1989
DOI: 10.1017/s0143385700005150
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Almost all interval exchange transformations with flips are nonergodic

Abstract: Here we prove that almost all interval exchange transformations which reverse orientation, in at least one interval, have a periodic point where the derivative is − 1. Therefore they are periodic in an open neighborhood of the periodic point.

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Cited by 31 publications
(44 citation statements)
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“…The proof is based on the argument presented in : first, we show that the exponential tail of the roof function implies that the corresponding Markov map is fast decaying in a sense of and then using check that this property implies the estimation we are interested in. Section 8 completes the proof of Theorem : we show the lower bound applying a construction related to the one described in Nogueira in . Using the same idea, we prove Theorem and Proposition .…”
Section: Introductionmentioning
confidence: 91%
See 1 more Smart Citation
“…The proof is based on the argument presented in : first, we show that the exponential tail of the roof function implies that the corresponding Markov map is fast decaying in a sense of and then using check that this property implies the estimation we are interested in. Section 8 completes the proof of Theorem : we show the lower bound applying a construction related to the one described in Nogueira in . Using the same idea, we prove Theorem and Proposition .…”
Section: Introductionmentioning
confidence: 91%
“…Sometimes, since we also work with the signed permutation trueπ̂, we can specify that α or α was the winner or the loser. Remark We will iterate the Rauzy induction map, however Rauzy induction is not defined everywhere, thus the iteration stops if we arrive to a point outside its domain of definition (see for an example when it stops). In case of an IET (without flips) which satisfies Keane's condition this never happens.…”
Section: Rauzy Inductionmentioning
confidence: 99%
“…We shall need the Rauzy induction [16,15,9] to obtain a minimal, self-similar IET whose associated matrix satisfies all the hypotheses of Theorem 1.…”
Section: The Interval Exchange Transformation Ementioning
confidence: 99%
“…This is the basic argument that drives the proof that almost every interval exchange map with flips is non-ergodic in [6]. Namely, the fact that the map acts locally as an isometry implies that, if we take d := min k=1,...,p {dist(T k · x 0 , D)}, where p is the period of x 0 and D is the discontinuity set, then the ball B d (x 0 ) follows the orbit of x 0 at all iterations and, therefore, the Lebesgue measure supported on the union of the iterates of B d (x 0 ) is an invariant measure, hence preventing any invariant non-atomic Borel measure from being ergodic.…”
Section: Introductionmentioning
confidence: 98%