Discrete Group Actions on Digital Objects and Fixed Point Sets by Isok(·)-Actions

Abstract

Given a digital image (or digital object) (X,k),X⊂Zn, this paper initially establishes a group structure of the set of self-k-isomorphisms of (X,k) with the function composition, denoted by Isok(X) or Autk(X). In particular, let Ckn,l be a simple closed k-curve with l elements in Zn. Then, the group Isok(Ckn,l) is proved to be isomorphic to the standard dihedral group Dl with order l. The calculation of this quantity Isok(Ckn,l) is a key step for obtaining many new results. Indeed, it is essential for exploring many features of Isok(X). Furthermore, this quantity is proved to be a digital topological invariant. After proceeding with an Isok(X)-action on (X,k), we investigate some properties of fixed point sets by this action. Finally, we explore various features of fixed point sets by this action from the viewpoint of digital k-curve theory. This paper only deals with k-connected digital images (X,k) whose cardinality is equal to or greater than 2. Besides, we discuss some errors that have appeared in the lilterature.