Digital Topological Properties of an Alignment of Fixed Point Sets

Sang-Eon Han

doi:10.3390/math8060921

Abstract

The present paper investigates digital topological properties of an alignment of fixed point sets which can play an important role in fixed point theory from the viewpoints of computational or digital topology. In digital topology-based fixed point theory, for a digital image ( X , k ) , let F ( X ) be the set of cardinalities of the fixed point sets of all k-continuous self-maps of ( X , k ) (see Definition 4). In this paper we call it an alignment of fixed point sets of ( X , k ) . Then we have the following unsolved problem. How many components are there in F ( X ) up to 2-connectedness? In particular, let C k n , l be a simple closed k-curve with l elements in Z n and X : = C k n , l 1 ∨ C k n , l 2 be a digital wedge of C k n , l 1 and C k n , l 2 in Z n . Then we need to explore both the number of components of F ( X ) up to digital 2-connectivity (see Definition 4) and perfectness of F ( X ) (see Definition 5). The present paper addresses these issues and, furthermore, solves several problems related to the main issues. Indeed, it turns out that the three models C 2 n n , 4 , C 3 n − 1 n , 4 , and C k n , 6 play important roles in studying these topics because the digital fundamental groups of them have strong relationships with alignments of fixed point sets of them. Moreover, we correct some errors stated by Boxer et al. in their recent work and improve them (see Remark 3). This approach can facilitate the studies of pure and applied topologies, digital geometry, mathematical morphology, and image processing and image classification in computer science. The present paper only deals with k-connected spaces in DTC. Moreover, we will mainly deal with a set X such that X ♯ ≥ 2 .

Highlights

A paper [1] initially raised the following query involving a fixed point set and a homotopy fixed point set from the viewpoint of typical fixed point theory
It can be of interest in digital topology-based fixed point theory
For a given digital image ( X, k), since each of the above quantities need not be 2-connected, the present paper will take another term to exactly characterize the “set of the cardinalities of fixed point sets of k-continuous self-maps of it”, i.e., the so-called “an alignment of fixed point sets” exactly defining the above set in mathematical sense and further, we denote it with F ( X ) as referred to in [4]

Summary

Introduction

For a given digital image ( X, k), since each of the above quantities need not be 2-connected, the present paper will take another term to exactly characterize the “set of the cardinalities of fixed point sets of k-continuous self-maps of it”, i.e., the so-called “an alignment of fixed point sets” exactly defining the above set in mathematical sense and further, we denote it with F ( X ) as referred to in [4] (see Definition 4). In the Rosenfeld’s digital topological setting for fixed point theory, regarding F ( X ) of ( X, k ), we may raise the following queries This issue was partially studied in [4]. This paper corrects and improves some results stated by Boxer et al in [4] (see Remark 3) and corrects several incorrect citations in [4] related to a digital wedge (see Section 2) and a generalization of digital k-connectivity and a normal adjacency for a digital product

Preliminaries

Conclusions and Further Work