Möbius disjointness for interval exchange transformations on three intervals

Jon Chaika; Alex Eskin

doi:10.3934/jmd.2019003

Article Contents

2019, Volume 14: 55-86. Doi: 10.3934/jmd.2019003

This volume Previous Article The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces Next Article Equidistribution of saddle connections on translation surfaces

Möbius disjointness for interval exchange transformations on three intervals

Jon Chaika^1, and
Alex Eskin^2,

1.
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
2.
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA

Received: June 12, 2017

Revised: May 26, 2018

Published: March 2019

JC: Supported in part by NSF grants DMS-135500 and DMS-1452762 and the Sloan foundation.
AE: Supported in part by NSF grant DMS 1201422 and the Simons Foundation.

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Abstract

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

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Mathematics Subject Classification: Primary: 37E05; Secondary: 28D20.

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Figure 1. The torus $\hat{X}$. A vertical segment of length $q_k$ intersects a horizontal slit of length $z$

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Figure 2. The torus $g_{\log(q_k)} \hat{X}$: A vertical segment $\gamma_1$ of length $1$ (drawn in red) intersects a horizontal slit $\gamma_2$ of length $q^k z$ (drawn in blue)

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Figure 3. Closing the curves. We complete the vertical segment $\gamma_1$ to a closed curve $\hat{\gamma_1}$ by adding a horizontal segment $\zeta_1$ (drawn in green). Simularly, we close up the horizontal slit $\gamma_2$ to obtain a closed curve $\hat{\gamma}_2$ by adding in a horizontal segment $\zeta_2$ and a vertical segment $\zeta_2'$ (drawn in purple)

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