Coincidence Point Sets in Digital Topology

TL;DR

This paper explores properties of coincidence point sets in digital topology, introducing invariants and examining effects of deformation, aiming to extend fixed point theory to digital images with a combinatorial approach.

Contribution

It introduces new topological invariants for digital images and analyzes how coincidence point sets behave under rigidity and deformation retraction.

Findings

Defined topological invariants related to coincidence points

Analyzed the impact of deformation retraction on coincidence sets

Proposed a concept of divergence degree in digital images

Abstract

In this article, we investigate some properties of the coincidence point set of digitally continuous maps. Following the Rosenfeld graphical model which seems more combinatorial than topological, we expect to achieve results that might not be analogous to the classical topological fixed point theory. We also introduce and study some topological invariants related to the coincidence and common fixed point sets for continuous maps on a digital image. Moreover, we study how these coincidence point sets are affected by rigidity and deformation retraction. Lastly, we present briefly a concept of divergence degree of a point in a digital image.