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December  2021, 41(12): 5579-5607. doi: 10.3934/dcds.2021089

Classification of transitive group actions

Kesong Yan a,b, , Qian Liu b,, and  Fanping Zeng a,b,

a. 

School of Information and Statistics, Guangxi University of Finance and Economics, Nanning, Guangxi, 530003, China
b. 

College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, 530004, China

* Corresponding author: Qian Liu

Received  September 2020 Revised  April 2021 Published  December 2021 Early access  June 2021

We investigate systematically several topological transitivity and mixing concepts for group actions via weak disjointness, return time sets and topological complexity functions.

Keywords: group actions, mixing, topological transitivity, weak disjointness, return time sets, topological complexity functions.
Mathematics Subject Classification: Primary: 37B05; Secondary: 37B20, 37C85.
Citation: Kesong Yan, Qian Liu, Fanping Zeng. Classification of transitive group actions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5579-5607. doi: 10.3934/dcds.2021089
References:
[1]

E. Akin, Recurrence in Topological Dynamical Systems: Furstenberg Families and Ellis Actions, Plenum, New York, 1997. doi: 10.1007/978-1-4757-2668-8.

[2]

E. Akin and E. Glasner, Residual properties and almost equicontinuity, J. Anal. Math., 84 (2001), 243-286.  doi: 10.1007/BF02788112.

[3]

J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153, North-Holland, Amsterdam, 1988.

[4]

M. Beiglb$\ddot{\mathrm{o}}$ck, V. Bergelson and A. Fish, Sumset phenomenon in countable amenable groups, Adv. Math., 223 (2010), 416-432.  doi: 10.1016/j.aim.2009.08.009.

[5]

V. Bergelson, Combinatorial and Diophantine applications of ergodic theory, Appendix A by A. Leibman and Appendix B by Anthony Quas and Mate Wierdl, in: Handbook of dynamical systems, Elsevier B. V., Amsterdam, 1 (2006), 745-869. doi: 10.1016/S1874-575X(06)80037-8.

[6]

V. Bergelson, N. Hindman and R. McCutcheon, Notions of size and combinatorial properties of quotient sets in semigroups, Topology Proceedings, 23 (1998), 23-60. 

[7]

V. Bergelson and R. McCutcheon, Recurrence for semigroup actions and a non-commutative Schur theorem, in: Topological Dynamics and Applications, Contemp. Math., 215 (1998), 205-222.

[8]

V. Bergelson and A. F. Moragues, Juxtaposing $d^*$ and $\overline{d}$, preprint, arXiv: 2003.03029.

[9]

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

[10]

G. Cairns, A. Kolganova and A. Nielsen, Topological transitivity and mixing notions for group actions, Rocky Mountain J. Math., 37 (2007), 371-397.  doi: 10.1216/rmjm/1181068757.

[11]

Z. Chen, J. Li and J. L$\ddot{\mathrm{u}}$, Point transitivity, $\Delta$-transitivity and multi-miminality, Ergodic Theory Dynam. Systems, 35 (2015), 1423-1442.  doi: 10.1017/etds.2013.106.

[12]

X. Dai and H. Liang, Realization of $IP$-sets of any discrete group $T$ via $IP$-recurrent points of some $T$-action topological dynamics, preprint, 2017.

[13]

A. Dooley and G. Zhang, Co-induction in dynamical systems, Ergodic Theory Dynam. Systems, 32 (2012), 919-940.  doi: 10.1017/S0143385711000083.

[14]

T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, J. Reine Angew. Math., 747 (2019), 277-298.  doi: 10.1515/crelle-2016-0025.

[15]

T. Downarowicz and G. Zhang, Symbolic extensions of amenable group actions and the comparison property, Memoirs of the Amerrican Mathematical Society (to apper), arXiv: 1901.01457.

[16]

E. Følner, On groups with full Banach mean value, Math. Scand., 3 (1955), 245-254.  doi: 10.7146/math.scand.a-10442.

[17]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Sys. Th., 1 (1967), 1-49.  doi: 10.1007/BF01692494.

[18] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981. 
[19]

E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, Vol. 101, American Mathematical Society, 2003. doi: 10.1090/surv/101.

[20]

E. Glasner, Classifying dynamical systems by their recurrence properties, Topol. Methods Nonlinear Anal., 24 (2004), 21-40.  doi: 10.12775/TMNA.2004.018.

[21]

W. Huang and X. Ye, An explicit scattering, non-weakly mixing example and weak disjointness, Nonlinearity, 15 (2002), 849-862.  doi: 10.1088/0951-7715/15/3/320.

