Reference request: Cut-and-project method gives rise to a fiber bundle over the torus

Question

I apologize in advance for how vague this request is.

A few weeks ago, I came upon a paper that (if I recall correctly) proves that the hull of a cut-and-project tiling is a fiber bundle over a torus. This is not the paper by Sadun cited below, but rather it specifically deals with cut-and-project tilings. If I recall correctly, it was published in a journal of physics, or the author was a physicist. I don't recall the name being one of the "big names" in tiling spaces (like Bellisard, Forrest, Hunton, Kellendonk, Putnam, etc.). I am interested mainly in the methods that they used in the paper, and not the result itself. Sadly, the computer on which I found the paper died shortly after I found it and I lost everything related to that paper. I have no author name, no paper title, and very vague search terms to go off of in my hunt.

Edit: it might be that the result I'm thinking of is actually related to something with bounded displacement rather than the hull being a fiber bundle.

Does anyone happen to have any idea what paper this might be?

References

Sadun, Lorenzo; Williams, R. F., Tiling spaces are Cantor set fiber bundles., Ergodic Theory Dyn. Syst. 23, No. 1, 307-316 (2003). ZBL1038.37014.

Answer 1

I have found the sought-after paper. It was Duneau, M., and Christophe O. "Displacive transformations and quasicrystalline symmetries." Journal de Physique 51.1 (1990): 5-19.

In section 3, they show that a cut and project pattern can be mapped onto a lattice by what they call (but never define) a "modulation".

Answer 2

You are probably looking for

Martin Schlottmann, "Generalized model sets and dynamical systems", In: "Directions in Mathematical Quesicrystals", M. Baake, R. Moody (eds.).

In section 4, he proves the following: given any cut-and-project scheme $(G, H, \mathcal{L})$, and any regular (i.e. boundary measure zero) aperiodic window $W$, there exists a mapping $\alpha$ from the hull $\mathbb X$ of $\Lambda(W)$ to the torus $\mathbb T= (G \times H)/\mathcal{L}$ with the following properties:

• $\alpha$ is continuous and commutes with the $G$-action.
• $\alpha$ is almost surely 1-1.

In other words, $(\mathbb X, G)$ is an almost 1-1 extension of $\mathbb T$.

P.S. This is also explained, and extended beyond regular windows in:

G. Keller, C. Richard, "Dynamics on the graph of the torus parametrization", Ergodic Theory and Dynamical Systems , Volume 38 , Issue 3 , May 2018 , pp. 1048 - 1085 DOI