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.

[2]

H. Davenport, On some infinite series involving arithmetical functions Ⅱ, Quat. J. Math., 8 (1937), 313–320.

[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.

[4]

T. Downarowicz and S. Kasjan, Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture, Studia Math., 229 (2015), 45–72.

[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.

[6]

E. H. El AbdalaouiJ. Kułaga-PrzymusM. 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.

[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.

[8]

A. Fan and Y. Jiang, Oscillating sequences, minimal mean attractability and minimal mean-Lyapunov-stability, Ergodic Theory and Dynamical Systems, 2017, to appear.

[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.

[10]

L. FlaminioK. FrączekJ. 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.

[11]

S. Fomin, On dynamical systems with a purely point spectrum}, Russian, Doklady Akad. Nauk SSSR (N.S.), 77 (1951), 29–32.

[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.

[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.

[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.

[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.

[16]

E. Lehrer, Topological mixing and uniquely ergodic systems, Israel J. Math., 57 (1987), 239–255. doi: 10.1007/BF02772176.

[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.

[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.

[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.

[20]

J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc., 58 (1952), 116–136. doi: 10.1090/S0002-9904-1952-09580-X.

[21]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, IAS, 2009.

[22]

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

[23]

W. A. Veech, Möbius orthogonality for generalized Morse-Kakutani flows, Amer. J. Math., 139 (2017), 1157-1203. doi: 10.1353/ajm.2017.0031.

[24]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[25]

F. Wei, Entropy of Arithmetic Functions and Sarnak's Möbius Disjointness Conjecture, Thesis (Ph.D.)–The University of Chinese Academy of Sciences, 2016.

show all references

References:
[1]

J. Auslander, Mean-$L$-stable systems, Illinois J. Math., 3 (1959), 566–579.

[2]

H. Davenport, On some infinite series involving arithmetical functions Ⅱ, Quat. J. Math., 8 (1937), 313–320.

[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.

[4]

T. Downarowicz and S. Kasjan, Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture, Studia Math., 229 (2015), 45–72.

[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.

[6]

E. H. El AbdalaouiJ. Kułaga-PrzymusM. 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.

[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.

[8]

A. Fan and Y. Jiang, Oscillating sequences, minimal mean attractability and minimal mean-Lyapunov-stability, Ergodic Theory and Dynamical Systems, 2017, to appear.

[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.

[10]

L. FlaminioK. FrączekJ. 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.

[11]

S. Fomin, On dynamical systems with a purely point spectrum}, Russian, Doklady Akad. Nauk SSSR (N.S.), 77 (1951), 29–32.

[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.

[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.

[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.

[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.

[16]

E. Lehrer, Topological mixing and uniquely ergodic systems, Israel J. Math., 57 (1987), 239–255. doi: 10.1007/BF02772176.

[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.

[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.

[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.

[20]

J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc., 58 (1952), 116–136. doi: 10.1090/S0002-9904-1952-09580-X.

[21]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, IAS, 2009.

[22]

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

[23]

W. A. Veech, Möbius orthogonality for generalized Morse-Kakutani flows, Amer. J. Math., 139 (2017), 1157-1203. doi: 10.1353/ajm.2017.0031.

[24]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[25]

F. Wei, Entropy of Arithmetic Functions and Sarnak's Möbius Disjointness Conjecture, Thesis (Ph.D.)–The University of Chinese Academy of Sciences, 2016.

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