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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 & 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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

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

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

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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.  Google Scholar

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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.  Google Scholar

[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. Google Scholar

[13]

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

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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.  Google Scholar

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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. Google Scholar

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

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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.  Google Scholar

[18] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981.   Google Scholar
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E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, Vol. 101, American Mathematical Society, 2003. doi: 10.1090/surv/101.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[35]

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

[36]

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

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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[13]

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

[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.  Google Scholar

[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. Google Scholar

[16]

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

[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.  Google Scholar

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

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[35]

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

[36]

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

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