January  2021, 41(1): 359-393. doi: 10.3934/dcds.2020167

Mean equicontinuity, complexity and applications

1. 

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

2. 

Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

3. 

Department of Mathematics, Shantou University, Shantou 515063, China

Received  December 2019 Published  January 2021 Early access  March 2020

We will review the recent development of the research related to mean equicontinuity, focusing on its characterizations, its relationship with discrete spectrum, topo-isomorphy, and bounded complexity. Particularly, the application of the complexity function in the mean metric to the Sarnak and the logarithmic Sarnak Möbius disjointness conjecture will be addressed.

Citation: Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167
References:
[1]

E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, in Convergence in Ergodic Theory and Probability, Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996, 25–40.

[2]

J. Auslander, Mean-$L$-stable systems, Illinois J. Math., 3 (1959), 566-579.  doi: 10.1215/ijm/1255455462.

[3]

J. Auslander and S. Glasner, Distal and highly proximal extensions of minimal flows, Indiana Univ. Math. J., 26 (1977), 731-749.  doi: 10.1512/iumj.1977.26.26057.

[4]

J. Auslander and J. Yorke, Interval maps, factors of maps, and chaos, Tohoku Math. J. (2), 32 (1980), 177-188. doi: 10.2748/tmj/1178229634.

[5]

F. BlanchardB. Host and A. Maass, Topological complexity, Ergodic Theory Dynam. Systems, 20 (2000), 641-662.  doi: 10.1017/S0143385700000341.

[6]

H. Davenport, On some infinite series involving arithmetical functions Ⅱ, Quat. J. Math., 8 (1937), 313-320.  doi: 10.1093/qmath/os-8.1.313.

[7]

T. Downarowicz and E. Glasner, Isomorphic extension and applications, Topol. Methods Nonlinear Anal., 48 (2016), 321-338.  doi: 10.12775/TMNA.2016.050.

[8]

T. Downarowicz and S. Kasjan, Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture, Studia Math., 229 (2015), 45-72.  doi: 10.4064/sm8314-12-2015.

[9]

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

[10]

S. Ferenczi, Measure-theoretic complexity of ergodic systems, Israel J. Math., 100 (1997), 189-207.  doi: 10.1007/BF02773640.

[11]

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

[12]

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

[13]

G. Fuhrmann, E. Glasner, T. Jäger and C. Oertel, Irregular model sets and tame dynamics, preprint, arXiv: 1811.06283.

[14]

G. Fuhrmann, M. Gröger and D. Lenz, The structure of mean equicontinuous group actions, preprint, arXiv: 1812.10219.

[15]

F. García-Ramos, Weak forms of topological and measure-theoretical equicontinuity: Relationships with discrete spectrum and sequence entropy, Ergodic Theory Dynam. Systems, 37 (2017), 1211-1237.  doi: 10.1017/etds.2015.83.

[16]

F. García-Ramos, T. Jäger and X. Ye, Mean equicontinuity, almost automorphy and regularity, preprint, arXiv: 1908.05207.

[17]

F. García-RamosJ. Li and R. Zhang, When is a dynamical system mean sensitive?, Ergodic Theory Dynam. Systems, 39 (2019), 1608-1636.  doi: 10.1017/etds.2017.101.

[18]

F. García-Ramos and B. Marcus, Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems, Discrete Contin. Dyn. Syst., 39 (2019), 729-746.  doi: 10.3934/dcds.2019030.

[19]

E. Glasner, On tame dynamical systems, Colloq. Math., 105 (2006), 283-295.  doi: 10.4064/cm105-2-9.

[20]

E. Glasner, The structure of tame minimal dynamical systems for general groups, Invent. Math., 211 (2018), 213-244.  doi: 10.1007/s00222-017-0747-z.

[21]

S. Glasner and D. Maon, Rigidity in topological dynamics, Ergodic Theory Dynam. Systems, 9 (1989), 309-320.  doi: 10.1017/S0143385700004983.

[22]

E. Glasner and M. Megrelishvili, Linear representations of hereditarily non-sensitive dynamical systems, preprint, arXiv: 0406192v1.

[23]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075.  doi: 10.1088/0951-7715/6/6/014.

[24]

B. Green and T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2), 175 (2010), 541-566. doi: 10.4007/annals.2012.175.2.3.

