Generalized interval exchanges and the 2–3 conjecture

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.