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

Sang-Eon Han

doi:10.3390/math8091617

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.

Highlights

Given a digital image ( X, k) in Ob( DTC (k )), a paper [2] explored some features of (k-homotopy) fixed point sets related to this issue in a DTC (k )
This paper explores some conditions supporting the perfectness of F (Conk ( X )) because the notion of perfectness can play an important role in mechatronics and digital geometry
After joining a simple k-path ( P, k) onto W to produce a digital wedge with a k-adjacency, denoted by (W ∨ P, k ), we prove that F (Conk (W ∨ P)) is perfect if and only if l ≤ 2d + 4, where d is the length of P

Summary

Introduction

Given a digital image ( X, k) in Ob( DTC (k )), a paper [2] explored some features of (k-homotopy) fixed point sets related to this issue in a DTC (k ). A recent paper [7] proved that a k-isomorphism preserves a k-homotopy, a k-homotopy equivalence, k-contractibility (for more details see Theorem 2 and Corollaries 1 and 2 of Reference [7]) This finding can facilitate fixed point theory and homotopy theory in a DTC (k ) setting. After joining a simple k-path ( P, k) onto W to produce a digital wedge with a k-adjacency, denoted by (W ∨ P, k ), we prove that F (Conk (W ∨ P)) is perfect if and only if l ≤ 2d + 4, where d is the length of P.

Preliminaries

Existence of a Perfect Alignment of Fixed Point Sets

Conclusions and a Further Work