Question. Did you ever see inverse limits to be used (or even seriousely considered) anywhere in metric geometry (but NOT in topology)?
The definition of inverse limit for metric spaces is given below. (It is usual inverse limit in the category with class of objects formed by metric spaces and class of morphisms formed by short maps.)
Definition.
Consider an inverse system of metric spaces $X_n$ and short maps $\phi_{m,n}:X_m\to X_n$ for $m\ge n$;
i.e.,(1) $\phi_{m,n}\circ \phi_{k,m}=\phi_{k,n}$ for any triple $k\ge m\ge n$ and (2) for any $n$, the map $\phi_{n,n}$ is identity map of $X_n$.
A metric space $X$ is called inverse limit of the system $(\phi_{m,n}, X_n)$ if its underlying space consists of all sequences $x_n\in X_n$ such that $\phi_{m,n}(x_m)=x_n$ for all $m\ge n$ and for any two such sequences $(x_n)$ and $(y_n)$ the distance is defined by
$$ | (x_n) (y_n)| = \lim_{n\to\infty} | x_n y_n | .$$
Why: I have a theorem, with little cheating you can stated it this way: The class of metric spaces which admit path-isometries to Euclidean $d$-spaces coincides with class of inverse limits of $d$-polyhedral spaces.
In the paper I write: it seems to be the first case when inverse limits help to solve a natural problem in metric geometry. But I can not be 100% sure, and if I'm wrong I still have time to change this sentence.
Best Answer
This paper by P.-E. Caprace, uses a "refined boundary" of a CAT(0) space. This boundary is constructed in the following way : given a point $\xi$ in the boundary at infinity of your space $X$, you construct a point $X_\xi$, which is the inverse limit of the horoballs centered at $\xi$. Here the maps $\phi_{m,n}$ are the CAT(0) projections. Then the space $X_\xi$ is itself CAT(0), and you can iterate the construction. Under reasonable hypotheses, the construction stops after a finite number of steps, and the refined boundary is the union of all the spaces you get.
(In the case of symmetric spaces, this construction has been already considered by Karpelevic in 1965, but with different definitions, and I don't think he saw it as inverse limits).