# American Institute of Mathematical Sciences

2019, 14: 277-290. doi: 10.3934/jmd.2019010

## Möbius disjointness for topological models of ergodic systems with discrete spectrum

 1 School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China 2 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA 3 School of Mathematical Sciences and Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  December 14, 2017 Revised  June 08, 2018 Published  March 2019

Fund Project: WH: Supported by NSFC (11431012 and 11731003).
ZW: Supported by NSF (DMS-1451247 and DMS-1501095).
GZ: Supported by NSFC (11671094, 11722103 and 11731003).

We provide a criterion for a point satisfying the required disjointness condition in Sarnak's Möbius Disjointness Conjecture. As a direct application, we have that the conjecture holds for any topological model of an ergodic system with discrete spectrum.

Citation: Wen Huang, Zhiren Wang, Guohua Zhang. Möbius disjointness for topological models of ergodic systems with discrete spectrum. Journal of Modern Dynamics, 2019, 14: 277-290. doi: 10.3934/jmd.2019010
##### References:
 [1] J. Auslander, Mean-$L$-stable systems, Illinois J. Math., 3 (1959), 566–579.  Google Scholar [2] H. Davenport, On some infinite series involving arithmetical functions Ⅱ, Quat. J. Math., 8 (1937), 313–320. Google Scholar [3] T. Downarowicz and E. Glasner, Isomorphic extensions and applications, Topological Methods in Nonlinear Analysis, 48 (2016), 321–338. doi: 10.12775/TMNA.2016.050.  Google Scholar [4] T. Downarowicz and S. Kasjan, Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture, Studia Math., 229 (2015), 45–72.  Google Scholar [5] E. H. El Abdalaoui, S. Kasjan and M. Lemańczyk, 0-1 sequences of the Thue-Morse type and Sarnak's conjecture, Proc. Amer. Math. Soc., 144 (2016), 161–176. doi: 10.1090/proc/12683.  Google Scholar [6] E. H. El Abdalaoui, J. Kułaga-Przymus, M. Lemańczyk and T. de la Rue, Möbius disjointness for models of an ergodic system and beyond, Israel J. Math., 228 (2018), 707-751.  doi: 10.1007/s11856-018-1784-z.  Google Scholar [7] E. H. El Abdalaoui, M. Lemańczyk and T. de la Rue, Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals, International Mathematics Research Notices, 14 (2017), 4350–4368. doi: 10.1093/imrn/rnw146.  Google Scholar [8] A. Fan and Y. Jiang, Oscillating sequences, minimal mean attractability and minimal mean-Lyapunov-stability, Ergodic Theory and Dynamical Systems, 2017, to appear. Google Scholar [9] S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk and C. Mauduit, Substitutions and Möbius disjointness, in Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby, Contemp. Math., 678, Amer. Math. Soc., Providence, RI, 2016, 151–173.  Google Scholar [10] L. Flaminio, K. Frączek, J. Kułaga-Przymus and M. Lemańczyk, Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds, Studia Math., 244 (2019), 43-97.  doi: 10.4064/sm170512-25-9.  Google Scholar [11] S. Fomin, On dynamical systems with a purely point spectrum}, Russian, Doklady Akad. Nauk SSSR (N.S.), 77 (1951), 29–32.  Google Scholar [12] 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.  Google Scholar [13] P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. Ⅱ, Ann. of Math. (2), 43 (1942), 332–350. doi: 10.2307/1968872.  Google Scholar [14] W. Huang, Z. Lian, S. Shao and X. Ye, Sequences from zero entropy noncommutative toral automorphisms and Sarnak Conjecture, J. Differential Equations, 263 (2017), 779–810. doi: 10.1016/j.jde.2017.02.046.  Google Scholar [15] H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004. doi: 10.1090/coll/053.  Google Scholar [16] E. Lehrer, Topological mixing and uniquely ergodic systems, Israel J. Math., 57 (1987), 239–255. doi: 10.1007/BF02772176.  Google Scholar [17] J. Li, S. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems, 35 (2015), 2587–2612. doi: 10.1017/etds.2014.41.  Google Scholar [18] K. Matomäki, M. Radziwiłl 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 [19] S. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Mathematical Society Lecture Note Series, No. 29, Cambridge University Press, Cambridge-New York-Melbourne, 1977.  Google Scholar [20] J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc., 58 (1952), 116–136. doi: 10.1090/S0002-9904-1952-09580-X.  Google Scholar [21] P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, IAS, 2009. Google Scholar [22] P. Sarnak, Möbius randomness and dynamics, Not. S. Afr. Math. Soc., 43 (2012), 89–97.  Google Scholar [23] W. A. Veech, Möbius orthogonality for generalized Morse-Kakutani flows, Amer. J. Math., 139 (2017), 1157-1203.  doi: 10.1353/ajm.2017.0031.  Google Scholar [24] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar [25] F. Wei, Entropy of Arithmetic Functions and Sarnak's Möbius Disjointness Conjecture, Thesis (Ph.D.)–The University of Chinese Academy of Sciences, 2016. Google Scholar

