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 AbdalaouiM. 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 AbdalaouiM. 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.

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