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We investigate the relation between the complexity function of a sequence, that is the number p(n) of its factors of length n, and the rank of the associated dynamical system, that is the number of Rokhlin towers required to approximate it. We prove that if the rank is one, then lim inf n→+∞ p(n) n 2 ≤ 1 2 , but give examples with lim sup n→+∞ p(n) G(n) = 1 for any prescribed function G with G(n) = o(a n ) for every a > 1. We give exact computations for examples of the "staircase" type, which are strongly mixing systems with quadratic complexity. Conversely, for minimal sequences, if p(n) < an + b for some a ≥ 1, the rank is at most 2 [a], with bounded strings of spacers, and the system is generated by a finite number of substitutions.Given a dynamical system, there are several notions indicating that it has a simple structure. One is the notion of rank, defined in [ORN-RUD-WEI] to formalize some constructions initiated by [CHA]; it is purely measuretheoretic, but leads to symbolic constructions, with systems defined by sequences on a finite alphabet. Another one, which may be tracked back to , is the combinatorial notion of complexity function, of languages or sequences; since the famous works of Hedlund and Morse, it is 1

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Abstract. This paper gives a complete characterization of those sequences of subword complexity (k − 1)n + 1 which are natural codings of orbits of k-interval exchange transformations, thereby answering an old question of Rauzy.

An overview of last seven years results concerning Sarnak's conjecture on Möbius disjointness is presented, focusing on ergodic theory aspects of the conjecture.1 Most often, however not always, T will be a homeomorphism. 2 µ stands for the arithmetic Möbius function, see next sections for explanations of notions that appear in Introduction.3 To be compared with Möbius Randomness Law by Iwaniec and Kowalski [95], page 338, that any "reasonable" sequence of complex numbers is orthogonal to µ.

Abstract. We define a new induction algorithm for k-interval exchange transformations associated to the "symmetric" permutation i → k − i + 1. Acting as a multi-dimensional continued fraction algorithm, it defines a sequence of generalized partial quotients given by an infinite path in a graph whose vertices, or states, are certain trees we call trees of relations. This induction is self-dual for the duality between the usual Rauzy induction and the da Rocha induction. We use it to describe those words obtained by coding orbits of points under a symmetric interval exchange, in terms of the generalized partial quotients associated with the vector of lengths of the k intervals. As a consequence, we improve a bound of Boshernitzan in a generalization of the three-distances theorem for rotations. However, a variant of our algorithm, applied to a class of interval exchange transformations with a different permutation, shows that the former bound is optimal outside the hyperelliptic class of permutations. PreliminariesInterval exchange transformations were originally introduced by Oseledec [27], following an idea of Arnold [2]; see also Katok and Stepin [20]. An exchange of k intervals, denoted throughout this paper by I, is given by a probability vector of k lengths (α 1 , . . . , α k ) together with a permutation π on k letters. The unit interval is partitioned into k subintervals of lengths α 1 , . . . , α k which are rearranged by I according to π. It was Rauzy [29] who first saw interval exchange transformations as a possible framework for generalizing the well-known interaction between circle rotations on one hand and Sturmian sequences on the other via the continued fraction algorithm (see, for example, the survey [7] [4]. But, in contrast with the Sturmian case, though this induction gives access to the symbolic dynamics of the trajectories, it is difficult to describe them explicitly and indeed the Rauzy induction was never actually used for that purpose. A partial description of the trajectories was given in [24], and it aroused interest by defining a different kind of induction; this da Rocha induction was then considered to be dual to the Rauzy induction: the Rauzy induction builds the points where the orbits of the discontinuities of I approximate 0, while the da Rocha induction builds the points where the orbit of 0 approximates the discontinuities of I −1 . Note that (non-constructive) combinatorial characterizations of the language of trajectories were given recently in [16] and [5].These inductions are also a fundamental tool in the study of the space of moduli of Riemann surfaces, and the various strata of its unit tangent bundle, through what has been a basic object of interest for the last 25 years, the Teichmüller flow on a stratum; in the case of interval exchanges, we consider the Teichmüller flow for the Riemann surface obtained by glueing parallel opposite sides of a polygon. The Rauzy induction chooses an initial segment of a horizontal separatrix and follows its vertical separatrix till it i...

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