[25]

P. Halmos and J. Von Neumann, Operator methods in classical mechanics. Ⅱ, Ann. of Math. (2), 43 (1942), 332-350. doi: 10.2307/1968872.

[26]

B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, Mathematical Surveys and Monographs, 236, American Mathematical Society, Providence, RI, 2018.

[27]

W. Huang, Tame systems and scrambled pairs under an abelian group action, Ergodic Theory Dynam. Systems, 26 (2006), 1549-1567.  doi: 10.1017/S0143385706000198.

[28]

W. Huang, J. Li, J. Thouvenot, L. Xu and X. Ye, Bounded complexity, mean equicontinuity and discrete spectrum, Ergodic Theory Dynam. Systems, (2019). doi: 10.1017/etds.2019.66.

[29]

W. HuangS. LiS. Shao and X. Ye, Null systems and sequence entropy pairs, Ergodic Theory Dynam. Systems, 23 (2003), 1505-1523.  doi: 10.1017/S0143385702001724.

[30]

W. Huang, Z. Lian, S. Shao and X. Ye, Reducing the Sarnak Conjecture to Toeplitz systems, preprint, arXiv: 1908.07554.

[31]

W. HuangP. Lu and X. Ye, Measure-theoretical sensitivity and equicontinuity, Israel J. Math., 183 (2011), 233-283.  doi: 10.1007/s11856-011-0049-x.

[32]

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.

[33]

W. HuangZ. Wang and G. Zhang, Möbius disjointness for topological models of ergodic systems with discrete spectrum, J. Mod. Dyn., 14 (2019), 227-290.  doi: 10.3934/jmd.2019010.

[34]

W. Huang and L. Xu, Special flow, weak mixing and complexity, Commun. Math. Stat., 7 (2019), 85-122.  doi: 10.1007/s40304-018-0166-5.

[35]

W. Huang, L. Xu and X. Ye, A distal skew product map on the torus with sub-exponential measure complexity, Ergodic Theory Dynam. Systems, to appear.

[36]

W. Huang, L. Xu and X. Ye, Polynomial mean complexity and Logarithmic Sarnak conjecture, to appear.

[37]

W. Huang and X. Ye, A local variational relation and applications, Israel J. Math., 151 (2006), 237-279.  doi: 10.1007/BF02777364.

[38]

H. Ju, J. Kim, S. Ri and P. Raith, $\mathcal{F}$-equicontinuity and an analogue of Auslander-Yorke dichotomy theorem, preprint, arXiv: 1910.00837.

[39]

A. Katok, Lyapunov exponents, entropy and the periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137–173.

[40]

D. Kerr and H. Li, Dynamical entropy in Banach spaces, Invent. Math., 162 (2005), 649-686.  doi: 10.1007/s00222-005-0457-9.

[41]

D. Kerr and H. Li, Independence in topological and $C^*$-dynamics, Math. Ann., 338 (2007), 869-926.  doi: 10.1007/s00208-007-0097-z.

[42]

A. Köhler, Enveloping semigroups for flows, Proc. Roy. Irish Acad. Sect. A, 95 (1995), 179-191. 

[43]

J. Kulaga-Przymus and M. Lemanczyk, Sarnak's conjecture from the ergodic theory point of view, Encyclopedia Complexity Systems Sci., to appear.

[44]

J. Li, Measure-theoretic sensitivity via finite partitions, Nonlinearity, 29 (2016), 2133-2144.  doi: 10.1088/0951-7715/29/7/2133.

[45]

J. LiP. OprochaY. Yang and T. Zeng, On dynamics of graph maps with zero topological entropy, Nonlinearity, 30 (2017), 4260-4276.  doi: 10.1088/1361-6544/aa8817.

[46]

J. LiS. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems, 35 (2015), 2587-2612.  doi: 10.1017/etds.2014.41.

[47]

J. Li and X. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.), 32 (2016), 83-114.  doi: 10.1007/s10114-015-4574-0.

[48]

J. Li, How chaotic is an almost mean equicontinuous system?, Discrete & Continuous Dynamical Systems - A, 38 (2018), 4727-4744.  doi: 10.3934/dcds.2018208.

[49]

J. Li, T. Yu and X. Ye, Equicontinuity and sensitivity in mean forms, preprint, 2020.

[50]

J. LiT. Yu and T. Zeng, Dynamics on sensitive and equicontinuous functions, Topol. Methods Nonlinear Anal., 51 (2018), 545-563.  doi: 10.12775/tmna.2017.054.

