Edmund Harris gave a talk in Orsay on his joint work with Pierre Arnoux, Maki Furukado and Shunji Ito. The original motivation of this line of study is the construction of Markov partitions for hyperbolic linear automorphisms (Bowen showed that the pieces of an irreducible Anosov diffeomorphisms beyond dimension 2 — I don’t know if more is known about them, eg, are there estimates on the Hausdorff dimension of the boundary?).

Gérard Rauzy found a very nice way to build such partitions for a simple example. It involves a substitution reflecting the dynamics along the expanding line (which is unique in his case and more generally whenever the characteristic polynomial is Pisot). The pieces of the Markov partition are obtained by a cut-and-project method linking this to tiling constructions.

In the non-Pisot case, the expanding space is multidimensional and one must find a correct notion of “two-dimensional substitution” to represent the “transversal dynamics” of Vershik. This is also related to a numeration system in the vein of Kamae.

Kenyon-Vershik and Leborgne had obtained somewhat weaker results (sofic representation which are only semiconjugate).

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