Clustering words

We characterize words which cluster under the Burrows-Wheeler transform as those words $w$ such that $ww$ occurs in a trajectory of an interval exchange transformation, and build examples of clustering words

Diagonal changes for every interval exchange transformation

International audienceWe give a geometric version of the induction algorithms defined in [10] and generalizing the self-dual induction of [17]. For all interval exchanges, whatever the permutation and the disposition of the discontinuities, we define diagonal changes which generalize those of [7]: they are exchange of unions of triangles on a set of triangulated polygons, which may be glued to cre- ate a translation surface. There are many possible algorithms depending on decisions at each step, and when the decision is fixed each diagonal change is a natural extension of the corresponding induction, which extends the result shown in [7] in the particular case of the hyperelliptic Rauzy class. Furthermore, for that class, we can define decisions such that we get an algorithm of diagonal changes which is a natural extension of the underlying algorithm of self-dual induction, and we can thus compute an invariant measure for the normalized induction. The diagonal changes allow us also to realize the self-duality of the induction in the hyperelliptic class, and to prove this does not hold outside that class

COMBINATORIAL METHODS FOR INTERVAL EXCHANGE TRANSFORMATIONS

International audienceThis is a survey on the big questions about interval exchanges (minimality, unique ergodicity, weak mixing, simplicity) with emphasis on how they can be tackled by mainly combina-torial methods. Interval exchange transformations, defined in Definition 1 below, constitute a famous class of dynamical systems; they were introduced by V. Oseledec [25], and have been extensively studied by many famous authors; up to now, the main results in this swifly-evolving field can be found in the two excellent courses [33] and [34]. To study interval exchanges, three kind of methods can be used: by definition, these systems are one-dimensional, and the first results on them naturally used one-dimensional techniques; then the strongest results on interval exchanges have been obtained by lifting the transformation to higher dimensions and using deep geometric methods. However, many of these results have been reproved by using zero-dimensional methods; these use the codings of orbits to replace the original dynamical system by a symbolic dynamical system, as in Definition 4 below. Now, most of the existing texts, including the two courses mentioned above, focus on the geometric methods; the present survey wants to emphasize what can be achieved by the two other kinds of methods, which have both a strong flavour of combinatorics. The one-dimensional methods yield the basic results, some of which the reader will find in Section 2 below, but also the famous Keane counterexamples described in Section 4, and a very nice new result of M. Bosher-nitzan which is the object of our Section 6; Sections 3 and 5 are devoted to the zero-dimensional methods; the necessary definitions of word combinatorics, symbolic and measurable dynamics are given in Section 1. All those sections are also retracing the colourful history of the theory of interval exchanges, made with big conjectures brilliantly solved after long waits; thus we finish the paper by explaining in Section 7 the last big open question in the domain. This paper stems from a course given during the summer school Dynamique en Cornouaille, which took place in Fouesnant in june 2011; the author is very grateful to the organizer, R. Lep-laideur, for having commandeered it

Billiards in regular 2n-gons and the self-dual induction

International audienceWe build a coding of the trajectories of billiards in regular 2n-gons, similar but different from the one in [16], by applying the self-dual induction [9] to the underlying one-parameter family of n-interval exchange transformations. This allows us to show that, in that family, for n = 3 non-periodicity is enough to guarantee weak mixing, and in some cases minimal self-joinings, and for every n we can build examples of n-interval exchange transformations with weak mixing, which are the first known explicitly for n > 6. In [16], see also [15], John Smillie and Corinna Ulcigrai develop a rich and original theory of billiards in the regular octagons, and more generally of billiards in the regular 2n-gons, first studied by Veech [17]: their aim is to build explicitly the symbolic trajectories, which generalize the famous Sturmian sequences (see for example [1] among a huge literature), and they achieve it through a new renormalization process which generalizes the usual continued fraction algorithm. In the present shorter paper, we show that similar results, with new consequences, can be obtained by using an existing, though recent, theory, the self-dual induction on interval exchange transformations. As in [16], we define a trajectory of a billiard in a regular 2n-gon as a path which starts in the interior of the polygon, and moves with constant velocity until it hits the boundary, then it re-enters the polygon at the corresponding point of the parallel side, and continues travelling with the same velocity; we label each pair of parallel sides with a letter of the alphabet (A 1 , ...A n), and read the labels of the pairs of parallel sides crossed by the trajectory as time increases; studying these trajectories is known to be equivalent to studying the trajectories of a one-parameter family of n-interval exchange transformations, and to this family we apply a slightly modified version of the self-dual induction defined in [9]. Now, the self-dual induction is in general not easy to manipulate, as its states are a family of graphs, and its typical itineraries, or paths in the so-called graph of graphs, are quite complicated to describe; but in our main Theorem 7 below, we show that for any non-periodic n-interval exchange in this particular family, after at most 2n − 2 steps our self-dual induction goes back, up to small modifications, to the initial state of another member of the family. This gives us a renormalization process, which differs from the one in [16] essentially because it is applied to lengths of intervals instead of angles, and allows us to compute the whole itinerary of the original interval exchange transformation under the self-dual induction in function of a single sequence of integers between 1 and 2n − 1, which act as the partial quotients of a continued fraction algorithm applied to initial lengths of subintervals

