Generalized interval exchanges and the 2–3 conjecture

Shmuel Friedland; Benjamin Weiss

Open Mathematics, Tome 3 (2005), p. 412-429 / Harvested from The Polish Digital Mathematics Library

Résumé

We introduce the notion of a generalized interval exchange $φ_{𝒜}$ induced by a measurable k-partition $𝒜 = \{A_{1}, . . ., A_{k}\}$ of [0,1). $φ_{𝒜}$ can be viewed as the corresponding restriction of a nondecreasing function $f_{𝒜}$ on ℝ with $f_{𝒜} (0) = 0, f_{𝒜} (k) = 1$ . A is called λ-dense if λ(A i∩(a, b))>0 for each i and any 0≤ a< b≤1. We show that the 2–3 Furstenberg conjecture is invalid if and only if there are 2 and 3 λ-dense partitions A and B of [0,1), such that $f_{𝒜} \circ f_{ℬ} = f_{ℬ} \circ f_{𝒜}$ . We give necessary and sufficient conditions for this equality to hold. We show that for each integer m≥2, such that 3∤2m+1, there exist 2 and 3 non λ-dense partitions A and B of [0,1), corresponding to the interval exchanges on 2m intervals, for which $f_{𝒜}$ and $f_{ℬ}$ commute.

Publié le : 2005-01-01

Zbl 1107.37004

EUDML-ID : urn:eudml:doc:268701

@article{bwmeta1.element.doi-10_2478_BF02475916,
     author = {Shmuel Friedland and Benjamin Weiss},
     title = {Generalized interval exchanges and the 2--3 conjecture},
     journal = {Open Mathematics},
     volume = {3},
     year = {2005},
     pages = {412-429},
     zbl = {1107.37004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02475916}
}

Shmuel Friedland; Benjamin Weiss. Generalized interval exchanges and the 2–3 conjecture. Open Mathematics, Tome 3 (2005) pp. 412-429. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02475916/

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