[Paper Review] Approximate Fixed Point Properties in Digital Topology

Laurence Boxer|arXiv (Cornell University)|Nov 18, 2019

Digital Image Processing Techniques11 references4 citations

TL;DR

This paper investigates the approximate fixed point property (AFPP) for both single-valued and multivalued continuous functions in digital topology, correcting errors in prior literature regarding universal multivalued functions. It establishes that a digital image has the multivalued AFPP if and only if its identity function is a weak universal, providing corrected characterizations and new examples of images with the AFPP.

ABSTRACT

We study the approximate fixed point property (AFPP) for continuous single-valued functions and for continuous multivalued functions in digital topology. We extend what is known about these notions and discuss errors that have appeared in the literature.

Motivation & Objective

• To clarify and correct misconceptions in the literature regarding the approximate fixed point property (AFPP) for multivalued functions in digital topology.
• To provide new examples of digital images that possess the approximate fixed point property (AFPP) for both single-valued and multivalued continuous functions.
• To distinguish between universal and weak universal multivalued functions, correcting prior errors that conflated the two concepts.
• To establish a precise characterization of the multivalued AFPP using the concept of weak universality of the identity function.
• To investigate open questions regarding the relationship between single-valued and multivalued AFPP, and the behavior of AFPP under multivalued retractions.

Proposed method

• Uses digital adjacency relations (e.g., $c_u$-adjacency) to define continuity and connectivity in digital images.
• Applies the concept of $r$-th subdivision of digital images to define multivalued continuity via induced functions on subdivided grids.
• Introduces and distinguishes between universal and weak universal multivalued functions, correcting prior misuse in the literature.
• Employs the identity function $1_X$ as a central tool to characterize the multivalued AFPP via weak universality.
• Applies topological reasoning based on path-connectedness and adjacency to verify AFPP conditions in specific digital image structures.
• Corrects flawed proofs in earlier works (e.g., [7]) by replacing universal functions with weak universal functions in key theorems.

Experimental results

Research questions

• RQ1 Does the identity function being a weak universal imply the multivalued AFPP in a digital image?
• RQ2 Is the converse of Proposition 2.12 true — does the single-valued AFPP imply the multivalued AFPP?
• RQ3 Does a product space $X = \prod_{i=1}^v [a_i, b_i]_{\mathbb{Z}}$ with $b_i > a_i$ for at least two indices have the single-valued AFPP under $c_v$-adjacency?
• RQ4 If a subset $Y$ is a $(c_v, c_v)$-continuous multivalued retract of a digital image $X$ with the multivalued AFPP, does $Y$ also have the multivalued AFPP?
• RQ5 If the identity function is a weak universal for single-valued maps on $X$, is it also a weak universal for multivalued maps on $X$?

Key findings

• The multivalued AFPP holds for a digital image $(X, u)$ if and only if the identity function $1_X$ is a weak universal for multivalued functions on $X$.
• The claim in Proposition 3.1 of [7] that the constant multivalued function $F(x) = Y$ is universal is incorrect; it is only weak universal, as shown by counterexample with a single-point image.
• The assertion in Proposition 4.1 of [7] that the identity is universal if and only if $X$ has the multivalued AFPP is false; the correct condition involves weak universality, not universality.
• The paper provides a corrected characterization of the multivalued AFPP using weak universality, resolving inconsistencies in prior literature.
• The identity function $1_X$ is a weak universal for $C(X, u)$ if and only if $(X, u)$ has the single-valued AFPP, as established in [4] and confirmed here.
• The paper identifies and corrects errors in [7] and [11], particularly the misuse of universal functions instead of weak universal functions in defining and proving AFPP results.