Figure 1.
The torus $\hat{X}$ . A vertical segment of length $q_k$ intersects a horizontal slit of length $z$
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The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces
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 |
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.
Mathematics Subject Classification:
Primary: 37E05; Secondary: 28D20.
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:
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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. |
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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. |
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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. |
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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. |
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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. Google Scholar |
| [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.
Google Scholar
|
| [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. Google Scholar |
| [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.
Google Scholar
|
| [13] |
Z. Wang,
Möbius disjointness for analytic skew products, Invent. Math., 209 (2017), 175-196.
doi: 10.1007/s00222-016-0707-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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