I am currently writing a paper in which I need to use the following fact: if $T \colon X \to X$ is a uniquely ergodic transformation of a compact metric space, and $\mathcal{A}$ is a continuous function from $X$ to the set of invertible, positive $2 \times 2$ real matrices, then the limit $$\lim_{n \to \infty} \frac{1}{n}\log\|\mathcal{A}(T^{n-1}x) \cdots \mathcal{A}(Tx)\mathcal{A}(x)\|$$ exists and is independent of $x$. This result essentially follows from a 1997 paper by Alex Furman, but in Furman's paper the additional assumption is made that $T$ is a homeomorphism. This hypothesis can be circumvented by lifting everything to the invertible natural extension of the dynamical system and applying Furman's theorem in the new context. However, I am having trouble writing this up in a concise manner, essentially because I don't know of any references which deal with the natural extension of a topological dynamical system.

I would ideally like to find a reasonably crisp reference for the following fact:

Let $T \colon X \to X$ be a uniquely ergodic transformation of a compact metric space. Then there exists a uniquely ergodic homeomorphism $\hat T$ of a compact metrisable space $\hat X$ and a continuous surjection $\pi \colon \hat X \to X$ such that $\pi \circ \hat T = T \circ \pi$.

This is not terribly hard to prove by developing the natural extension from first principles, but when all the details are put in it takes up a whole page, which seems to me quite a lot of journal space to take up with a result which is not in any sense original. On the other hand, I am reluctant to leave all the details to the reader, since the purpose of the paper is to prove a result in numerical linear algebra, and as such a large part of the paper's intended audience will not be dynamicists. A crisp reference therefore seems desirable, but I can't seem to find a suitable resource. Does anyone know where I might find this, or something similar?