Papers › Rauzy induction of polygon partitions and toral ℤ²-rotations
Rauzy induction of polygon partitions and toral ℤ²-rotations
Sébastien Labbé
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We extend the notion of Rauzy induction of interval exchange transformations to the case of toral ℤ²-rotation, i.e., ℤ²-action defined by rotations on a 2-torus. If 𝒳_(𝒫,R) denotes the symbolic dynamical system corresponding to a partition 𝒫 and ℤ²-action R such that R is Cartesian on a sub-domain W, we express the 2-dimensional configurations in 𝒳_(𝒫,R) as the image under a 2-dimensional morphism (up to a shift) of a configuration in 𝒳_(𝒫|_W,R|_W) where 𝒫|_W is the induced partition and R|_W is the induced ℤ²-action on W. We focus on one example 𝒳_(𝒫₀,R₀) for which we obtain an eventually periodic sequence of 2-dimensional morphisms. We prove that it is the same as the substitutive structure of the minimal subshift X₀ of the Jeandel-Rao Wang shift computed in an earlier work by the author. As a consequence, 𝒫₀ is a Markov partition for the associated toral ℤ²-rotation R₀. It also implies that the subshift X₀ is uniquely ergodic and is isomorphic to the toral ℤ²-rotation R₀ which can be seen as a generalization for 2-dimensional subshifts of the relation between Sturmian sequences and irrational rotations on a circle. Batteries included: the algorithms and code to reproduce the proofs are provided.
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