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

Author

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  • Sang-Eon Han

    (Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju 54896, Jeonbuk, Korea)

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.

Suggested Citation

  • Sang-Eon Han, 2021. "Discrete Group Actions on Digital Objects and Fixed Point Sets by Iso k (·)-Actions," Mathematics, MDPI, vol. 9(3), pages 1-25, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:290-:d:491140
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    References listed on IDEAS

    as
    1. Sang-Eon Han, 2020. "Fixed Point Sets of Digital Curves and Digital Surfaces," Mathematics, MDPI, vol. 8(11), pages 1-25, October.
    2. Sang-Eon Han, 2020. "The Most Refined Axiom for a Digital Covering Space and Its Utilities," Mathematics, MDPI, vol. 8(11), pages 1-21, October.
    3. Sang-Eon Han, 2019. "Remarks on the Preservation of the Almost Fixed Point Property Involving Several Types of Digitizations," Mathematics, MDPI, vol. 7(10), pages 1-14, October.
    4. Han, Sang-Eon, 2019. "Estimation of the complexity of a digital image from the viewpoint of fixed point theory," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 236-248.
    Full references (including those not matched with items on IDEAS)

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