Fixed Point Sets of Digital Curves and Digital Surfaces

Abstract

Given a digital image (or digital object) (X,k), we address some unsolved problems related to the study of fixed point sets of k-continuous self-maps of (X,k) from the viewpoints of digital curve and digital surface theory. Consider two simple closed k-curves with li elements in Zn, i∈{1,2},l1⪈l2≥4. After initially formulating an alignment of fixed point sets of a digital wedge of these curves, we prove that perfectness of it depends on the numbers li,i∈{1,2}, instead of the k-adjacency. Furthermore, given digital k-surfaces, we also study an alignment of fixed point sets of digital k-surfaces and digital wedges of them. Finally, given a digital image which is not perfect, we explore a certain condition that makes it perfect. In this paper, each digital image (X,k) is assumed to be k-connected and X♯≥2 unless stated otherwise.