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題名 關於黃金分割樹上的子平移之拓樸熵研究
Topological Entropy of Golden-Mean Tree-Shift
作者 蔡承育
Tsai, Cheng-Yu
貢獻者 班榮超
Ban, Jung-Chao
蔡承育
Tsai, Cheng-Yu
關鍵詞 條型法
拓樸及條型熵
黃金分割子平移樹
Strip method
Topological and strip entropy
Golden-mean tree-shift
日期 2022
上傳時間 1-七月-2022 16:20:42 (UTC+8)
摘要 於2019 年,Petersen 和Salama [1, 2] 給出了在樹上拓樸熵的定義並證明其存在且等於最大下界,此外,證明在k-tree 上考慮黃金子平移,條型熵h_n^{(k)}會收斂到拓樸熵h^{(k)}。

此工作擴展了Petersen 和Salama 的結果,藉由考慮有限字母集A在黃
金分割樹T上利用條型法去計算其拓樸熵h(T_A)。首先,給出一個實數值矩
陣M^∗ 用來描述在高度為n條型樹上的複雜度。其次,找到兩個實數值矩
陣C, D 使得b_{n−2}D ≤ M^∗ ≤ b_{n−2}C, 其中b_{n−2}是指所有在黃金分割樹上的子平移高度為n − 2 的著色數。最後,證明在黃金分割樹上的子平移,條型熵h_n(T_A) 將收斂到拓樸熵h(T_A)。
In 2019, Petersen and Salama [1, 2] showed that the limit in their definition of tree-shift topological entropy is actually the infimum and also proved that the site specific strip approximation entropies h_n^{(k)} converges to the entropy h^{(k)} of the golden-mean shift of finite type on the k-tree.

In this article, we prove that the preceding work of Petersen and Salama can be extended to consider a golden-mean tree T with finite alphabet A and use the strip method to calculate its topological entropy h(T_A). First, a real matrix M which describe the complexity of strip method tree with hight n is introduced. Second, two real matrices C and D are constructed for which b_{n−2}D ≤ M^∗ ≤ b_{n−2}C, where b_{n−2} is the number of all different labeling of subtree of the golden-mean tree-shift with level n − 2. Finally, we shown that the n-strip entropy h_n(T_A) will converge to the topological entropy h(T_A) of golden-mean tree-shift T_A.
參考文獻 [1] Karl Petersen and Ibrahim Salama. Tree shift topological entropy. Theoretical Computer Science, 743:64–71, 2018.

[2] Karl Petersen and Ibrahim Salama. Entropy on regular trees. Discrete & Continuous Dynamical Systems, 40(7):4453, 2020.

[3] Nathalie Aubrun and Marie-Pierre Béal. Tree-shifts of finite type. Theoretical Computer Science, 459:16–25, 2012.

[4] Nathalie Aubrun and Marie-Pierre Béal. Sofic tree-shifts. Theory of Computing Systems, 53(4):621–644, 2013.

[5] Nishant Chandgotia and Brian Marcus. Mixing properties for hom-shifts and the distance between walks on associated graphs. Pacific Journal of Mathematics, 294(1):41–69, 2018.

[6] Roy L Adler, Alan G Konheim, and M Harry McAndrew. Topological entropy. Transactions of the American Mathematical Society, 114(2):309–319, 1965.

[7] Tomasz Downarowicz. Entropy in dynamical systems, volume 18. Cambridge University Press, 2011.

[8] Douglas Lind, Brian Marcus, Lind Douglas, Marcus Brian, et al. An introduction to symbolic dynamics and coding. Cambridge university press, 1995.

[9] Jung-Chao Ban and Chih-Hung Chang. Mixing properties of tree-shifts. Journal of Mathematical Physics, 58(11):112702, 2017.

[10] Jung-Chao Ban and Chih-Hung Chang. Tree-shifts: Irreducibility, mixing, and the chaos of tree-shifts. Transactions of the American Mathematical Society, 369(12):8389–8407, 2017.

