Natural coding of minimal rotations of the torus, induction and exduction

I2M, Aix-Marseille Université

Date(s) : 28/09/2021   iCal
11 h 00 min - 12 h 00 min

The notion of natural coding of rotation, sometimes better called natural coding of translation of the torus, goes back to the works of Morse and Hedlund [MH40], and to the seed paper of Rauzy [Rau82] (study of the Tribonacci word) in dimension 2. Nonetheless, the terminology appears later (for instance in [CFZ01] and [Fog02]).
Roughly speaking, a ”natural” coding of rotation denotes a word obtained as the coding trajectory of a point of the torus, under the action of a rotation, with respect to a remarkable partition that can be covered such that the induced rotation on the associated fundamental domain coincides, on each covered piece, with a translation.
Although much studied, this notion has never been well defined; in particular, there is no consensus on the topological/metrical conditions that should satisfy the pieces of the partition – without such conditions, the definition is trivial.
The aim of this talk is to introduce a new topological definition, with a minimal set of hypotheses.
In particular, this definition should allow the study of words with infinite imbalance (like those constructed in [CFZ01], [DHS13] or in my thesis).

Short references:
[CFZ00] Cassaigne, Ferenczi, Zamboni. “Imbalances in Arnoux-Rauzy sequences” (2000)
[DHS13] Delecroix, Hejda, Steiner. “Balancedness of Arnoux-Rauzy and Brun words” (2013)
[Fog02] Pytheas Fogg. “Substitutions in dynamics, arithmetics and combinatorics” (book, 2002)
[MH40] Morse and Hedlund. “Symbolic dynamics ii. Sturmian trajectories” (1940)
[Rau82] Rauzy. “Nombres algébriques et substitutions” (1982)

Campus de Luminy, Marseille


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