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

Sang-Eon Han
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Sang-Eon Han: Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju 54896, Jeonbuk, Korea

Mathematics, 2021, vol. 9, issue 3, 1-25

Abstract: Given a digital image (or digital object) ( X , k ) , X ? Z n , this paper initially establishes a group structure of the set of self- k -isomorphisms of ( X , k ) with the function composition, denoted by I s o k ( X ) or A u t k ( X ) . In particular, let C k n , l be a simple closed k -curve with l elements in Z n . Then, the group I s o k ( C k n , l ) is proved to be isomorphic to the standard dihedral group D l with order l . The calculation of this quantity I s o k ( C k n , l ) is a key step for obtaining many new results. Indeed, it is essential for exploring many features of I s o k ( X ) . Furthermore, this quantity is proved to be a digital topological invariant. After proceeding with an I s o k ( 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.

Keywords: Iso k (·)-action; Aut k ( X )-action; simple closed k-curve; dihedral group; digital wedge; fixed point set; perfect (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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