Li, Jian; Tu, Siming; Ye, Xiangdong Mean equicontinuity and mean sensitivity. (English) Zbl 1356.37016 Ergodic Theory Dyn. Syst. 35, No. 8, 2587-2612 (2015). Summary: Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the latter property must have zero entropy. Cited in 3 ReviewsCited in 77 Documents MSC: 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) 28D20 Entropy and other invariants 37B20 Notions of recurrence and recurrent behavior in topological dynamical systems 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems Keywords:ergodic invariant measure; dichotomy; mean equicontinuity; positive entropy × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Huang, Proc. Steklov Inst. Math. 244 pp 280– (2004) [2] DOI: 10.1007/BF02777364 · Zbl 1122.37013 · doi:10.1007/BF02777364 [3] DOI: 10.1007/s11856-011-0049-x · Zbl 1257.37018 · doi:10.1007/s11856-011-0049-x [4] DOI: 10.1016/j.jfa.2014.01.005 · Zbl 1364.37016 · doi:10.1016/j.jfa.2014.01.005 [5] DOI: 10.1017/S0143385702001724 · Zbl 1134.37308 · doi:10.1017/S0143385702001724 [6] DOI: 10.1515/9781400855162 · doi:10.1515/9781400855162 [7] DOI: 10.1090/S0002-9947-1967-0207959-1 · doi:10.1090/S0002-9947-1967-0207959-1 [8] Fomin, Dokl. Akad. Nauk SSSR 77 pp 29– (1951) [9] DOI: 10.1088/0951-7715/6/6/014 · Zbl 0790.58025 · doi:10.1088/0951-7715/6/6/014 [10] DOI: 10.1090/S0002-9947-1960-0123635-1 · doi:10.1090/S0002-9947-1960-0123635-1 [11] DOI: 10.1007/s11856-008-0032-3 · Zbl 1147.22003 · doi:10.1007/s11856-008-0032-3 [12] DOI: 10.1090/S0002-9939-2013-11717-X · Zbl 1304.37008 · doi:10.1090/S0002-9939-2013-11717-X [13] DOI: 10.1017/S0143385700004983 · Zbl 0661.58027 · doi:10.1017/S0143385700004983 [14] DOI: 10.1017/S0143385707000296 · Zbl 1127.37011 · doi:10.1017/S0143385707000296 [15] Blanchard, Bull. Soc. Math. France 121 pp 465– (1993) · Zbl 0814.54027 · doi:10.24033/bsmf.2216 [16] DOI: 10.1112/jlms/s1-11.2.139 · Zbl 0014.05501 · doi:10.1112/jlms/s1-11.2.139 [17] DOI: 10.2748/tmj/1178229634 · Zbl 0448.54040 · doi:10.2748/tmj/1178229634 [18] Auslander, Minimal Flows and Their Extensions (1988) [19] Auslander, Illinois J. Math. 3 pp 566– (1959) [20] Akin, Convergence in Ergodic Theory and Probability (Columbus, OH, 1993) pp 25– (1996) [21] DOI: 10.1088/0951-7715/21/7/012 · Zbl 1153.37322 · doi:10.1088/0951-7715/21/7/012 [22] DOI: 10.1007/978-1-4612-5775-2 · doi:10.1007/978-1-4612-5775-2 [23] DOI: 10.1112/jlms/s2-26.3.451 · doi:10.1112/jlms/s2-26.3.451 [24] DOI: 10.1090/S0002-9904-1952-09580-X · Zbl 0046.11504 · doi:10.1090/S0002-9904-1952-09580-X [25] Ornstein, Proc. Steklov Inst. Math. 244 pp 295– (2004) [26] DOI: 10.1016/j.jmaa.2014.02.021 · Zbl 1317.37026 · doi:10.1016/j.jmaa.2014.02.021 [27] DOI: 10.1007/s00208-007-0097-z · Zbl 1131.46046 · doi:10.1007/s00208-007-0097-z [28] DOI: 10.1016/S0166-8641(01)00025-6 · Zbl 0997.54061 · doi:10.1016/S0166-8641(01)00025-6 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.