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  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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[36]

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

[37]

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

[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. Google Scholar

[39]

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

[40]

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

[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.  Google Scholar

[42]

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

[43]

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

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

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

[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.  Google Scholar

[51]

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

[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.  Google Scholar

[53]

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

[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.  Google Scholar

[55]

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

[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.  Google Scholar

[57]

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

[58]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[62]

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

[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.  Google Scholar

[64]

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

[65]

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

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[36]

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

[37]

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

[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. Google Scholar

[39]

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

[40]

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

[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.  Google Scholar

[42]

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

[43]

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

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

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

[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.  Google Scholar

[51]

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

[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.  Google Scholar

[53]

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

[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.  Google Scholar

[55]

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

[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.  Google Scholar

[57]

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

[58]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[62]

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

[63]

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