[11] Jung-Chao Ban and Chih-Hung Chang. Tree-shifts: The entropy of tree-shifts of finite type. Nonlinearity, 30(7):2785, 2017.

[12] Jung-Chao Ban, Chih-Hung Chang, Wen-Guei Hu, and Yu-Liang Wu. Topological entropy for shifts of finite type over Z and tree. arXiv preprint arXiv:2006.13415, 2020.

[13] Jung-Chao Ban and Chih-Hung Chang. Characterization for entropy of shifts of finite type on cayley trees. Journal of Statistical Mechanics: Theory and Experiment, 2020(7): 073412, 2020.

[14] Jung-Chao Ban, Chih-Hung Chang, and Yu-Hsiung Huang. Complexity of shift spaces on semigroups. Journal of Algebraic Combinatorics, 53(2):413–434, 2021.

[15] Wei-Lin Lin. On the strip entropy of the golden-mean tree shift. Master’s thesis, National Chengchi University, 2021.

[16] Itai Benjamini and Yuval Peres. Markov chains indexed by trees. The annals of probability, pages 219–243, 1994.

[17] Hans-Otto Georgii. Gibbs measures and phase transitions. de Gruyter, 2011.
描述 碩士
國立政治大學
應用數學系
109751002
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0109751002
資料類型 thesis
dc.contributor.advisor 班榮超zh_TW
dc.contributor.advisor Ban, Jung-Chaoen_US
dc.contributor.author (作者) 蔡承育zh_TW
dc.contributor.author (作者) Tsai, Cheng-Yuen_US
dc.creator (作者) 蔡承育zh_TW
dc.creator (作者) Tsai, Cheng-Yuen_US
dc.date (日期) 2022en_US
dc.date.accessioned 1-七月-2022 16:20:42 (UTC+8)-
dc.date.available 1-七月-2022 16:20:42 (UTC+8)-
dc.date.issued (上傳時間) 1-七月-2022 16:20:42 (UTC+8)-
dc.identifier (其他 識別碼) G0109751002en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/140659-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學系zh_TW
dc.description (描述) 109751002zh_TW
dc.description.abstract (摘要) 於2019 年,Petersen 和Salama [1, 2] 給出了在樹上拓樸熵的定義並證明其存在且等於最大下界,此外,證明在k-tree 上考慮黃金子平移,條型熵h_n^{(k)}會收斂到拓樸熵h^{(k)}。

此工作擴展了Petersen 和Salama 的結果,藉由考慮有限字母集A在黃
金分割樹T上利用條型法去計算其拓樸熵h(T_A)。首先,給出一個實數值矩
陣M^∗ 用來描述在高度為n條型樹上的複雜度。其次,找到兩個實數值矩
陣C, D 使得b_{n−2}D ≤ M^∗ ≤ b_{n−2}C, 其中b_{n−2}是指所有在黃金分割樹上的子平移高度為n − 2 的著色數。最後,證明在黃金分割樹上的子平移,條型熵h_n(T_A) 將收斂到拓樸熵h(T_A)。
zh_TW
dc.description.abstract (摘要) In 2019, Petersen and Salama [1, 2] showed that the limit in their definition of tree-shift topological entropy is actually the infimum and also proved that the site specific strip approximation entropies h_n^{(k)} converges to the entropy h^{(k)} of the golden-mean shift of finite type on the k-tree.