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##### References:
 [1] J. Auslander, Mean-$L$-stable systems, Illinois J. Math., 3 (1959), 566–579.  Google Scholar [2] H. Davenport, On some infinite series involving arithmetical functions Ⅱ, Quat. J. Math., 8 (1937), 313–320. Google Scholar [3] T. Downarowicz and E. Glasner, Isomorphic extensions and applications, Topological Methods in Nonlinear Analysis, 48 (2016), 321–338. doi: 10.12775/TMNA.2016.050.  Google Scholar [4] T. Downarowicz and S. Kasjan, Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture, Studia Math., 229 (2015), 45–72.  Google Scholar [5] E. H. El Abdalaoui, S. Kasjan and M. Lemańczyk, 0-1 sequences of the Thue-Morse type and Sarnak's conjecture, Proc. Amer. Math. Soc., 144 (2016), 161–176. doi: 10.1090/proc/12683.  Google Scholar [6] E. H. El Abdalaoui, J. Kułaga-Przymus, M. Lemańczyk and T. de la Rue, Möbius disjointness for models of an ergodic system and beyond, Israel J. Math., 228 (2018), 707-751.  doi: 10.1007/s11856-018-1784-z.  Google Scholar [7] E. H. El Abdalaoui, M. Lemańczyk and T. de la Rue, Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals, International Mathematics Research Notices, 14 (2017), 4350–4368. doi: 10.1093/imrn/rnw146.  Google Scholar [8] A. Fan and Y. Jiang, Oscillating sequences, minimal mean attractability and minimal mean-Lyapunov-stability, Ergodic Theory and Dynamical Systems, 2017, to appear. Google Scholar [9] S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk and C. Mauduit, Substitutions and Möbius disjointness, in Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby, Contemp. Math., 678, Amer. Math. Soc., Providence, RI, 2016, 151–173.  Google Scholar [10] L. Flaminio, K. Frączek, J. Kułaga-Przymus and M. Lemańczyk, Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds, Studia Math., 244 (2019), 43-97.  doi: 10.4064/sm170512-25-9.  Google Scholar [11] S. Fomin, On dynamical systems with a purely point spectrum}, Russian, Doklady Akad. Nauk SSSR (N.S.), 77 (1951), 29–32.  Google Scholar [12] 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.  Google Scholar [13] P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. Ⅱ, Ann. of Math. (2), 43 (1942), 332–350. doi: 10.2307/1968872.  Google Scholar [14] W. Huang, Z. Lian, S. Shao and X. Ye, Sequences from zero entropy noncommutative toral automorphisms and Sarnak Conjecture, J. Differential Equations, 263 (2017), 779–810. doi: 10.1016/j.jde.2017.02.046.  Google Scholar [15] H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004. doi: 10.1090/coll/053.  Google Scholar [16] E. Lehrer, Topological mixing and uniquely ergodic systems, Israel J. Math., 57 (1987), 239–255. doi: 10.1007/BF02772176.  Google Scholar [17] J. Li, S. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems, 35 (2015), 2587–2612. doi: 10.1017/etds.2014.41.  Google Scholar [18] K. Matomäki, M. Radziwiłl 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 [19] S. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Mathematical Society Lecture Note Series, No. 29, Cambridge University Press, Cambridge-New York-Melbourne, 1977.  Google Scholar [20] J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc., 58 (1952), 116–136. doi: 10.1090/S0002-9904-1952-09580-X.  Google Scholar [21] P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, IAS, 2009. Google Scholar [22] P. Sarnak, Möbius randomness and dynamics, Not. S. Afr. Math. Soc., 43 (2012), 89–97.  Google Scholar [23] W. A. Veech, Möbius orthogonality for generalized Morse-Kakutani flows, Amer. J. Math., 139 (2017), 1157-1203.  doi: 10.1353/ajm.2017.0031.  Google Scholar [24] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar [25] F. Wei, Entropy of Arithmetic Functions and Sarnak's Möbius Disjointness Conjecture, Thesis (Ph.D.)–The University of Chinese Academy of Sciences, 2016. Google Scholar
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