Non-ergodic interval exchange transformations

Keane, Michaël

doi:10.1007/bf03007668

Cited by 144 publications

(127 citation statements)

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“…A mainly combinatorial proof of this result, quite in the spirit of the present paper, can be deduced from [8] and [9]. When the solutions are non-uniquely ergodic, the analysis has been conducted in [22]: our induction path corresponds to any symmetric k-interval exchange whose vector of lengths lies in a convex set S, with extremal points [31]. Each 3 Induction in the symmetric and hyperelliptic cases 3.1 Induction using composite tree of relations.…”

Section: Proofmentioning

confidence: 96%

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Structure of K-interval exchange transformations: Induction, trajectories, and distance theorems

Ferenczi

Zamboni

2010

JAMA

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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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Section: Proofmentioning

confidence: 96%

“…• the first one is to imitate the reasoning of Theorem 6 of [22] to get that l 1,1 , l 2,1 , l 3,1 and r 1 have no rational relations except the normalization; • the other one is to take a n , b n and c n constant, with a 1 , b 1 and c 1 all different.…”

Section: A Family Of Examplesmentioning

confidence: 99%

Structure of K-interval exchange transformations: Induction, trajectories, and distance theorems

Ferenczi

Zamboni

2010

JAMA

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“…'s are minimal (this is guaranteed by a condition due to Keane [Ke1], which is automatically dealt with if the interval lengths are rationally independent) but note that ergodic properties of minimal i.e.m. 's can differ substantially from those of circle rotations: first they need not be uniquely ergodic [Ke2,KN,Co], and second, being ergodic they can be weakly mixing [KS,V3,V4]. On the other hand uniquely ergodic i.e.m.…”

Section: Introductionmentioning

confidence: 97%

The cohomological equation for Roth-type interval exchange maps

Marmi

Moussa

Yoccoz³

2005

J. Amer. Math. Soc.

124

235

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We exhibit an explicit class of minimal interval exchange maps (i.e.m.’s) T T for which the cohomological equation \[ Ψ − Ψ ∘ T = Φ \Psi -\Psi \circ T=\Phi \] has a bounded solution Ψ \Psi provided that the datum Φ \Phi belongs to a finite codimension subspace of the space of functions having on each interval a derivative of bounded variation. The proof is purely dynamical and is based on a renormalization argument and on Gottshalk-Hedlund’s theorem. If the datum is more regular the loss of differentiability in solving the cohomological equation will be the same. The class of interval exchange maps is characterized in terms of a diophantine condition of Roth type imposed to an acceleration of the Rauzy-Veech-Zorich continued fraction expansion associated to T T . More precisely one must impose a growth rate condition for the matrices appearing in the continued fraction algorithm together with a spectral gap condition (which guarantees unique ergodicity) and a coherence condition. We also prove that the set of Roth-type interval exchange maps has full measure. In the appendices we construct concrete examples of Roth-type i.e.m.’s and we show how the growth rate condition alone does not imply unique ergodicity.

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“…Though it uses a very heavy measure-theoretic and geometric machinery [2], we recall that if k ≥ 3, for every "nontrivial" permutation π and almost all (for the Lebesgue measure) probability vectors, k-interval exchange transformations are weakly mixing for one invariant measure, while the i.d.o.c. condition implies minimality [13], thus by Theorem 2.2 their maximal pattern complexity is k n . This last result could also be deduced from topological strong mixing for a residual set of 4-interval exchange transformations, see [4].…”

Section: Example 2: Doubled Sturmian Wordsmentioning

confidence: 92%