Fixed point property for digital spaces

Abstract

The paper compares the fixed point property (FPP for short) of a compact Euclidean plane with its digital versions associated with Khalimsky and Marcus-Wyse topology. More precisely, by using a Khalimsky and a Marcus-Wyse topological digitization, the paper studies digital versions of the FPP for Euclidean topological spaces. Besides, motivated by the digital homotopy fixed point property (DHFP for brevity) [O. Ege, I. Karaca, C. R. Math. Acad. Sci. Paris, 353 (2015), 1029-1033], the present paper establishes the digital homotopy almost fixed point property (DHAFP for short) which is more generalized than the DHFP.

Remarks on the Preservation of the Almost Fixed Point Property Involving Several Types of Digitizations

“…Then, we may pose a query about the FPP of another shape of a diamond, as follows: Proof. According to Proposition 5, since the FPP in MTC is an M-topological invariant property [8], we may prove that (Y, γ Y ) has the FPP. For any M-continuous self-map f of (Y, γ Y ), we prove that there is always a point y ∈ Y such that f (y) = y.…”

“…The author in [8,10] proved the FPP of the smallest open neighborhood of (Z n , κ n ) [10] and the non-FPP of a compact M-topological plane in (Z 2 , γ) [8]. Thus, we may now pose the following queries about the AFPP of compact M-topological plane X and the preservation of the AFPP of a compact n-dimensional Euclidean space (or cube) into that of each of K-, M-, Uand L-digitization, as follows:…”

“…Conversely, if k = 3 n − 1, then we first prove that (X, k) does not have the AFPP. For instance, in Z 2 , consider the digital image ([−1, 1] 2 Z := X, 4) instead of (X, 8). Let us consider a self-map of (X, 4).…”

“…A simple closed M-curve with l elements on (Z 2 , γ), denoted by SC 2,l M , is defined as a finite sequence < x i > i∈[0,l−1] Z in Z 2 [8], where x i and x j are M-adjacent if and only if |i − j| = ±1(mod l).…”

The Fixed Point Property of the Infinite M-Sphere

“…Indeed, it is obvious that the usual topological space (R n , U) is not an Alexandroff space. As an Alexandroff topological space [4,5], the M-topological space was proposed [6] and the study of various properties of it includes the papers [1,[6][7][8][9][10][11][12][13]. Regarding digital spaces [14] in Z 2 , we will follow the concept of a digital k-neighborhood of a point p ∈ Z 2 .…”