[22]

W. Huang and X. Ye, Generic eigenvalues, generic factors and weak disjointness,, in: Dynamical Systems and Group Actions, Contemp. Math., 567 (2012), 119-142. doi: 10.1090/conm/567/11232.

[23]

W. Huang and X. Ye, Topological complexity, return times and weak disjointness, Ergodic Theory Dynam. Systems, 24 (2004), 825-846.  doi: 10.1017/S0143385703000543.

[24]

W. Huang and X. Ye, Dynamical system disjoint from any minimal system, Trans. Amer. Math. Soc., 375 (2005), 669-694.  doi: 10.1090/S0002-9947-04-03540-8.

[25]

W. Huang, X. Ye and G. Zhang, Local entropy theory for a countable discrete amenable group action, J. Funct. Anal., 261 (2011), 1028-1082.  doi: 10.1016/j.jfa.2011.04.014.

[26]

H. B. Keynes and J. B. Robertson, On ergodicity and mixing in topological transformation groups, Duke Math. J., 35 (1968), 809-819.  doi: 10.1215/S0012-7094-68-03585-0.

[27]

J. Li, Transitive points via Furstenberg family, Topology Appl., 158 (2011), 2221-2231.  doi: 10.1016/j.topol.2011.07.013.

[28]

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

[29]

R. Peleg, Weak disjointness of transformation groups, Proc. Amer. Math. Soc., 33 (1972), 165-170.  doi: 10.1090/S0002-9939-1972-0298642-2.

[30]

S. Shao and X. Ye, $\mathcal{F}$-mixing and weakly disjointness, Topology Appl., 135 (2004), 231-247.  doi: 10.1016/S0166-8641(03)00166-4.

[31]

H. Wang, Z. Chen and H. Fu, $M$-systems and scattering systems of semigroup actions, Semigroup Forum, 91 (2015), 699-717.  doi: 10.1007/s00233-015-9736-y.

[32]

Z. Wang and G. Zhang, Chaotic behavior of group actions,, in: Dynamics and Numbers, Contemp. Math., 669 (2016), 299-315. doi: 10.1090/conm/669/13434.

[33]

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

[34]

X. Yan and L. He, Topological complexity of semigroup actions, J. Korean Math. Soc., 45 (2008), 221-228.  doi: 10.4134/JKMS.2008.45.1.221.

[35]

R. Yang, Topological sequence complexity and mixing, Chinese Ann. Math. Ser. A, 25 (2004), 809-816. 

[36]

G. Zhang, Relativization of complexity and sensitivity, Ergodic Theory Dynam. Systems, 27 (2007), 1349-1371.  doi: 10.1017/S0143385706000988.

show all references

References:
[1]

E. Akin, Recurrence in Topological Dynamical Systems: Furstenberg Families and Ellis Actions, Plenum, New York, 1997. doi: 10.1007/978-1-4757-2668-8.

[2]

E. Akin and E. Glasner, Residual properties and almost equicontinuity, J. Anal. Math., 84 (2001), 243-286.  doi: 10.1007/BF02788112.

[3]

J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153, North-Holland, Amsterdam, 1988.

[4]

M. Beiglb$\ddot{\mathrm{o}}$ck, V. Bergelson and A. Fish, Sumset phenomenon in countable amenable groups, Adv. Math., 223 (2010), 416-432.  doi: 10.1016/j.aim.2009.08.009.

[5]

V. Bergelson, Combinatorial and Diophantine applications of ergodic theory, Appendix A by A. Leibman and Appendix B by Anthony Quas and Mate Wierdl, in: Handbook of dynamical systems, Elsevier B. V., Amsterdam, 1 (2006), 745-869. doi: 10.1016/S1874-575X(06)80037-8.

[6]

V. Bergelson, N. Hindman and R. McCutcheon, Notions of size and combinatorial properties of quotient sets in semigroups, Topology Proceedings, 23 (1998), 23-60. 

[7]

V. Bergelson and R. McCutcheon, Recurrence for semigroup actions and a non-commutative Schur theorem, in: Topological Dynamics and Applications, Contemp. Math., 215 (1998), 205-222.

[8]

V. Bergelson and A. F. Moragues, Juxtaposing $d^*$ and $\overline{d}$, preprint, arXiv: 2003.03029.

[9]

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

[10]

G. Cairns, A. Kolganova and A. Nielsen, Topological transitivity and mixing notions for group actions, Rocky Mountain J. Math., 37 (2007), 371-397.  doi: 10.1216/rmjm/1181068757.

