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2005 | 3 | 3 | 412-429
Tytuł artykułu

Generalized interval exchanges and the 2–3 conjecture

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Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
3
Numer
3
Strony
412-429
Opis fizyczny
Daty
wydano
2005-09-01
online
2005-09-01
Twórcy
Bibliografia
  • [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.
  • [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
  • [3] B. Host: “Nombres normaux, entropie, translations”, Israel J. Math., Vol. 91, (1995), pp. 419–428.
  • [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
  • [5] G. Margulis: “Problems and conjectures in rigidity theory”, In: Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 161–174.
  • [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.
  • [8] Jean-Paul Thouvenot: private communication.
  • [9] P. Walters: An Introduction to Ergodic Theory, Springer-Verlag, 1982.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_BF02475916
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