Möbius disjointness for interval exchange transformations on three intervals

Abstract

We show that Sarnak's conjecture on Mobius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.

The logarithmic Sarnak conjecture for ergodic weights

“…Note that the Sarnak conjecture for a system implies the logarithmic Sarnak conjecture for the same system. The Sarnak conjecture has been proved for a variety of systems, for example nilsystems [34], some horocycle flows [9] and more general zero entropy systems arising from homogeneous dynamics [59], certain distal systems, in particular some extensions of a rotation by a torus [47,51,70], a large class of rank one transformations [3,8,20], systems generated by various substitutions [1,14,19,55], all automatic sequences [57], some interval exchange transformations [8,12,20], some systems of number theoretic origin [7,30], and more... The survey article [18] contains an up to date list of relevant bibliography.…”

Sarnak’s Conjecture: What’s New

“…(33) holds. 35 If we have all ergodic measures giving discrete spectrum but we have too many ergodic measures then the argument above does not go through. Consider…”

“…also [90]. 35 We use here the standard result in the theory of unitary operators that mutual singularity of spectral measures implies orthogonality. Recall also the classical result in ergodic theory that spectral disjointness implies disjointness.…”

“…Generalizing these methods, it is shown in [63] that Möbius disjointness holds for examples of exchanges of k intervals for every k ě 2 and every Rauzy class, with a PNT in the weak mixing case. A breakthrough came with [35], for a large subclass of exchange of 3 intervals:…”

“…Theorem 5.10 ( [35]). For (Lebesgue)-almost all pα 1 , α 2 q, Sarnak's conjecture holds for exchanges of 3 intervals with permutation πi " 3´i and probability vector pα 1 , α 2 , 1´α 1´α2 q.…”