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Title: 關於黃金平均樹上子平移的條型熵研究
On the Strip Entropy of the Golden-Mean Tree Shift
Authors: 林韋霖
Lin, Wei-Lin
Contributors: 班榮超
Ban, Jung-Chao
Lin, Wei-Lin
Keywords: 樹平移
tree shifts
topological entropy
golden mean tree
strip entropy
Date: 2021
Issue Date: 2021-08-04 15:39:35 (UTC+8)
Abstract: 在2019 年,彼得森跟莎拉曼[11] 證明在樹平移上拓樸熵的存在性。之後他們取最左邊的樹枝當作基底,用條型法[12] 去估計黃金平均樹上子平移的熵。以{0, 1} 為字母,並且不接受連續兩個1,他們證明在k 維樹上,hn^(k) 會收斂到h^(k)。
週期路徑上,稱作hn(T)。我們證明hn(T) 會收斂到到在黃金平均樹上的熵
In 2019, Petersen and Salama [11] demonstrated the existence of topological entropy for tree shifts. Later, they took the most left branch as a fixed base, and use the strip method [12] to evaluate the entropy of the golden mean tree shift. By alphabet {0, 1} with no adjacent 1’s, they proved that hn^(k) converges to h^(k) on the k-tree shift.
In this paper, the periodic paths on the golden mean tree are considered, and the strip entropy, said hn(T), is defined in these periodic paths. We prove that hn(T) converges to the topological entropy h(T) on the golden mean tree.
Reference: [1] Nathalie Aubrun and Marie-Pierre Béal. Tree-shifts of finite type. Theoretical Computer Science, 459:16–25, 2012.
[2] Nathalie Aubrun and Marie-Pierre Béal. Sofic tree-shifts. Theory of Computing Systems, 53(4):621–644, 2013.
[3] Jung-Chao Ban and Chih-Hung Chang. Mixing properties of tree-shifts. Journal of Mathematical Physics, 58(11):112702, 2017.
[4] 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.
[5] Jung-Chao Ban and Chih-Hung Chang. Tree-shifts: The entropy of tree-shifts of finite type. Nonlinearity, 30(7):2785, 2017.
[6] 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.
[7] 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.
[8] Jung-Chao Ban, Chih-Hung Chang, and Nai-Zhu Huang. Entropy bifurcation of neural networks on cayley trees. International Journal of Bifurcation and Chaos, 30(01):2050015, 2020.
[9] 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.
[10] Douglas Lind and Brian Marcus. An introduction to symbolic dynamics and coding. Cambridge university press, 2021.
[11] Karl Petersen and Ibrahim Salama. Tree shift topological entropy. Theoretical Computer Science, 743:64–71, 2018.
[12] Karl Petersen and Ibrahim Salama. Entropy on regular trees. Discrete & Continuous Dynamical Systems, 40(7):4453, 2020.
[13] Cheng-Yu Tsai. Strip entropy of some tree-shifts.Master’s thesis.
Description: 碩士
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Data Type: thesis
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