# American Institute of Mathematical Sciences

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

## Möbius disjointness for interval exchange transformations on three intervals

 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 13, 2017 Revised  May 27, 2018 Published  March 2019

Fund Project: 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.

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.

Citation: Jon Chaika, Alex Eskin. Möbius disjointness for interval exchange transformations on three intervals. Journal of Modern Dynamics, 2019, 14: 55-86. doi: 10.3934/jmd.2019003
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##### References:
The torus $\hat{X}$. A vertical segment of length $q_k$ intersects a horizontal slit of length $z$
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)
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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