DYNAMICAL GENERALIZATIONS OF THE LAGRANGE SPECTRUM

International audienceWe compute two invariants of topological conjugacy, the upper and lower limits of the inverse of Boshernitzan's ne n , where e n is the smallest measure of a cylinder of length n, for three families of symbolic systems, the natural codings of rotations and three-interval exchanges and the Arnoux-Rauzy systems. The sets of values of these invariants for a given family of systems generalize the Lagrange spectrum, which is what we get for the family of rotations with the upper limit of 1 nen. The Lagrange spectrum is the set of finite values of L(α) for all irrational numbers α, where L(α) is the largest constant c such that |α − p q | ≤ 1 cq 2 for infinitely many integers p and q (a variant is known as the Markov spectrum, see Section 1.3 below). It was recently remarked that this arithmetic definition can be replaced by a dynamical definition involving the irrational rotations of angle α, through their natural coding by the partition {[0, 1 − α[, [1 − α, 1[}. Namely, as we prove in Theorem 2.4 below which was never written before, L(α) is also the upper limit of the inverse of the so-called Boshernitzan's ne n , where e n is the smallest (Lebesgue) measure of the nonempty cylinders of length n. Thus, for any symbolic dynamical system, it is interesting to compute two new invariants of topological conjugacy, lim sup n→+∞ 1 ne

An algorithm for the word entropy

For any infinite word $w$ on a finite alphabet $A$, the complexity function $p_w$ of $w$ is the sequence counting, for each non-negative $n$, the number $p_w(n)$ of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$ and the the entropy of $w$ is the quantity $E(w)=\lim\limits_{n\to\infty}\frac 1n\log p_w(n)$. For any given function $f$ with exponential growth, Mauduit and Moreira introduced in [MM17] the notion of word entropy $E_W(f) = \sup \{E(w), w \in A^{{\mathbb N}}, p_w \le f \}$ and showed its links with fractal dimensions of sets of infinite sequences with complexity function bounded by $f$. The goal of this work is to give an algorithm to estimate with arbitrary precision $E_W(f)$ from finitely many values of $f$

ON SARNAK'S CONJECTURE AND VEECH'S QUESTION FOR INTERVAL EXCHANGES

International audienceUsing a criterion due to Bourgain [10] and the generalization of the self-dual induction defined in [18], for each primitive permutation we build a large family of k-interval exchanges satisfying Sarnak's conjecture, and, for at least one permutation in each Rauzy class, smaller families for which we have weak mixing, which implies a prime number theorem, and simplicity in the sense of Veech

THREE COMPLEXITY FUNCTIONS

International audienceFor an extensive range of infinite words, and the associated symbolic dynamical systems, we compute, together with the usual language complexity function counting the finite words, the minimal and maximal complexity functions we get by replacing finite words by finite patterns, or words with holes. Given a language L on a finite alphabet A, the complexity function p L (n) counts for every n the number of factors of length n of L; this is a very useful notion, both inside word combinatorics and for the study of symbolic dynamical systems, see for example the survey [7]; of particular interest are the infinite words which are determined by the complexity of their language, those words for which p L (n) ≤ n for at least one n are ultimately periodic [15], while the Sturmian words, of complexity n + 1 for all n, are natural codings of rotations, see [6, 16], or Chapter 6 of [17], and Section 4 below. Note that the complexity is exponential when the language has positive topological entropy, and has not been widely used for that range of languages. To study further the combinatorial properties of infinite words, the notion of maximal pattern complexity, denoted by p

Eigenvalues and simplicity of interval exchange transformations

International audienceFor a class of d-interval exchange transformations, whichwe call the symmetric class,we deÞne a new self-dual induction process in which the system is successively induced on a union ofsub-intervals. This algorithm gives rise to an underlying graph structure which reßects the dynamicalbehavior of the system, through theRokhlin towers of the induced maps. We apply it to build a wide as-sortment of explicit examples on four intervals having different dynamical properties: these include theÞrst nontrivial examples with eigenvalues (rational or irrational), the Þrst ever example of an exchangeon more than three intervals satisfying VeechÕs simplicity (though this weakening of the notion of min-imal self-joinings was designed in 1982 to be satisÞed by interval exchange transformations), and anunexpected example which is non uniquely ergodic, weakly mixing for one invariant ergodic measurebut has rational eigenvalues for the other invariant ergodic measure