  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 $g$.