[51]

E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.  doi: 10.1007/s002220100162.

[52]

E. Lindenstrauss and M. Tsukamoto, From rate distortion theory to metric mean dimension: Variational principle, IEEE Trans. Inform. Theory, 64 (2018), 3590-3609.  doi: 10.1109/TIT.2018.2806219.

[53]

M. Morse and G. Hedlund, Symbolic dynamics Ⅱ. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42.  doi: 10.2307/2371431.

[54]

D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.

[55]

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

[56]

J. Qiu and J. Zhao, A note on mean equicontinuity, J. Dynam. Differential Equations, 32 (2020), 101-116.  doi: 10.1007/s10884-018-9716-5.

[57]

J. Qiu and J. Zhao, Null systems in non-minimal case, Ergodic Theory Dynam. Systems, (2019). doi: 10.1017/etds.2019.38.

[58]

P. Sarnak, Three Lectures on the Möbius Functions, Randomness and Dynamics, Lecture Notes in Mathematics, IAS, 2009.

[59]

B. Scarpellini, Stability properties of flows with pure point spectrum, J. London Math. Soc. (2), 26 (1982), 451-464. doi: 10.1112/jlms/s2-26.3.451.

[60]

T. Tao, Equivalence of the logarithmically averaged Chowla and Sarnak conjectures, in Number Theory–Diophantine Problems, Uniform Distribution and Applications, Springer, Cham, 2017,391–421. doi: 10.1007/978-3-319-55357-3_21.

[61]

A. VershikP. Zatitskiy and F. Petrov, Geometry and dynamics of admissible metrics in measure spaces, Cent. Eur. J. Math., 11 (2013), 379-400.  doi: 10.2478/s11533-012-0149-9.

[62]

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

[63]

T. Yu, Measure-theoretic mean equicontinuity and bounded complexity, J. Differential Equations, 267 (2019), 6152-6170.  doi: 10.1016/j.jde.2019.06.017.

[64]

T. Yu, G. Zhang and R. Zhang, Discrete spectrum for amenable group action, preprint, arXiv: 1908.08434.

[65]

B. Zhu, X. Huang and Y. Lian, The systems with almost Banach mean equicontinuity for abelian group actions, preprint, arXiv: 1909.00920.

[66]

R. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math., 20 (1976), 555-588.  doi: 10.1215/ijm/1256049648.

show all references

References:
[1]

E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, in Convergence in Ergodic Theory and Probability, Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996, 25–40.

[2]

J. Auslander, Mean-$L$-stable systems, Illinois J. Math., 3 (1959), 566-579.  doi: 10.1215/ijm/1255455462.

[3]

J. Auslander and S. Glasner, Distal and highly proximal extensions of minimal flows, Indiana Univ. Math. J., 26 (1977), 731-749.  doi: 10.1512/iumj.1977.26.26057.

[4]

J. Auslander and J. Yorke, Interval maps, factors of maps, and chaos, Tohoku Math. J. (2), 32 (1980), 177-188. doi: 10.2748/tmj/1178229634.

[5]

F. BlanchardB. Host and A. Maass, Topological complexity, Ergodic Theory Dynam. Systems, 20 (2000), 641-662.  doi: 10.1017/S0143385700000341.

[6]

H. Davenport, On some infinite series involving arithmetical functions Ⅱ, Quat. J. Math., 8 (1937), 313-320.  doi: 10.1093/qmath/os-8.1.313.

[7]

T. Downarowicz and E. Glasner, Isomorphic extension and applications, Topol. Methods Nonlinear Anal., 48 (2016), 321-338.  doi: 10.12775/TMNA.2016.050.

[8]

T. Downarowicz and S. Kasjan, Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture, Studia Math., 229 (2015), 45-72.  doi: 10.4064/sm8314-12-2015.

[9]

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

[10]

S. Ferenczi, Measure-theoretic complexity of ergodic systems, Israel J. Math., 100 (1997), 189-207.  doi: 10.1007/BF02773640.

