August  2021, 41(8): 3531-3553. doi: 10.3934/dcds.2021006

Möbius disjointness for skew products on a circle and a nilmanifold

1. 

CAS Wu Wen-Tsun Key Laboratory of Mathematics & Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

2. 

School of Mathematics & Data Science Institute, Shandong University, Jinan, Shandong 250100, China

* Corresponding author: Ke Wang

Received  September 2020 Published  August 2021 Early access  January 2021

Let $ \mathbb{T} $ be the unit circle and $ \Gamma \backslash G $ the $ 3 $-dimensional Heisenberg nilmanifold. We prove that a class of skew products on $ \mathbb{T} \times \Gamma \backslash G $ are distal, and that the Möbius function is linearly disjoint from these skew products. This verifies the Möbius Disjointness Conjecture of Sarnak.

Citation: Wen Huang, Jianya Liu, Ke Wang. Möbius disjointness for skew products on a circle and a nilmanifold. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3531-3553. doi: 10.3934/dcds.2021006
References:
[1]

J. Bourgain, On the correlation of the Möbius function with rank-one systems, J. Anal. Math., 120 (2013), 105-130.  doi: 10.1007/s11854-013-0016-z.  Google Scholar

[2]

J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Möbius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., vol. 28, Springer, New York, 2013, 67–83. doi: 10.1007/978-1-4614-4075-8_5.  Google Scholar

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H. Davenport, On some infinite series involving arithmetical functions, II, Quart. J. Math., 8 (1937), 313-350.  doi: 10.1093/qmath/os-8.1.313.  Google Scholar

[4]

A. de Faveri, Möbius disjiontness for $C^{1+\epsilon}$ skew products, preprint, arXiv: 2002.01076. Google Scholar

[5]

A.-H. Fan and Y. Jiang, Oscillating sequences, MMA and MMLS flows and Sarnak's conjecture, Ergodic Theory Dynam. Systems, 38 (2018), 1709-1744.  doi: 10.1017/etds.2016.121.  Google Scholar

[6]

S. Ferenczi, J. Kulaga-Przymus and M. Lemanczyk, Sarnak's conjecture: What's new, in Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics, Lecture Notes in Math., vol. 2213, Springer, Cham, 2018,163–235.  Google Scholar

[7]

H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601.  doi: 10.2307/2372899.  Google Scholar

[8]

H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.  doi: 10.2307/2373137.  Google Scholar

[9]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math., 175 (2012), 465-540.  doi: 10.4007/annals.2012.175.2.2.  Google Scholar

[10]

B. Green and T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math., 175 (2012), 541-566.  doi: 10.4007/annals.2012.175.2.3.  Google Scholar

[11]

L. K. Hua, Additive theory of prime numbers, Transl. Math. Monogr. 13, Amer. Math. Soc., Providence, 1965.  Google Scholar

[12]

W. HuangZ. Wang and X. Ye, Measure complexity and Möbius disjointness, Adv. Math., 347 (2019), 827-858.  doi: 10.1016/j.aim.2019.03.007.  Google Scholar

[13]

W. HuangZ. Wang and G. Zhang, Möbius disjointness for topological model of any ergodic system with discrete spectrum, J. Mod. Dyn., 14 (2019), 277-290.  doi: 10.3934/jmd.2019010.  Google Scholar

[14]

A. Kanigowski, M. Lemanczyk and M. Radziwill, Rigidity in dynamics and Möbius disjointness, preprint, arXiv: 1905.13256v2. Google Scholar

[15]

M. Litman and Z. Wang, Möbius disjointness for skew products on the Heisenberg nilmanifold, Proc. Amer. Math. Soc., 147 (2019), 2033-2043.  doi: 10.1090/proc/14259.  Google Scholar

[16]

J. Liu and P. Sarnak, The Möbius function and distal flows, Duke Math. J., 164 (2015), 1353-1399.  doi: 10.1215/00127094-2916213.  Google Scholar

[17]

J. Liu and P. Sarnak, The Möbius disjointness conjecture for distal flows, in Proceedings of the Sixth International Congress of Chinese Mathematician, Vol. I, Adv. Lect. Math. (ALM) 36, Int. Press, Somerville, MA, 2017, 327-335.  Google Scholar

[18]

A. I. Mal'cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949), 9-32.   Google Scholar

[19]

K. MatomäkiM. Radziwill and T. Tao, An averaged form of Chowla's conjecture, Algebra Number Theory, 9 (2015), 2167-2196.  doi: 10.2140/ant.2015.9.2167.  Google Scholar

[20]

W. Parry, Zero entropy of distal and related transformations, Topological Dynamics, (Symposium, Colorado State Univ., Ft. Collins, Colo., 1967), 383–389.  Google Scholar

[21]

R. Peckner, Möbius disjointness for homogeneous dynamics, Duke Math. J., 167 (2018), 2745-2792.  doi: 10.1215/00127094-2018-0026.  Google Scholar

[22]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, IAS Lecture Notes, 2009; https://publications.ias.edu/sites/default/files/MobiusFunctionsLectures(2).pdf. Google Scholar

[23]

