Reference request: natural extensions of topological dynamical systems

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?

• I believe that Jewett-Krieger theorem does not answer this exactly, but came into my mind and may help. – rpotrie Nov 5 '10 at 17:35
• Perhaps, it would help if you revealed the nature of $T$? It's just the shift on a Sturmian system if I understand correctly. And -- at least geometrically -- its natural extension looks straightforward: one could simply take the $\mathbb Z$-orbits instead of $\mathbb N$-ones. (Under the irrational rotation by $\gamma$, for all $x\in[0,1)$.) I'm not sure what it does in terms of symbolic sequences though. – Nikita Sidorov Nov 5 '10 at 20:17
• Hi Nik! Yes, in this particular case it is the Sturmian system, and on balance the way to go is probably to use the fact that we know explicitly what the natural extension is. I'd still like to know the general answer though! – Ian Morris Nov 5 '10 at 20:28
• Perhaps, there isn't one. As a symbolic kind of guy, I'm always inclined to encode a given system as a subshift so the natural extension becomes really natural. ;) Otherwise one has to deal with all these $\sigma$-algebras and their preimages -- and what good has ever come out of this, honestly? By the way, in your case you have the (unique) measure as well, not just a topological model. So the natural extension is totally canonical. – Nikita Sidorov Nov 5 '10 at 20:39
• Is Proposition 3.5 in the paper "Topological mixing and uniquely ergodic systems" by Lehrer relevant? tau.ac.il/~lehrer/Papers/topological%20mixing.pdf – Lasse Rempe-Gillen Aug 24 '15 at 11:30