[11]

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

[12]

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

[13]

G. Fuhrmann, E. Glasner, T. Jäger and C. Oertel, Irregular model sets and tame dynamics, preprint, arXiv: 1811.06283.

[14]

G. Fuhrmann, M. Gröger and D. Lenz, The structure of mean equicontinuous group actions, preprint, arXiv: 1812.10219.

[15]

F. García-Ramos, Weak forms of topological and measure-theoretical equicontinuity: Relationships with discrete spectrum and sequence entropy, Ergodic Theory Dynam. Systems, 37 (2017), 1211-1237.  doi: 10.1017/etds.2015.83.

[16]

F. García-Ramos, T. Jäger and X. Ye, Mean equicontinuity, almost automorphy and regularity, preprint, arXiv: 1908.05207.

[17]

F. García-RamosJ. Li and R. Zhang, When is a dynamical system mean sensitive?, Ergodic Theory Dynam. Systems, 39 (2019), 1608-1636.  doi: 10.1017/etds.2017.101.

[18]

F. García-Ramos and B. Marcus, Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems, Discrete Contin. Dyn. Syst., 39 (2019), 729-746.  doi: 10.3934/dcds.2019030.

[19]

E. Glasner, On tame dynamical systems, Colloq. Math., 105 (2006), 283-295.  doi: 10.4064/cm105-2-9.

[20]

E. Glasner, The structure of tame minimal dynamical systems for general groups, Invent. Math., 211 (2018), 213-244.  doi: 10.1007/s00222-017-0747-z.

[21]

S. Glasner and D. Maon, Rigidity in topological dynamics, Ergodic Theory Dynam. Systems, 9 (1989), 309-320.  doi: 10.1017/S0143385700004983.

[22]

E. Glasner and M. Megrelishvili, Linear representations of hereditarily non-sensitive dynamical systems, preprint, arXiv: 0406192v1.

[23]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075.  doi: 10.1088/0951-7715/6/6/014.

[24]

B. Green and T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2), 175 (2010), 541-566. doi: 10.4007/annals.2012.175.2.3.

[25]

P. Halmos and J. Von Neumann, Operator methods in classical mechanics. Ⅱ, Ann. of Math. (2), 43 (1942), 332-350. doi: 10.2307/1968872.

[26]

B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, Mathematical Surveys and Monographs, 236, American Mathematical Society, Providence, RI, 2018.

[27]

W. Huang, Tame systems and scrambled pairs under an abelian group action, Ergodic Theory Dynam. Systems, 26 (2006), 1549-1567.  doi: 10.1017/S0143385706000198.

[28]

W. Huang, J. Li, J. Thouvenot, L. Xu and X. Ye, Bounded complexity, mean equicontinuity and discrete spectrum, Ergodic Theory Dynam. Systems, (2019). doi: 10.1017/etds.2019.66.

[29]

W. HuangS. LiS. Shao and X. Ye, Null systems and sequence entropy pairs, Ergodic Theory Dynam. Systems, 23 (2003), 1505-1523.  doi: 10.1017/S0143385702001724.

[30]

W. Huang, Z. Lian, S. Shao and X. Ye, Reducing the Sarnak Conjecture to Toeplitz systems, preprint, arXiv: 1908.07554.

[31]

W. HuangP. Lu and X. Ye, Measure-theoretical sensitivity and equicontinuity, Israel J. Math., 183 (2011), 233-283.  doi: 10.1007/s11856-011-0049-x.

[32]

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.

[33]

W. HuangZ. Wang and G. Zhang, Möbius disjointness for topological models of ergodic systems with discrete spectrum, J. Mod. Dyn., 14 (2019), 227-290.  doi: 10.3934/jmd.2019010.

[34]

W. Huang and L. Xu, Special flow, weak mixing and complexity, Commun. Math. Stat., 7 (2019), 85-122.  doi: 10.1007/s40304-018-0166-5.

[35]

W. Huang, L. Xu and X. Ye, A distal skew product map on the torus with sub-exponential measure complexity, Ergodic Theory Dynam. Systems, to appear.

[36]

W. Huang, L. Xu and X. Ye, Polynomial mean complexity and Logarithmic Sarnak conjecture, to appear.

[37]

W. Huang and X. Ye, A local variational relation and applications, Israel J. Math., 151 (2006), 237-279.  doi: 10.1007/BF02777364.

[38]

H. Ju, J. Kim, S. Ri and P. Raith, $\mathcal{F}$-equicontinuity and an analogue of Auslander-Yorke dichotomy theorem, preprint, arXiv: 1910.00837.

