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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo

IMPACT FACTOR 2018: 0.726
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ICV 2018: 152.31

Open Access
Online
ISSN
2391-5455
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Volume 3, Issue 3

# Generalized interval exchanges and the 2–3 conjecture

Shmuel Friedland
/ Benjamin Weiss
Published Online: 2005-09-01 | DOI: https://doi.org/10.2478/BF02475916

## Abstract

We introduce the notion of a generalized interval exchange $$\phi _\mathcal{A}$$ induced by a measurable k-partition $$\mathcal{A} = \left\{ {A_1 ,...,A_k } \right\}$$ of [0,1). $$\phi _\mathcal{A}$$ can be viewed as the corresponding restriction of a nondecreasing function $$f_\mathcal{A}$$ on ℝ with $$f_\mathcal{A} (0) = 0, f_\mathcal{A} (k) = 1$$ . 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 $$f_\mathcal{A} \circ f_\mathcal{B} = f_\mathcal{B} \circ f_\mathcal{A}$$ . 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 $$f_\mathcal{A}$$ and $$f_\mathcal{B}$$ commute.

Keywords: 37A05; 37A35

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Published Online: 2005-09-01

Published in Print: 2005-09-01

Citation Information: Open Mathematics, Volume 3, Issue 3, Pages 412–429, ISSN (Online) 2391-5455,

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