Characterisation of types of order satisfying $α_0+α_1 = α_1+α_0$

N. Aronszajn

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Generalized interval exchanges and the 2–3 conjecture

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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 $f_{𝒜} \circ f_{ℬ} = f_{ℬ} \circ f_{𝒜}$ . We give necessary and sufficient conditions for this equality to hold. We...