關於黃金平均樹上子平移的條型熵研究

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)。\n本篇文章會考慮在黃金平均樹上的週期路徑，並且定義條型熵在這些\n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\nh(T)。

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.\nIn 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.