Research Of Finite And Infinite Symbolic Sequences

The kncading sequences in discrete dynamical system, Sturmian sequences in formal language and DNA sequence in bioinformatics play an important, even central role in their respective fields. As sequences in finite alphabets, they are also research objects of Combinatorics on Words. In this thesis we make some studies and discussions for the symbolic sequences of anti-symmetric cube maps in discrete dynamical systems, Sturmian sequences, generalized Fibonacci sequences and DNA sequences.In the first chapter, we study combinatorial properties of symbolic sequences of antisymmetric cubic maps in discrete dynamical systems. For a given length, we study the relationship between the kneading sequences of antisymmetric cubic maps and the equivalence classes of primitive necklaces on three colors under the fixing one of them and exchange of the others, then determine their cardinalities and try to find a combinatorial interpretation.In the second chapter, we study combinatorial properties of Sturmian sequences. By a rotation sequence, an equivalent definition of Sturmian sequences, we first construct a scries of sequences with complexity kn + 1 for any positive integer n and k, then, from these sequences, we reconstruct other Sturmian sequences by some coarse-grained processes, and determine their relationships with the original Sturmian sequence.The third chapter is contributed to the Wall numbers of the it-step Fibonacci sequences. It is introduced from a viewpoint of algebra. A general property of Wall number is given. Two problems are posed.In the last chapter, we present a new representation of a DNA sequence and give a corresponding numerical characterization. Based on them, similarity analysis of the coding sequences of cxon 1 of beta-globin gene of eleven species is made.