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S-adic Sequences: A Bridge Between Dynamics, Arithmetic, and Geometry

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Substitution and Tiling Dynamics: Introduction to Self-inducing Structures

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2273))

Abstract

A Sturmian sequence is an infinite nonperiodic string over two letters with minimal subword complexity. In two papers, the first written by Morse and Hedlund in 1940 and the second by Coven and Hedlund in 1973, a surprising correspondence was established between Sturmian sequences on one side and rotations by an irrational number on the unit circle on the other. In 1991 Arnoux and Rauzy observed that an induction process (invented by Rauzy in the late 1970s), related with the classical continued fraction algorithm, can be used to give a very elegant proof of this correspondence. This process, known as the Rauzy induction, extends naturally to interval exchange transformations (this is the setting in which it was first formalized). It has been conjectured since the early 1990s that these correspondences carry over to rotations on higher dimensional tori, generalized continued fraction algorithms, and so-called S-adic sequences generated by substitutions. The idea of working towards such a generalization is known as Rauzy’s program. Recently Berthé, Steiner, and Thuswaldner made some progress on Rauzy’s program and were indeed able to set up the conjectured generalization of the above correspondences. Using a generalization of Rauzy’s induction process in which generalized continued fraction algorithms show up, they proved that under certain natural conditions an S-adic sequence gives rise to a dynamical system which is measurably conjugate to a rotation on a higher dimensional torus. Moreover, they established a metric theory which shows that counterexamples like the one constructed in 2000 by Cassaigne, Ferenczi, and Zamboni are rare. It is the aim of the present chapter to survey all these ideas and results.

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Notes

  1. 1.

    This implies that u and v have Σx and Σy in their orbit where x and y are the two limit sequences of σ. This interesting fact, which is not needed in this proof, should be proved by the reader.

  2. 2.

    We could also have started with writing out 1 and going to the right from the origin. This would have produced the second limit sequence of (σ) which coincides with w save for the first two letters.

  3. 3.

    A sequence is called linearly recurrent if there is a constant K such that each of its factors u occurs infinitely often in the sequence with gaps bounded by K|u|.

  4. 4.

    A sequence is called uniformly recurrent if each of its factors occurs infinitely often in the sequence with bounded gaps.

  5. 5.

    If we impose the additional property of primitivity on the coding sequence of a sequence \(w\in \mathcal {A}^{\mathbb {N}}\) it turns out that X w depends only on the directive sequence σ defining the S-adic sequence w and we have X σ = X w. This will be worked out precisely in Sect. 3.5.

  6. 6.

    Only the first letter a n in the argument of σ [0,n) is relevant for the limit. However, since we use the topology on \(\mathcal {A}^{\mathbb {N}}\) and \(\sigma _{[0,n)}(a_{n})\not \in \mathcal {A}^{\mathbb {N}}\) we have to write σ [0,n)(a na n…).

  7. 7.

    In [111] a space of tilings is equipped with a topology by saying that two tilings are close to each other if their tiles are close to each other in Hausdorff metric inside a large ball around the origin. Although \(\mathcal {C}_{\mathbf {v}}\) and \(\mathcal {C}^{(n_k)}_{\mathbf {v}}\) are no tilings, an analogous topology can be used here: \(\mathcal {C}_{\mathbf {v}}\) and are said to be close to each other if Γ(v) and \(\varGamma ({\mathbf {v}}^{(n_k)})-{\mathbf {y}}_k\) coincide inside a large ball B around the origin and the tiles associated to an element of [y, i] ∈ Γ(v) ∩ B in each of these two collections are close to each other in Hausdorff metric.

  8. 8.

    We note that in [48] the dual \(E_1^*(\sigma )\) is defined using suffixes of the images of σ instead of prefixes. Nevertheless, this difference does not change the behavior of \(E_1^*(\sigma )\) significantly and in Fig. 3.19 we get the same image as the authors obtained in [48, Figure 1].

  9. 9.

    Note that there are two kinds of shifts: the one just defined acts on the sequence of substitutions \(S^{\mathbb {N}}\), the other one (the S-adic shift) acts on the set of sequences X σ which is defined in terms of a single sequence of substitutions \({\boldsymbol {\sigma }}\in S^{\mathbb {N}}\). It should cause no confusion that both of these shift mappings are denoted by Σ.

  10. 10.

    Not to be confused with the level n subtiles introduced in Sect. 3.7.1.

  11. 11.

    Here ∥⋅∥2 is the operator norm w.r.t. the Euclidean norm on \(\mathbb {R}^d\).

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Acknowledgements

I warmly thank Shigeki Akiyama and Pierre Arnoux for inviting me to contribute to this volume. I very much appreciate their constant encouragement and support, and the many discussions I had with them. Moreover, I am indebted to Valérie Berthé, Sébastien Labbé, and Wolfgang Steiner for their suggestions.

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  1. Department Mathematics and Information Technology, University of Leoben, Leoben, Austria

    Jörg M. Thuswaldner

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  1. Institute of Mathematics, University of Tsukuba, Tsukuba, Japan

    Shigeki Akiyama

  2. Institut de Mathématiques de Marseille (I2M), Aix-Marseille University, Marseille Cedex 09, France

    Pierre Arnoux

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Thuswaldner, J.M. (2020). S-adic Sequences: A Bridge Between Dynamics, Arithmetic, and Geometry. In: Akiyama, S., Arnoux, P. (eds) Substitution and Tiling Dynamics: Introduction to Self-inducing Structures. Lecture Notes in Mathematics, vol 2273. Springer, Cham. https://doi.org/10.1007/978-3-030-57666-0_3

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