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

Abstract

top
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 ∘ f ℬ = f ℬ ∘ 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.

How to cite

top

Shmuel 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

top
  1. [1] P. Arnoux, D.S. Ornstein and B. Weiss: “Cutting and stacking, interval exchanges and geometric models”, Israel J. Math., Vol. 50, (1985), pp. 160–168. Zbl0558.58019
  2. [2] H. Furstenberg: “Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation”, Math. Sys. Theory, Vol. 1, (1967), pp. 1–49. http://dx.doi.org/10.1007/BF01692494 Zbl0146.28502
  3. [3] B. Host: “Nombres normaux, entropie, translations”, Israel J. Math., Vol. 91, (1995), pp. 419–428. 
  4. [4] A. Johnson and D.J. Rudolph: “Convergence under ×q of ×p invariant measures on the circle”, Adv. Math., Vol. 115, (1995), pp. 117–140. http://dx.doi.org/10.1006/aima.1995.1052 Zbl0934.28010
  5. [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. [6] W. Parry: “In general a degree two map is an automorphism”, Contemporary Math., Vol. 135, (1992), pp. 219–224. 
  7. [7] D. Rudolph: “×2 and ×3 invariant measures and entropy”, Ergodic Theory & Dynamical Systems, Vol. 10, (1990), pp. 395–406. Zbl0709.28013
  8. [8] Jean-Paul Thouvenot: private communication. 
  9. [9] P. Walters: An Introduction to Ergodic Theory, Springer-Verlag, 1982. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.