P. Sarnak, Möbius randomness and dynamics, Not. S. Afr. Math. Soc., 43 (2012), 89-97.   Google Scholar

[24]

R. Tolimieri, Analysis on the Heisenberg manifold, Trans. Amer. Math. Soc., 288 (1977), 329-343.  doi: 10.2307/1998533.  Google Scholar

[25]

Z. Wang, Möbius disjointness for analytic skew products, Invent. Math., 209 (2017), 175-196.  doi: 10.1007/s00222-016-0707-z.  Google Scholar

show all references

References:
[1]

J. Bourgain, On the correlation of the Möbius function with rank-one systems, J. Anal. Math., 120 (2013), 105-130.  doi: 10.1007/s11854-013-0016-z.  Google Scholar

[2]

J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Möbius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., vol. 28, Springer, New York, 2013, 67–83. doi: 10.1007/978-1-4614-4075-8_5.  Google Scholar

[3]

H. Davenport, On some infinite series involving arithmetical functions, II, Quart. J. Math., 8 (1937), 313-350.  doi: 10.1093/qmath/os-8.1.313.  Google Scholar

[4]

A. de Faveri, Möbius disjiontness for $C^{1+\epsilon}$ skew products, preprint, arXiv: 2002.01076. Google Scholar

[5]

A.-H. Fan and Y. Jiang, Oscillating sequences, MMA and MMLS flows and Sarnak's conjecture, Ergodic Theory Dynam. Systems, 38 (2018), 1709-1744.  doi: 10.1017/etds.2016.121.  Google Scholar

[6]

S. Ferenczi, J. Kulaga-Przymus and M. Lemanczyk, Sarnak's conjecture: What's new, in Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics, Lecture Notes in Math., vol. 2213, Springer, Cham, 2018,163–235.  Google Scholar

[7]

H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601.  doi: 10.2307/2372899.  Google Scholar

[8]

H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.  doi: 10.2307/2373137.  Google Scholar

[9]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math., 175 (2012), 465-540.  doi: 10.4007/annals.2012.175.2.2.  Google Scholar

[10]

B. Green and T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math., 175 (2012), 541-566.  doi: 10.4007/annals.2012.175.2.3.  Google Scholar

[11]

L. K. Hua, Additive theory of prime numbers, Transl. Math. Monogr. 13, Amer. Math. Soc., Providence, 1965.  Google Scholar

[12]

W. HuangZ. Wang and X. Ye, Measure complexity and Möbius disjointness, Adv. Math., 347 (2019), 827-858.  doi: 10.1016/j.aim.2019.03.007.  Google Scholar

[13]

W. HuangZ. Wang and G. Zhang, Möbius disjointness for topological model of any ergodic system with discrete spectrum, J. Mod. Dyn., 14 (2019), 277-290.  doi: 10.3934/jmd.2019010.  Google Scholar

[14]

A. Kanigowski, M. Lemanczyk and M. Radziwill, Rigidity in dynamics and Möbius disjointness, preprint, arXiv: 1905.13256v2. Google Scholar

[15]

M. Litman and Z. Wang, Möbius disjointness for skew products on the Heisenberg nilmanifold, Proc. Amer. Math. Soc., 147 (2019), 2033-2043.  doi: 10.1090/proc/14259.  Google Scholar

[16]

J. Liu and P. Sarnak, The Möbius function and distal flows, Duke Math. J., 164 (2015), 1353-1399.  doi: 10.1215/00127094-2916213.  Google Scholar

[17]

J. Liu and P. Sarnak, The Möbius disjointness conjecture for distal flows, in Proceedings of the Sixth International Congress of Chinese Mathematician, Vol. I, Adv. Lect. Math. (ALM) 36, Int. Press, Somerville, MA, 2017, 327-335.  Google Scholar

[18]

A. I. Mal'cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949), 9-32.   Google Scholar

[19]

K. MatomäkiM. Radziwill and T. Tao, An averaged form of Chowla's conjecture, Algebra Number Theory, 9 (2015), 2167-2196.  doi: 10.2140/ant.2015.9.2167.  Google Scholar

[20]

W. Parry, Zero entropy of distal and related transformations, Topological Dynamics, (Symposium, Colorado State Univ., Ft. Collins, Colo., 1967), 383–389.  Google Scholar

[21]

R. Peckner, Möbius disjointness for homogeneous dynamics, Duke Math. J., 167 (2018), 2745-2792.  doi: 10.1215/00127094-2018-0026.  Google Scholar

[22]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, IAS Lecture Notes, 2009; https://publications.ias.edu/sites/default/files/MobiusFunctionsLectures(2).pdf. Google Scholar

[23]

P. Sarnak, Möbius randomness and dynamics, Not. S. Afr. Math. Soc., 43 (2012), 89-97.   Google Scholar

[24]

R. Tolimieri, Analysis on the Heisenberg manifold, Trans. Amer. Math. Soc., 288 (1977), 329-343.  doi: 10.2307/1998533.  Google Scholar

[25]

Z. Wang, Möbius disjointness for analytic skew products, Invent. Math., 209 (2017), 175-196.  doi: 10.1007/s00222-016-0707-z.  Google Scholar

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