Generalized interval exchanges and the 2–3 conjecture

• Autorzy: Shmuel Friedland , Benjamin Weiss
• 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.

Słowa kluczowe

EN

Generalized interval exchange; entropy; 2–3 conjecture; 37A05; 37A35

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2005
• Tom: 3
• Numer: 3
• Strony: 412-429

Daty

wydano

2005-09-01

online

2005-09-01

Twórcy

autor

Shmuel Friedland

• University of Illinois at Chicago

autor

Benjamin Weiss

• Hebrew University

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.

Identyfikatory

DOI

10.2478/BF02475916