵近似 、收斂速率 、Li-Yorke 混沌 、馬可夫樹移位 、具單位元半群作用 |
| 外文關鍵詞： | Topological entropy, strip entropy approximation, rate of convergence, Li-Yorke chaos, Markov tree shifts, monoid actions |
| 相關次數： | 點閱：120 下載：1 |
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本文研究了擴張樹與非擴張樹子平移（tree shifts）熵與混合性質。
對於擴張樹子平移，我們證明了條型熵（strip entropy）近似對於黃金平均樹移位的所有路徑，以及某些類別的馬可夫樹移位的部分路徑成立。我們進一步證明，對於與本原矩陣相關聯的d-元樹移位，其收斂速率是線性的。此外，我們表明條型熵近似的收斂速率與對應馬可夫樹移位的擴張常數密切相關。
對於非擴張樹子平移，我們在具單位元半群作用（monoid actions）下推導了熵公式與混合性質，並引入了多種Li-Yorke 混沌形式。特別地，我們證明正熵等價於局部Li-Yorke 混沌，而這比傳統的Li-Yorke 混沌更強。
In this article, we investigate the entropy and mixing behaviors of both expandable and non-expandable tree shifts.
For expandable tree shifts, we establish that the strip entropy approximation holds for every path in a golden-mean tree shift and for certain paths in a class of Markov tree shifts. We further show that, for d-tree shifts associated with primitive matrices, the rates of convergence are linear. Moreover, the expanding constant of the corresponding Markov tree shift plays a key role in determining the convergence rate of the strip entropy approximation.
For non-expandable tree shifts, we derive entropy formulas and mixing properties under monoid actions, and we introduce various forms of Li-Yorke chaos. In particular, we prove that positive entropy is equivalent to locally Li-Yorke chaos, which is a stronger form than the classical Li-Yorke chaos.
中文摘要 i
Abstract ii
Contents iii
1 Introduction 1
1.1 Monoid actions on shifts 1
1.2 Zd shift of finite type 4
2 Fundamental Definitions and Results 7
2.1 1-dimensional shift spaces 7
2.1.1 Higher block shifts 9
2.1.2 Sliding block codes 11
2.1.3 Shifts of finite type 13
2.2 Topological structure of 1-dimensional shift spaces 17
2.2.1 Cylinder sets in a shift space 18
2.3 Topological entropy of 1-dimensional shift spaces 20
2.3.1 Computing entropy 23
3 Main Results and Examples 25
3.1 Non-expandable Markov tree shifts 26
3.1.1 Mixing 26
3.1.2 Topological entropy 28
3.1.3 Li-Yorke chaos 35
3.2 Expandable Markov tree shifts 38
3.2.1 d-tree shifts 38
3.2.2 Golden mean tree shifts 44
3.2.3 Complete recursive tree shifts 45
4 Proof of Main Results 47
4.1 Proof of Theorem 3.1.1 47
4.2 Proof of Theorem 3.1.3 49
4.3 Proof of Theorem 3.1.8 57
4.4 Proof of Theorem 3.2.1 58
4.5 Proof of Theorem 3.2.3 63
4.6 Proof of Theorem 3.2.5 68
4.7 Proof of Theorem 3.2.6 74
4.8 Proof of Theorem 3.2.7 80
5 Conclusion 84
References 86
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