[11]

Z. Chen, J. Li and J. L$\ddot{\mathrm{u}}$, Point transitivity, $\Delta$-transitivity and multi-miminality, Ergodic Theory Dynam. Systems, 35 (2015), 1423-1442.  doi: 10.1017/etds.2013.106.

[12]

X. Dai and H. Liang, Realization of $IP$-sets of any discrete group $T$ via $IP$-recurrent points of some $T$-action topological dynamics, preprint, 2017.

[13]

A. Dooley and G. Zhang, Co-induction in dynamical systems, Ergodic Theory Dynam. Systems, 32 (2012), 919-940.  doi: 10.1017/S0143385711000083.

[14]

T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, J. Reine Angew. Math., 747 (2019), 277-298.  doi: 10.1515/crelle-2016-0025.

[15]

T. Downarowicz and G. Zhang, Symbolic extensions of amenable group actions and the comparison property, Memoirs of the Amerrican Mathematical Society (to apper), arXiv: 1901.01457.

[16]

E. Følner, On groups with full Banach mean value, Math. Scand., 3 (1955), 245-254.  doi: 10.7146/math.scand.a-10442.

[17]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Sys. Th., 1 (1967), 1-49.  doi: 10.1007/BF01692494.

[18] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981. 
[19]

E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, Vol. 101, American Mathematical Society, 2003. doi: 10.1090/surv/101.

[20]

E. Glasner, Classifying dynamical systems by their recurrence properties, Topol. Methods Nonlinear Anal., 24 (2004), 21-40.  doi: 10.12775/TMNA.2004.018.

[21]

W. Huang and X. Ye, An explicit scattering, non-weakly mixing example and weak disjointness, Nonlinearity, 15 (2002), 849-862.  doi: 10.1088/0951-7715/15/3/320.

[22]

W. Huang and X. Ye, Generic eigenvalues, generic factors and weak disjointness,, in: Dynamical Systems and Group Actions, Contemp. Math., 567 (2012), 119-142. doi: 10.1090/conm/567/11232.

[23]

W. Huang and X. Ye, Topological complexity, return times and weak disjointness, Ergodic Theory Dynam. Systems, 24 (2004), 825-846.  doi: 10.1017/S0143385703000543.

[24]

W. Huang and X. Ye, Dynamical system disjoint from any minimal system, Trans. Amer. Math. Soc., 375 (2005), 669-694.  doi: 10.1090/S0002-9947-04-03540-8.

[25]

W. Huang, X. Ye and G. Zhang, Local entropy theory for a countable discrete amenable group action, J. Funct. Anal., 261 (2011), 1028-1082.  doi: 10.1016/j.jfa.2011.04.014.

[26]

H. B. Keynes and J. B. Robertson, On ergodicity and mixing in topological transformation groups, Duke Math. J., 35 (1968), 809-819.  doi: 10.1215/S0012-7094-68-03585-0.

[27]

J. Li, Transitive points via Furstenberg family, Topology Appl., 158 (2011), 2221-2231.  doi: 10.1016/j.topol.2011.07.013.

[28]

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

[29]

R. Peleg, Weak disjointness of transformation groups, Proc. Amer. Math. Soc., 33 (1972), 165-170.  doi: 10.1090/S0002-9939-1972-0298642-2.

[30]

S. Shao and X. Ye, $\mathcal{F}$-mixing and weakly disjointness, Topology Appl., 135 (2004), 231-247.  doi: 10.1016/S0166-8641(03)00166-4.

[31]

H. Wang, Z. Chen and H. Fu, $M$-systems and scattering systems of semigroup actions, Semigroup Forum, 91 (2015), 699-717.  doi: 10.1007/s00233-015-9736-y.

[32]

Z. Wang and G. Zhang, Chaotic behavior of group actions,, in: Dynamics and Numbers, Contemp. Math., 669 (2016), 299-315. doi: 10.1090/conm/669/13434.

[33]

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

[34]

X. Yan and L. He, Topological complexity of semigroup actions, J. Korean Math. Soc., 45 (2008), 221-228.  doi: 10.4134/JKMS.2008.45.1.221.

[35]

R. Yang, Topological sequence complexity and mixing, Chinese Ann. Math. Ser. A, 25 (2004), 809-816. 

[36]

G. Zhang, Relativization of complexity and sensitivity, Ergodic Theory Dynam. Systems, 27 (2007), 1349-1371.  doi: 10.1017/S0143385706000988.

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