Ling, Bin;Nie, Xiaoxiao;Yin, Jiandong

  • Received : 2018.01.08
  • Accepted : 2018.05.11
  • Published : 2019.01.01


The aim of this paper is to introduce the notions of (quasi) weakly almost periodic point, measure center and minimal center of attraction of amenable group actions, explore the connections of levels of the orbit's topological structure of (quasi) weakly almost periodic points and study chaotic dynamics of transitive systems with full measure centers. Actually, we showed that weakly almost periodic points and quasiweakly almost periodic points have distinct orbit's topological structure and proved that there exists at least countable Li-Yorke pairs if the system contains a proper (quasi) weakly almost periodic point and that a transitive but not minimal system with a full measure center is strongly ergodically chaotic.


weakly almost periodic points;measure centers;amenable group actions;chaotic dynamics


  1. L. N. Argabright and C. O. Wilde, Semigroups satisfying a strong Flner condition, Proc. Amer. Math. Soc. 18 (1967), 587-591.
  2. J. Auslander, Minimal flows and their extensions, North-Holland Mathematics Studies, 153, North-Holland Publishing Co., Amsterdam, 1988.
  3. Z. Chen and X. Dai, Chaotic dynamics of minimal center of attraction of discrete amenable group actions, J. Math. Anal. Appl. 456 (2017), no. 2, 1397-1414.
  4. A. H. Dooley and V. Ya. Golodets, The spectrum of completely positive entropy actions of countable amenable groups, J. Funct. Anal. 196 (2002), no. 1, 1-18.
  5. N. Higson and J. Roe, Amenable group actions and the Novikov conjecture, J. Reine Angew. Math. 519 (2000), 143-153.
  6. W. Huang, X. Ye, and G. Zhang, Local entropy theory for a countable discrete amenable group action, J. Funct. Anal. 261 (2011), no. 4, 1028-1082.
  7. Y. Huang and Z. Zhou, Two new recurrent levels for $C^0$-flows, Acta Appl. Math. 118 (2012), 125-145.
  8. E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math. 146 (2001), no. 2, 259-295.
  9. C. C. Moore, Amenable subgroups of semisimple groups and proximal flows, Israel J. Math. 34 (1979), no. 1-2, 121-138 (1980).
  10. P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 1982.
  11. J. Yin and Z. Zhou, Weakly almost periodic points and some chaotic properties of dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 25 (2015), no. 9, 1550115, 10 pp.
  12. Z. L. Zhou, Weakly almost periodic point and measure centre, Sci. China Ser. A 36 (1993), no. 2, 142-153.
  13. Z. L. Zhou and W. H. He, Level of the orbit's topological structure and topological semiconjugacy, Sci. China Ser. A 38 (1995), no. 8, 897-907.


Supported by : National Natural Science Foundation of China