自相似序列的因子谱性质及相关分形结构

The factor spectra properties and related fractal structures of the self-similar sequences will be studied in this proposal. We mainly study two problems: (1) The traditional theory of combinatorics on words studies whether a word appears in a sequence as a factor, the frequencies of the factors, and so on. Since it lacks the tool, we cannot study the positions where the factors locate, but this property plays an important role in the factor structure. We will study the factors which satisfy some combinatorics property, and introduce the notion of spectra, which considers factor and the position where the factor appears as two variables. That will give a new idea and some new methods for studying in this area. For studying the positions where the factors appears, we introduce some tools, such as the gap sequences, kernel and envelope words of a factor, reflexivity. The properties of them are various in different self-similar sequences. (2) The fractal structure derived from the positions of factors, gives a new relation between the self-similar sequence and fractal. As an interesting application, by the chain structure and the tree structure we founded, we can solve a problem in theoretical of computer science: how many factors in a certain type occur in a given block of sequence. This project is at the intersection of fractal theory, combinatorics on words, and theoretical of computer science. In this project, we wish to obtain some rather general results.

本项目的研究对象是自相似序列的因子谱性质以及相关的分形结构。(1)传统的词上组合性质仅研究满足某一性质的某个因子是否出现、因子出现的频率等性质。但由于缺乏工具，没有研究因子逐次出现的位置这一重要性质。本项目研究满足某一组合性质的因子性质：同时考虑因子与位置两个变量，可以获得诸如任意因子在序列中每次出现的位置、相互关系等信息，称为因子谱性质。它提供了全新的研究视角。为了研究因子谱性质，我们引入了间隔序列、核词、包络词、自反性等有力的工具，它们在不同的自相似序列中表现出不同的作用。(2)因子位置的分形结构，建立了自相似序列与分形的新联系。作为一个重要应用，利用我们已经发现的链结构和树结构，可以解决理论计算机领域广泛关注的因子计数问题，即给定属性的因子在给定序列片段中出现的次数。本项目属于分形、词上组合和理论计算机的交叉领域。通过本项目，我们希望在现有研究基础上获得一般性结论。

本项目的研究对象是自相似序列的因子谱性质以及相关的分形结构。主要研究内容包括三个部分：因子谱、因子位置的分形结构和分形网络。(1)本项目研究满足某一组合性质的因子性质：同时考虑因子与位置两个变量，可以获得诸如任意因子在序列中每次出现的位置、相互关系等信息，称为因子谱性质。它提供了全新的研究视角。为了研究因子谱性质，我们引入了间隔序列、核词、包络词、自反性等有力的工具，它们在不同的自相似序列中表现出不同的作用。2019年12月,我们在CSCD收录期刊《数学研究及应用》发表综述性文章《The Factor Spectrum and Derived Sequence》。(2)因子位置的分形结构，建立了自相似序列与分形的新联系。作为一个重要应用，利用我们已经发现的链结构和树结构，可以解决理论计算机领域广泛关注的因子计数问题，即给定属性的因子在给定序列片段中出现的次数。2018年，我们解决了Fibonacci序列中高次方词的计数问题。撰写论文两篇：《The Number of Fractional Powers in the Fibonacci Word》和《Enumeration Properties of Repetitions in the Fibonacci Word》。(3)具有分形结构的复杂网络，简称：分形网络。2020年底在SCI收录期刊《Physica A-statistical mechanics and its applications》（JCR分区Q2）发表文章《Scale-free and small-world properties of a multiple-hub network with fractal structure》。2020年底获得SCI收录期刊《Fractals: complex geometry, patterns, and scaling in nature and society》（JCR分区Q1）接收函，文章题为《Fractal networks on Dürer-type polygon》。