Substitution With Cofinal Fixed Points

Finite words and infinite sequences are of increasing importance in various fields of math,physics bioinformatics and so on.The combinatorial properties of these sequences are of great interest in various subject.Z.-X.Wen and Z.-Y Wen provided that the monoid of invertible substitutions over a 2-letter alphabet is finitely generated.But for a larger alphabet,the situation will become more complicated.Z.-X.Wen and Y.-P.Zhang showed that the monoid of invertible substitutions over a 3-letter alphabet is not finitely generated.This paper mainly summarizes,on the two alphabets,which kind of substitutions will have two cofinal fixed points.We also characterize the non-primitive substitutions over 3-letter alphabet with three cofinal fixed points.This thesis is divided into three chapters.The first chapter mainly introduces the research background of the substitution over the 2-letter alphabet,introduces the work done by the predecessors and the corresponding conclusions and introduces the research questions of this paper.The second chapter introduces the relevant preliminary knowledge,introduces the definitions of words,cofinal,and fixed points,and introduces related properties and conclusions.The third chapter is the main part of thesis.The first section of this chapter mainly summarizes the fixed points of a primitive invertible substitution??<?,?>.The second section summarizes the primitive substitutions with two cofinal fixed points.The third section discusses the non-primitive substitutions with three cofinal fixed points over the alphabet S={a,b,c}.