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 and 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.

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[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.

[16]

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

[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.

[18]

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

[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.

[20]

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

[21]

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

[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.

[23]

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

[24]

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

[25]

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]

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.

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[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.

[16]

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

[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.

[18]

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

[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.

[20]

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

[21]

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

[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.

[23]

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

[24]

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

[25]

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

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