# 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
##### References:
 [1] M. Boshernitzan and A. Nogueira, Generalized eigenfunctions of interval exchange maps, Ergodic Theory Dynam. Sys., 24 (2004), 697-705. doi: 10.1017/S0143385704000021. [2] J. Bourgain, On the correlation of the Moebius function with rank-one systems, J. Anal. Math., 120 (2013), 105-130. doi: 10.1007/s11854-013-0016-z. [3] J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Moebius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., 28, Springer, New York, 2013, 67â€"83. doi: 10.1007/978-1-4614-4075-8_5. [4] H. Davenport, On some infinite series involving arithmetical functions (Ⅱ), Quart. J. of Math., 8 (1937), 313-320. doi: 10.1093/qmath/os-8.1.313. [5] E. H. El Abdalaoui, M. Lemańczyk and T. de la Rue, On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal., 266 (2014), 284-317. doi: 10.1016/j.jfa.2013.09.005. [6] S. Ferenczi and C. Mauduit, On Sarnak's conjecture and Veech's question for interval exchanges, J. Anal. Math., 134 (2018), 545-573. doi: 10.1007/s11854-018-0017-z. [7] B. Green and T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2), 175 (2012), 541-566. doi: 10.4007/annals.2012.175.2.3. [8] I. Katai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225. doi: 10.1007/BF01949145. [9] A. Khinchin, Continued Fractions, with a preface by B. V. Gnedenko, translated from the third (1961) Russian edition, reprint of the 1964 translation, Dover, 1997. [10] A. Harper, A different proof of a finite version of Vinogradov's bilinear sum inequality, online notes, 2011. [11] M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030. [12] D. Rudolph, Fundamentals of Measurable Dynamics. Ergodic Theory on Lebesgue Spaces, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990. [13] Z. Wang, Möbius disjointness for analytic skew products, Invent. Math., 209 (2017), 175-196. doi: 10.1007/s00222-016-0707-z.

show all references

##### References:
 [1] M. Boshernitzan and A. Nogueira, Generalized eigenfunctions of interval exchange maps, Ergodic Theory Dynam. Sys., 24 (2004), 697-705. doi: 10.1017/S0143385704000021. [2] J. Bourgain, On the correlation of the Moebius function with rank-one systems, J. Anal. Math., 120 (2013), 105-130. doi: 10.1007/s11854-013-0016-z. [3] J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Moebius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., 28, Springer, New York, 2013, 67â€"83. doi: 10.1007/978-1-4614-4075-8_5. [4] H. Davenport, On some infinite series involving arithmetical functions (Ⅱ), Quart. J. of Math., 8 (1937), 313-320. doi: 10.1093/qmath/os-8.1.313. [5] E. H. El Abdalaoui, M. Lemańczyk and T. de la Rue, On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal., 266 (2014), 284-317. doi: 10.1016/j.jfa.2013.09.005. [6] S. Ferenczi and C. Mauduit, On Sarnak's conjecture and Veech's question for interval exchanges, J. Anal. Math., 134 (2018), 545-573. doi: 10.1007/s11854-018-0017-z. [7] B. Green and T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2), 175 (2012), 541-566. doi: 10.4007/annals.2012.175.2.3. [8] I. Katai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225. doi: 10.1007/BF01949145. [9] A. Khinchin, Continued Fractions, with a preface by B. V. Gnedenko, translated from the third (1961) Russian edition, reprint of the 1964 translation, Dover, 1997. [10] A. Harper, A different proof of a finite version of Vinogradov's bilinear sum inequality, online notes, 2011. [11] M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030. [12] D. Rudolph, Fundamentals of Measurable Dynamics. Ergodic Theory on Lebesgue Spaces, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990. [13] Z. Wang, Möbius disjointness for analytic skew products, Invent. Math., 209 (2017), 175-196. doi: 10.1007/s00222-016-0707-z.
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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