The Research Of The Inverse Of Cantor-like Sequences

It's been a long history of the study of the automatic sequence and the classical Cantor sequence is just a 3-automatic sequence. Finite automaton sequence is defined as infinite sequence based on finite character set. There have been many useful conclusions and automaton sequence is also widely applied to number theory, ergodic theory, fractal geometry,theoretical physics, information theory and so on. Then later, Allouche and Shallit extended the automaton sequence to the regular sequence and studied the relationship between the regular sequence and the automaton sequence. They also studied the properties of regular sequences deeply and drew many useful conclusions, based on which, the regular sequence can be the coefficients of the power series and in the sense of the domain, we studied the inverse on the field in the later research.Cantor-like sequences are classical sequences of automatic sequence, especially the typical Cantor sequences. Studying Cantor-like sequences has an important significance to fractal geometry and substitution dynamical system. On the background of automatic sequences and regular sequences, this article mainly discusses the inverse structure of Cantorlike sequences.The organization of the paper is as blow. The first chapter introduces the research status of automatic sequences, regular sequences and Cantor sequences, and the organization arrangement of the paper. In the preliminary knowledge, the definitions and related properties of word and substitution, automatic sequences, regular sequences respectively are introduced. The third chapter presents some theorems and typical examples of k-regular sequences, in order to deepen our understanding of regularity. The fourth chapter is the main content of the paper, the research is concern on transform from 3-constant length,4-constant length and 5-constant length to the Cantor-like sequences, extended to any constant length transform, then the general Cantor-like sequences is obtained. Finally, the promotion of the relevant conclusions is introduced.