Interval exchange transformations

Keane, Michaël

doi:10.1007/bf01236981

Cited by 386 publications

(365 citation statements)

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“…Since µ = T µ and Supp(µ) ⊂F is equivalent to µ = ηµ, one is lead to studying the dynamics of η which appears to be locally affine and 1/2-contracting. This class of maps can be seen as a contracting variant of the family of Interval Exchange Transformations, namely the bijective piecewise translations of the Interval, introduced by Keane [15]. These considerations are the main reason for the present work.…”

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confidence: 99%

“…First, only branches starting from V N i, with (i, ) ∈ J can be weighted by a T −invariant Borel probability measure. Following an idea of Keane [15], let µ 1 , · · · , µ q be q T -invariant ergodic diffusive Borel probability measures. Let (A i ) 1≤i≤q be disjoint T −invariant Borel sets such that µ i (A j ) = δ i,j .…”

Section: Abstract Structure Of General Injective Quasi-contractionsmentioning

confidence: 99%

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Dynamics of injective quasi-contractions

Brémont¹

2005

Ergod. Th. Dynam. Sys.

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We study injective locally contracting maps of the Interval. After giving an upper-bound on the number of ergodic components, we show that generically finitely many periodic orbits attract the whole dynamics and that this picture is stable under perturbation. In relation with the problem of maximizing measures for regular maps, we next consider a class of probability measures on the Circle invariant by ×p generalizing the family of Sturm measures and show its generic periodic character. In a second half we detail the structure of order-preserving locally contracting maps on the Circle. The rotation number is shown to be generically rational and the transformations having a given rational rotation number are explicited. We also count the periodic attractors. We then deduce for a model with three pieces on the Interval the existence of measurable conjugacies with three-intervals exchange transformations in non-periodic cases.

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confidence: 99%

Section: Abstract Structure Of General Injective Quasi-contractionsmentioning

confidence: 99%

Dynamics of injective quasi-contractions

Brémont¹

2005

Ergod. Th. Dynam. Sys.

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“…Cantor minimal systems The notion of an interval exchange transformation was introduced by Keane [8]. For a more complete discussion of interval exchange transformations we refer to [2,Chapter 5].…”

Section: Minimal Interval Exchange Transformations and Their Associatedmentioning

confidence: 99%

Bratteli-Vershik models for Cantor minimal systems associated to interval exchange transformations

Gjerde¹,

Johansen²

2002

MATH. SCAND.

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“…In particular, from the article of Keane [11], one can deduce what the intervals of points with the same itinerary are. We summarize it as the following lemma.…”

Section: Itineraries In Exchange Of Three Intervalsmentioning

confidence: 99%