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: |
[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. Blanchard, B. 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-Ramos, J. 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. Huang, S. Li, S. 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. Huang, P. 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. Huang, Z. 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. Huang, Z. 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. Li, P. Oprocha, Y. 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. 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.![]() ![]() ![]() |
[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. Li, T. 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. Vershik, P. 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.![]() ![]() ![]() |