Descriptive Fixed Set Properties for Ribbon Complexes

TL;DR

This paper explores fixed set properties in descriptive proximity spaces within planar ribbon complexes, establishing fixed points for certain classes of maps and their invariance under proximal descriptive conjugacy.

Contribution

It introduces the concept of descriptive fixed sets in proximity spaces and proves their invariance under proximal descriptive conjugacy in ribbon complexes.

Findings

Fixed sets are characterized by matching descriptions of a set and its image.

Every amenable ribbon in a CW space has a fixed point.

Fixed subsets are preserved under proximal descriptive conjugacy.

Abstract

This article introduces descriptive fixed sets and their properties in descriptive proximity spaces viewed in the context of planar ribbon complexes. These fixed sets are a byproduct of descriptive proximally continuous maps that spawn fixed subsets, eventual fixed subsets and almost fixed subsets of the maps. For descriptive continuous map $f$ on a descriptive proximity space $X$, a subset $A$ of $X$ is fixed, provided the description of $f(A)$ matches the desription of $A$. In terms ribbon complexes in a CW space, an Abelian group representation of a ribbon is Day-amenable and each amenable ribbon has a fixed point. A main result in this paper is that if $h$ is a proximal descriptive conjugacy between maps $f,g$, then if $A$ is an [ordinary, eventual, almost] descriptively fixed subset of $f$, then $h(A)$ is a descriptively fixed subset of $g$.