In this article, we prove that the preceding work of Petersen and Salama can be extended to consider a golden-mean tree T with finite alphabet A and use the strip method to calculate its topological entropy h(T_A). First, a real matrix M which describe the complexity of strip method tree with hight n is introduced. Second, two real matrices C and D are constructed for which b_{n−2}D ≤ M^∗ ≤ b_{n−2}C, where b_{n−2} is the number of all different labeling of subtree of the golden-mean tree-shift with level n − 2. Finally, we shown that the n-strip entropy h_n(T_A) will converge to the topological entropy h(T_A) of golden-mean tree-shift T_A.
en_US
dc.description.tableofcontents 中文摘要 i

Abstract ii

Contents iii

1 Introduction 1

2 Preliminaries and main result 5
2.1 Notations and definitions . . . . . . . . . . . . . . 5
2.2 Main results . . . . . . . . . . . . . . . . . . . . 7

3 Conclusion 11

References 14
zh_TW
dc.format.extent 720851 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0109751002en_US
dc.subject (關鍵詞) 條型法zh_TW
dc.subject (關鍵詞) 拓樸及條型熵zh_TW
dc.subject (關鍵詞) 黃金分割子平移樹zh_TW
dc.subject (關鍵詞) Strip methoden_US
dc.subject (關鍵詞) Topological and strip entropyen_US
dc.subject (關鍵詞) Golden-mean tree-shiften_US
dc.title (題名) 關於黃金分割樹上的子平移之拓樸熵研究zh_TW
dc.title (題名) Topological Entropy of Golden-Mean Tree-Shiften_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] Karl Petersen and Ibrahim Salama. Tree shift topological entropy. Theoretical Computer Science, 743:64–71, 2018.

[2] Karl Petersen and Ibrahim Salama. Entropy on regular trees. Discrete & Continuous Dynamical Systems, 40(7):4453, 2020.

[3] Nathalie Aubrun and Marie-Pierre Béal. Tree-shifts of finite type. Theoretical Computer Science, 459:16–25, 2012.

[4] Nathalie Aubrun and Marie-Pierre Béal. Sofic tree-shifts. Theory of Computing Systems, 53(4):621–644, 2013.

[5] Nishant Chandgotia and Brian Marcus. Mixing properties for hom-shifts and the distance between walks on associated graphs. Pacific Journal of Mathematics, 294(1):41–69, 2018.

[6] Roy L Adler, Alan G Konheim, and M Harry McAndrew. Topological entropy. Transactions of the American Mathematical Society, 114(2):309–319, 1965.

[7] Tomasz Downarowicz. Entropy in dynamical systems, volume 18. Cambridge University Press, 2011.

[8] Douglas Lind, Brian Marcus, Lind Douglas, Marcus Brian, et al. An introduction to symbolic dynamics and coding. Cambridge university press, 1995.

[9] Jung-Chao Ban and Chih-Hung Chang. Mixing properties of tree-shifts. Journal of Mathematical Physics, 58(11):112702, 2017.

[10] Jung-Chao Ban and Chih-Hung Chang. Tree-shifts: Irreducibility, mixing, and the chaos of tree-shifts. Transactions of the American Mathematical Society, 369(12):8389–8407, 2017.

[11] Jung-Chao Ban and Chih-Hung Chang. Tree-shifts: The entropy of tree-shifts of finite type. Nonlinearity, 30(7):2785, 2017.

[12] Jung-Chao Ban, Chih-Hung Chang, Wen-Guei Hu, and Yu-Liang Wu. Topological entropy for shifts of finite type over Z and tree. arXiv preprint arXiv:2006.13415, 2020.

[13] Jung-Chao Ban and Chih-Hung Chang. Characterization for entropy of shifts of finite type on cayley trees. Journal of Statistical Mechanics: Theory and Experiment, 2020(7): 073412, 2020.

[14] Jung-Chao Ban, Chih-Hung Chang, and Yu-Hsiung Huang. Complexity of shift spaces on semigroups. Journal of Algebraic Combinatorics, 53(2):413–434, 2021.

[15] Wei-Lin Lin. On the strip entropy of the golden-mean tree shift. Master’s thesis, National Chengchi University, 2021.

[16] Itai Benjamini and Yuval Peres. Markov chains indexed by trees. The annals of probability, pages 219–243, 1994.

[17] Hans-Otto Georgii. Gibbs measures and phase transitions. de Gruyter, 2011.
zh_TW
dc.identifier.doi (DOI) 10.6814/NCCU202200470en_US