Fixed Point Sets of k-Continuous Self-Maps of m-Iterated Digital Wedges

Abstract

Let Ckn,l be a simple closed k-curves with l elements in Zn and W:=Ckn,l∨⋯∨Ckn,l︷m-times be an m-iterated digital wedges of Ckn,l, and F(Conk(W)) be an alignment of fixed point sets of W. Then, the aim of the paper is devoted to investigating various properties of F(Conk(W)). Furthermore, when proceeding with this work, this paper addresses several unsolved problems. To be specific, we firstly formulate an alignment of fixed point sets of Ckn,l, denoted by F(Conk(Ckn,l)), where l(≥7) is an odd natural number and k≠2n. Secondly, given a digital image (X,k) with X♯=n, we find a certain condition that supports n−1,n−2∈F(Conk(X)). Thirdly, after finding some features of F(Conk(W)), we develop a method of making F(Conk(W)) perfect according to the (even or odd) number l of Ckn,l. Finally, we prove that the perfectness of F(Conk(W)) is equivalent to that of F(Conk(Ckn,l)). This can play an important role in studying fixed point theory and digital curve theory. This paper only deals with k-connected digital images (X,k) such that X♯≥2.