[39]

A. Katok, Lyapunov exponents, entropy and the periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137–173.

[40]

D. Kerr and H. Li, Dynamical entropy in Banach spaces, Invent. Math., 162 (2005), 649-686.  doi: 10.1007/s00222-005-0457-9.

[41]

D. Kerr and H. Li, Independence in topological and $C^*$-dynamics, Math. Ann., 338 (2007), 869-926.  doi: 10.1007/s00208-007-0097-z.

[42]

A. Köhler, Enveloping semigroups for flows, Proc. Roy. Irish Acad. Sect. A, 95 (1995), 179-191. 

[43]

J. Kulaga-Przymus and M. Lemanczyk, Sarnak's conjecture from the ergodic theory point of view, Encyclopedia Complexity Systems Sci., to appear.

[44]

J. Li, Measure-theoretic sensitivity via finite partitions, Nonlinearity, 29 (2016), 2133-2144.  doi: 10.1088/0951-7715/29/7/2133.

[45]

J. LiP. OprochaY. Yang and T. Zeng, On dynamics of graph maps with zero topological entropy, Nonlinearity, 30 (2017), 4260-4276.  doi: 10.1088/1361-6544/aa8817.

[46]

J. LiS. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems, 35 (2015), 2587-2612.  doi: 10.1017/etds.2014.41.

[47]

J. Li and X. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.), 32 (2016), 83-114.  doi: 10.1007/s10114-015-4574-0.

[48]

J. Li, How chaotic is an almost mean equicontinuous system?, Discrete & Continuous Dynamical Systems - A, 38 (2018), 4727-4744.  doi: 10.3934/dcds.2018208.

[49]

J. Li, T. Yu and X. Ye, Equicontinuity and sensitivity in mean forms, preprint, 2020.

[50]

J. LiT. Yu and T. Zeng, Dynamics on sensitive and equicontinuous functions, Topol. Methods Nonlinear Anal., 51 (2018), 545-563.  doi: 10.12775/tmna.2017.054.

[51]

E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.  doi: 10.1007/s002220100162.

[52]

E. Lindenstrauss and M. Tsukamoto, From rate distortion theory to metric mean dimension: Variational principle, IEEE Trans. Inform. Theory, 64 (2018), 3590-3609.  doi: 10.1109/TIT.2018.2806219.

[53]

M. Morse and G. Hedlund, Symbolic dynamics Ⅱ. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42.  doi: 10.2307/2371431.

[54]

D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.

[55]

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

[56]

J. Qiu and J. Zhao, A note on mean equicontinuity, J. Dynam. Differential Equations, 32 (2020), 101-116.  doi: 10.1007/s10884-018-9716-5.

[57]

J. Qiu and J. Zhao, Null systems in non-minimal case, Ergodic Theory Dynam. Systems, (2019). doi: 10.1017/etds.2019.38.

[58]

P. Sarnak, Three Lectures on the Möbius Functions, Randomness and Dynamics, Lecture Notes in Mathematics, IAS, 2009.

[59]

B. Scarpellini, Stability properties of flows with pure point spectrum, J. London Math. Soc. (2), 26 (1982), 451-464. doi: 10.1112/jlms/s2-26.3.451.

[60]

T. Tao, Equivalence of the logarithmically averaged Chowla and Sarnak conjectures, in Number Theory–Diophantine Problems, Uniform Distribution and Applications, Springer, Cham, 2017,391–421. doi: 10.1007/978-3-319-55357-3_21.

[61]

A. VershikP. Zatitskiy and F. Petrov, Geometry and dynamics of admissible metrics in measure spaces, Cent. Eur. J. Math., 11 (2013), 379-400.  doi: 10.2478/s11533-012-0149-9.

[62]

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

[63]

T. Yu, Measure-theoretic mean equicontinuity and bounded complexity, J. Differential Equations, 267 (2019), 6152-6170.  doi: 10.1016/j.jde.2019.06.017.

[64]

T. Yu, G. Zhang and R. Zhang, Discrete spectrum for amenable group action, preprint, arXiv: 1908.08434.

[65]

B. Zhu, X. Huang and Y. Lian, The systems with almost Banach mean equicontinuity for abelian group actions, preprint, arXiv: 1909.00920.

[66]

R. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math., 20 (1976), 555-588.  doi: 10.1215/ijm/1256049648.

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