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| 研究生： |
黃迺筑 Huang, Nai-Zhu |
|---|---|
| 論文名稱： |
樹上馬可夫系統的拓樸性質及統計量 Markov Systems on Trees : Topological Properties and Statistic Quantities |
| 指導教授： |
班榮超
Ban, Jung-Chao |
| 口試委員: |
陳國璋
Chen, Kuo-Chang 許正雄 Hsu, Cheng-Hsiung 張志鴻 Chang, Chih-Hung 曾睿彬 Tseng, Jui-Pin |
| 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： | 2023 |
| 畢業學年度： | 111 |
| 語文別： | 英文 |
| 論文頁數： | 114 |
| 中文關鍵詞： | 動態系統 、馬可夫系統 、樹 、混合性質 、熵 、交換性 |
| 外文關鍵詞： | Dynamical systems, Markov systems, Tree, Mixing property, Entropy, Commutativity |
| 相關次數： | 點閱：80 下載：19 |
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本篇博士論文研究樹上馬可夫系統的拓樸性質和統計量。我們針對在多元樹上馬可夫系統的拓樸性質，包含混合性質、CPC混合性質、稠密軌道的存在性，最小系統和重現點，都提供了由馬可夫系統的鄰接矩陣判別的刻畫條件。其中幾乎所有的矩陣刻畫條件都是有限步以內可檢查的。
第二部分我們討論樹上馬可夫系統的增長率的一階估計（degree）和二階估計(熵)。得到了馬可夫樹上的馬可夫系統的degree是馬可夫樹的鄰接矩陣的譜半徑取對數的結果。關於馬可夫樹上的馬可夫系統的熵，我們提供了一個由鄰接矩陣表示的遞迴計算公式。最後得到了在非週期回歸樹上的馬可夫系統都具備熵的可交換性的結果。
This dissertation presents the equivalent matrix-conditions for the topological properties of a Markov system on a $d$-tree, based on its adjacency matrix. These characterizations include various kinds of mixing properties, mixing properties in the sense of CPC (complete prefix code), the existence of dense orbits, minimal systems, and the recurrence. Notably, almost all of these equivalent matrix-conditions are finitely checkable.
In the second part of this dissertation, we consider two statistical quantities, the degree and the entropy, of a Markov system on trees. It is showed that the degree of any Markov system on the Markov tree $\mathcal{T}_D$ with adjacency matrix $D$ is the logarithm of the spectral radius of $D$. An algorithm is provided to compute the entropy of a Markov system on the Markov tree. Finally, we proved that the entropy is commutative for nonautonomous $p$-periodic Markov systems on an aperiodic recursive tree.
致謝 i
中文摘要 ii
Abstract iii
Contents iv
1 Introductions p.1
2 Preliminaries p.5
2.1 Non-NegativeMatrices p.5
2.2 Trees p.11
2.2.1 d-Trees, Markov Trees and Recursive Trees p.11
2.2.2 Complete Recursive Markov Trees p.25
2.3 Markov Systems on Trees p.34
3 Topological Properties p.40
3.1 Mixing Properties on d-Trees p.40
3.1.1 TraditionalMixingProperties p.40
3.1.2 CPC-Mixing Properties p.47
3.2 Dense Orbit on d-Trees p.58
3.2.1 Dense Orbit p.58
3.2.2 MinimalSystems p.75
3.2.3 Recurrences p.79
4 Statistic Quantities p.83
4.1 Degree and Entropy on Markov Trees p.83
4.1.1 Degree p.83
4.1.2 Entropy p.91
4.2 The Commutativity of Entropy on Aperiodic Recursive Trees p.97
References p.112
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