關於黃金分割樹上的子平移之拓樸熵研究

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)}。\n\n此工作擴展了Petersen 和Salama 的結果，藉由考慮有限字母集A在黃\n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\n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\n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.\n\nIn 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.