Fixed point sets in digital topology, 2

Abstract

We continue the work of [10], studying properties of digital images determined by fixed point invariants. We introduce pointed versions of invariants that were introduced in [10]. We introduce freezing sets and cold sets to show how the existence of a fixed point set for a continuous self-map restricts the map on the complement of the fixed point set.

Fixed poin sets in digital topology, 1

“…Definition 2.5. [2] Let (X, κ) be a digital image. We say A ⊂ X is a freezing set for X if given f ∈ C(X, κ), A ⊂ Fix(f ) implies f = id X .…”

“…• In particular, a bounding curve need not be equal to Bd(D). E.g., in the disk D shown in Figure 2(i), (2, 2) is a point of the bounding curve; however, all of the points c 1 -adjacent to (2,2) are members of D, so by Definition 2.1, (2, 2) ∈ Bd(D). Thus, a bounding curve for D need not be contained in Bd(D).…”

“…Remark 3.2. Example 5.16 of [2] shows that the freezing set of Theorem 3.1 need not be minimal for n = 3.…”

“…The following provides a negative answer to this question. (3,2), (3,4), (3,6), (0, 6)} (see Figure 5) is a minimal freezing set for (D, c 1 ). Note (2, 2) and (2, 4) are endpoints of maximal horizontal and vertical bounding segments of D and are not members of A.…”

“…Note (2, 2) and (2, 4) are endpoints of maximal horizontal and vertical bounding segments of D and are not members of A. While (2, 2) and (2,4) are members of a bounding curve for D, they are not members of a minimal bounding curve, which includes edges from (3,4) to (2,3) and from (2, 3) to (3, 2). Figure 5: There are distinct boundary curves for the disk D that contain the horizontal segments from (0, 0) to (3, 0) and from (0, 6) to (3,6); and vertical segments from (0, 0) to (0, 6), from (3, 0) to (3,2), and from (3,4) to (3,6).…”