Suppose $X$ is a complete bounded metric space, and we have two contractive self-embeddings $f, g : X \to X$ onto closed, disjoint subsets. Is there a natural way of assigning to the triple $(X, f, g)$ a map $h : X \to X$ such that the fixed points of $h$ are homeomorphic to the Cantor set?
I was inspired by consideration of a "double Droste effect," like here. In the case of a single Droste, the "limiting point" of the diagram is plainly the fixed point of the contractive map. However, in the case of a double Droste, the only way I can seem to describe the points is as the $2^\mathbb{N}$ points arising from the following: Let $s$ be a sequence in the set $\{f, g\}$, and $x \in X$. Then the limit
$$x, ~~~~~s(0)(x), ~~~~~s(0) \circ s(1) (x), ~~~~~s(0) \circ s(1) \circ s(2)(x), ~~~~~...~~\to ~~x_s$$
exists, is independent of $x$, and $s \mapsto x_s$ is injective with image subspatially homeomorphic to the Cantor set.
I suppose I don't have any deeper reason to assume there's a description of the points as fixed points, but perhaps I'm missing a nice construction.
I think the natural "fixed-points" description is in terms of sets, not points. In particular, let $Y$ be the set of nonempty compact subsets of $X$, equipped with the Hausdorff metric (which is complete since $X$ is complete). There is then a map $h:Y\to Y$ given by $h(A)=f(A)\cup g(A)$, which is easily seen to be a contraction since $f$ and $g$ are. The unique fixed point of this map $h$ is then exactly the Cantor set you describe.