Möbius disjointness for interval exchange transformations on three intervals

Jon Chaika,Alex Eskin

doi:10.3934/jmd.2019003

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.

Highlights

Let μ : N → {−1, 0, 1} denote the Möbius function, namely, μ(n) = 0 if n is not square-free, μ(n) = 1 if n is square-free and has an even number of prime factors, and μ(n) = −1 if n is square-free and has an odd number of prime factors.Let X be a topological space, and let T : X → X be an invertible map
In Appendix B, we prove that for almost every 3-IET, T, T n is disjoint from T m for all 0 < n < m
Most closely related to this work is [4], where Vinogradov’s circle method is used to prove that every rotation (2IET) is disjoint from Möbius; [2] which shows a set of 3-IETs satisfying a certain measure 0 condition are disjoint from Möbius, [6] which builds families of d IETs for d > 3 that are disjoint from Möbius, and [3] where a slightly stronger version of our criterion is introduced to show that the time-1 map of horocycle flows are disjoint from Möbius

Summary

INTRODUCTION

Most closely related to this work is [4], where Vinogradov’s circle method is used to prove that every rotation (2IET) is disjoint from Möbius; [2] which shows a set of 3-IETs satisfying a certain measure 0 condition are disjoint from Möbius, [6] which builds families of d IETs for d > 3 that are disjoint from Möbius, and [3] where a slightly stronger version of our criterion is introduced to show that the time-1 map of horocycle flows are disjoint from Möbius Appendix B proves that almost every 3-IET has the property that all of its distinct positive powers are disjoint

DISJOINTNESS CRITERION

FACTS ABOUT ROTATIONS

APPLYING THE CRITERION

RENORMALIZATION