一类常长代换序列的结构

The research of substitution sequence has a long history, it is deeply contacted with fractal geometry, harmonic analysis, dynamic system, number theory, automatic and theoretical physics. Since the substitution sequences with constant length has its special structure and is applied successfully in many fields, it attracts many researchers' attention and its research is very active. In this project, we will mainly apply some technique and skill of combinatorics to study various properties of substitution and its sequence. There are mainly two problems in this project. One is that combinatorial properties of substitution sequences with constant length. In particular, we are interested in some properties relative to subsequence of generalized Morse sequence, singular words and singular decomposition of period doubling sequence. The other is about the characterization of substitutions with admissible sequences as their fixed points. Although it is well-known that admissible sequences are very useful in applications, there are few substitution sequences with some special structures are admissible. So it is worthy to do some research in this field.

代换序列的研究有很长的历史，它与分形几何、调和分析、动力系统、数论、自动机理论及理论物理等学科之间有深刻联系。由于常长代换序列结构的特殊性及广泛应用，因此吸引了众多学者的重视。本项目主要利用组合的方法和技巧，研究代换序列及代换本身的各类性质。主要研究以下两个方向的问题：其一是常长代换序列的组合性质，如广义Morse序列的子序列中的若干问题及加倍周期序列的奇异词和奇异分解；其二是不动点为可允许序列的代换的刻画。尽管可允许序列具有较广泛应用，但是目前仅知道几个特殊的代换序列为可允许序列，因此我们希望对该问题进行深入研究。

代换序列与分形几何、调和分析、动力系统、数论、自动机理论及理论物理等学科之间有深刻联系。本项目主要利用组合的方法和技巧，研究代换序列及代换本身的各类性质。主要解决以下三个方向的问题：其一是常长代换序列的性质，如广义Morse序列复杂度和排列复杂度，广义Morse子序列和加倍周期子序列中的若干问题；其二是不动点为可允许序列的代换的刻画，即我们分别给出常长代换序列和非常长代换序列为可允许序列的充分必要条件；其三是给出一类代换序列的无理指数的上界。