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| Title: | 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 |
| Authors: | 蔡承育
Tsai, Cheng-Yu |
| Contributors: | 班榮超
Ban, Jung-Chao
蔡承育
Tsai, Cheng-Yu |
| Keywords: | 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 |
| Date: | 2022 |
| Issue Date: | 2022-07-01 16:20:42 (UTC+8) |
| Abstract: | 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. |
| Reference: | [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. |
| Description: | 碩士
國立政治大學
應用數學系
109751002 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0109751002 |
| Data Type: | thesis |
| DOI: | 10.6814/NCCU202200470 |
| Appears in Collections: | [應用數學系] 學位論文
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