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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(Ai∩(a, b))&gt;0 for each i and any 0≤ a&lt; 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.

Generalized interval exchanges and the 2–3 conjecture