Posted by: matheuscmss | February 6, 2019

Examples of Rauzy classes (after Yoccoz)

This week I attended the mini-conference Autour des surfaces de translation organized by Corentin Boissy and Slavyana Geninska at Toulouse.

One of the main objectives of this meeting was to discuss in details a somewhat long (66 pages) text by Jean-Christophe Yoccoz containing new notions and tools allowing to efficiently describe certain combinatorial objects known as Rauzy diagrams.

In fact, this text was still in preliminary format when Jean-Christophe passed away and, for this reason, Corentin and I spend a certain time discussing the insertion of footnotes in order to clarify several portions of Jean-Christophe’s text. After Corentin and I found that the text was finally “accessible” (to anyone with a certain familiarity with Jean-Christophe’s survey here, say), it was decided that we should “celebrate” the occasion with a meeting around this matter.

In any case, one of the outcomes of the mini-conference is that Jean-Christophe’s text entitled Examples of Rauzy classes with footnotes by Corentin is finally publicly available here.

In a nutshell, the first part of Jean-Christophe’s text is devoted to the notions of heightbi-monotonous cycles, etc., allowing to explore a given Rauzy diagram starting from a certain subgraph whose vertices consist of the so-called standard permutations; then, the second part of Jean-Christophe’s text is a sort of “proof of concept” where several Rauzy diagrams are described (including some containing several thousands of vertices). Here, it is worth to notice that he did the corresponding calculations by hand (mostly during winter vacations at Loctudy as he told me)! Corentin wrote a few Sage programs to double check some of these calculations and, as expected, they turned out to be correct.

Closing this short post, let me try to explain below some of Jean-Christophe’s motivations to get a systematic description of Rauzy diagrams.

First of all, recall that the study of the dynamics of interval exchange transformations and translation flows often relies on a renormalization scheme (“continued fraction algorithm”) called Rauzy–Veech induction: for a detailed exposition of this topic, the reader can consult Yoccoz’s survey here.

Roughly speaking, the Rauzy–Veech induction serves to encode the renormalization of interval exchange transformations and translation flows via topological Markov shifts induced by Rauzy diagrams: more concretely, a Rauzy diagram is a special type of oriented graph ${\mathcal{D}}$ and the dynamics of the renormalization procedure is described by the topological Markov shift consisting of the shift dynamics ${(e_n)_{n\in\mathbb{Z}}\mapsto (e_{n+1})_{n\in\mathbb{Z}}}$ on the space of bi-infinite paths (i.e., concatenations of edges) on ${\mathcal{D}}$.

In general, Rauzy diagrams are defined as follows. We take an abstract finite alphabet ${\mathcal{A}}$ on ${d\geq 2}$ letters. A permutation ${\pi=(\pi_t, \pi_b)}$ is a pair of bijections ${\pi_t,\pi_b:\mathcal{A}\rightarrow\{1,\dots,d\}}$ (normally we would like to say that ${\pi_b\circ\pi_t^{-1}}$ is a permutation of ${\{1,\dots, d\}}$, but the data ${\pi=(\pi_t,\pi_b)}$ provides a more “symmetric” way to describe permutations). In the literature, ${\pi}$ is often denoted as a list of the form

$\displaystyle \pi=\left(\begin{array}{ccc} \pi_t^{-1}(1) & \dots & \pi_t^{-1}(d) \\ \pi_b^{-1}(1) & \dots & \pi_b^{-1}(d) \end{array}\right)$

and the first, resp. last letter of the top and bottom rows are denoted ${_{t}\alpha=\pi_t^{-1}(1)}$ and ${_{b}\alpha = \pi_b^{-1}(1)}$, resp. ${\alpha_{t}=\pi_t^{-1}(d)}$ and ${\alpha_{b}= \pi_b^{-1}(d)}$.

The top operation ${\mathcal{R}_t}$ maps a permutation ${\pi=(\pi_t,\pi_b)}$ to ${\mathcal{R}_t(\pi)=(\pi_t,\pi_b')}$ where ${\pi_b'}$ is obtained from ${\pi_b}$ by performing a cyclic permutation of the letters appearing after ${\alpha_t}$ on the bottom row of ${\pi}$. Similarly, one can define the bottom operation ${\mathcal{R}_b}$ by symmetry (i.e., essentially by exchanging the roles of top and bottom rows). In this setting, a Rauzy diagram ${\mathcal{D}}$ is the oriented graph whose vertices correspond to the orbit of a given permutation ${\pi}$ under the top and bottom operations, and whose oriented edges have the form ${\kappa\rightarrow\mathcal{R}_t(\kappa)}$ and ${\kappa\rightarrow\mathcal{R}_b(\kappa)}$.

Exercise 1 Draw the three Rauzy diagrams associated to the following three permutations: ${\left(\begin{array}{cc} A & B \\ B & A \end{array}\right)}$, ${\left(\begin{array}{ccc} A & B & C \\ C & B & A \end{array}\right)}$, ${\left(\begin{array}{cccc} A & B & C & D \\ D & C & B & A \end{array}\right)}$.

Among many other results in this topic, our recent work with Avila and Yoccoz on the partial solution of the so-called Zorich conjecture (previously discussed in this post here) relies upon the precise knowledge of the geometry of Rauzy diagrams.

For this reason, right after partially solving Zorich’s conjecture, Jean-Christophe started his detailed study of arbitrary Rauzy diagrams in hope to solve Zorich’s conjecture in full generality.

As it turns out, Zorich’s conjecture was recently solved in full generality by Rodolfo Gutiérrez-Romo while bypassing many fine aspects of Rauzy diagrams (see the original article here and/or this post here), but it is clear that Jean-Christophe’s text on Rauzy diagrams will pave a way for further applications of the fascinating combinatorial objects.

Responses

1. Thanks for this expository entry. Is there any similar combinatorial survey for Rauzy classes of generalized permutations?

Felipe

• Dear Felipe,

I’m not aware of a similar survey in the case of generalized permutations / quadratic differentials. In particular, I think that one of the best options to learn about this topic is still to consult the original article [C. Boissy, E. Lanneau, Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials. Ergodic Theory Dynam. Systems 29 (2009), no. 3, 767–816].

Best regards,

Matheus

• Ok, thanks!

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