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We look at d-point extensions of a rotation of angle α with r marked points, generalizing the examples of Veech 1969 and Sataev 1975, together with the square-tiled interval exchanges of [4]. We give conditions for minimality, solving the problem of minimality for Veech 1969, and show that minimality implies unique ergodicity when α has bounded partial quotients. Then we study the property of rigidity, in function of the Ostrowski expansions of the marked points by α: the most interesting case is when α has bounded partial quotients but the natural coding of the rotation with marked points is not linearly recurrent; it is only partially solved but allows us to build the first examples of non linearly recurrent and non rigid interval exchanges. In a famous paper of 1969 [9], much ahead of its time, W.A. Veech defines an extension of a rotation of angle α to two copies of the torus with a marked point β, the change of copy occurring on the interval [0, β[ (resp. [β, 1[ on a variant, thus there are two types of Veech 1969 systems, see Definition 2.1 below): for particular α with big partial quotients, these provide examples of minimal non uniquely ergodic interval exchanges. These were defined again independently, in a generalized way, by E.A. Sataev in 1975, in a beautiful but not very well known paper [8]: by taking r marked points and r + 1 copies of the torus, he defines minimal interval exchanges with a prescribed number of ergodic invariant measures. A more geometric model of Veech 1969 was given later by H. Masur and J. Smillie, where the transformation appears as a first return map of a directional flow on a surface made with two tori glued along one edge, see Lemma 2.1 below. In the present paper, we study slightly more general systems, by marking r points and taking d copies of the torus, for any r ≥ 1, d ≥ 2; also, though in general our marked points are not in Z(α), we allow one of them to be 1 − α, so that our systems generalize also the square-tiled interval exchanges we define in [4]. The geometric model generalizes also, to d glued tori. We study first the minimality of Veech 1969: it is proved in [9] that if β is not in Z(α), T is minimal; for the remaining cases, Veech could prove only (p. 6 of [9]) that if α and β are irrational, at least one of the two types of Veech 1969 defined by α and β is minimal. As far as we know, this result has not been improved in the last fifty years; we can now give a rather unexpected necessary and sufficient condition which implies it, see Theorem 3.1 below. In the general case, we give a sufficient condition for minimality, and prove, in contrast with Veech's cases, that whenever α has bounded partial quotients T is uniquely ergodic. We turn now to the measure-theoretic property of rigidity, meaning that for some sequence q n the q n-th powers of the transformation converge to the identity (Definition 1.7 below). Experimentally, in the class of interval exchanges, the absence of rigidity is difficult to achieve (indeed, by Veech [10] it is true only for a set of measure zero of parameters) and all known examples satisfy also the word-combinatorial property of linear recurrence (Definition 1.6 below) for their natural coding.
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hal-02120157 , version 1 (05-05-2019)


  • HAL Id : hal-02120157 , version 1


Sébastien Ferenczi, Pascal Hubert. GENERALIZED VEECH 1969 AND SATAEV 1975 EXTENSIONS OF ROTATIONS. 2019. ⟨hal-02120157⟩
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