# Generalized interval exchanges and the 2–3 conjecture

Shmuel Friedland; Benjamin Weiss

Open Mathematics (2005)

- Volume: 3, Issue: 3, page 412-429
- ISSN: 2391-5455

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topShmuel Friedland, and Benjamin Weiss. "Generalized interval exchanges and the 2–3 conjecture." Open Mathematics 3.3 (2005): 412-429. <http://eudml.org/doc/268701>.

@article{ShmuelFriedland2005,

abstract = {We introduce the notion of a generalized interval exchange \[\phi \_\mathcal \{A\} \]
induced by a measurable k-partition \[\mathcal \{A\} = \left\lbrace \{A\_1 ,...,A\_k \} \right\rbrace \]
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.},

author = {Shmuel Friedland, Benjamin Weiss},

journal = {Open Mathematics},

keywords = {37A05; 37A35},

language = {eng},

number = {3},

pages = {412-429},

title = {Generalized interval exchanges and the 2–3 conjecture},

url = {http://eudml.org/doc/268701},

volume = {3},

year = {2005},

}

TY - JOUR

AU - Shmuel Friedland

AU - Benjamin Weiss

TI - Generalized interval exchanges and the 2–3 conjecture

JO - Open Mathematics

PY - 2005

VL - 3

IS - 3

SP - 412

EP - 429

AB - We introduce the notion of a generalized interval exchange \[\phi _\mathcal {A} \]
induced by a measurable k-partition \[\mathcal {A} = \left\lbrace {A_1 ,...,A_k } \right\rbrace \]
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.

LA - eng

KW - 37A05; 37A35

UR - http://eudml.org/doc/268701

ER -

## References

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- [5] G. Margulis: “Problems and conjectures in rigidity theory”, In: Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 161–174. Zbl0952.22005
- [6] W. Parry: “In general a degree two map is an automorphism”, Contemporary Math., Vol. 135, (1992), pp. 219–224.
- [7] D. Rudolph: “×2 and ×3 invariant measures and entropy”, Ergodic Theory & Dynamical Systems, Vol. 10, (1990), pp. 395–406. Zbl0709.28013
- [8] Jean-Paul Thouvenot: private communication.
- [9] P. Walters: An Introduction to Ergodic Theory, Springer-Verlag, 1982.

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