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

Classification of transitive group actions

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.

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$\ddot{\mathrm{o}}$ckV. 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. BergelsonN. 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. BlanchardB. Host and A. Maass, Topological complexity, Ergodic Theory Dynam. Systems, 20 (2000), 641-662.  doi: 10.1017/S0143385700000341.

[10]

G. CairnsA. 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. ChenJ. 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. DownarowiczD. 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. HuangX. 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. WangZ. 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$\ddot{\mathrm{o}}$ckV. 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. BergelsonN. 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. BlanchardB. Host and A. Maass, Topological complexity, Ergodic Theory Dynam. Systems, 20 (2000), 641-662.  doi: 10.1017/S0143385700000341.

[10]

G. CairnsA. 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. ChenJ. 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. DownarowiczD. 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. HuangX. 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. WangZ. 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.

[1]

Yujun Ju, Dongkui Ma, Yupan Wang. Topological entropy of free semigroup actions for noncompact sets. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 995-1017. doi: 10.3934/dcds.2019041

[2]

John Banks, Brett Stanley. A note on equivalent definitions of topological transitivity. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1293-1296. doi: 10.3934/dcds.2013.33.1293

[3]

Xiaojun Huang, Zhiqiang Li, Yunhua Zhou. A variational principle of topological pressure on subsets for amenable group actions. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2687-2703. doi: 10.3934/dcds.2020146

[4]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547

[5]

Kengo Matsumoto. K-groups of the full group actions on one-sided topological Markov shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3753-3765. doi: 10.3934/dcds.2013.33.3753

[6]

Marcelo Sobottka. Topological quasi-group shifts. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 77-93. doi: 10.3934/dcds.2007.17.77

[7]

Yang Cao, Song Shao. Topological mild mixing of all orders along polynomials. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1163-1184. doi: 10.3934/dcds.2021150

[8]

Wen Huang, Zhiren Wang, Guohua Zhang. Möbius disjointness for topological models of ergodic systems with discrete spectrum. Journal of Modern Dynamics, 2019, 14: 277-290. doi: 10.3934/jmd.2019010

[9]

Bin Chen, Xiongping Dai. On uniformly recurrent motions of topological semigroup actions. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 2931-2944. doi: 10.3934/dcds.2016.36.2931

[10]

Michel Coornaert, Fabrice Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 779-793. doi: 10.3934/dcds.2005.13.779

[11]

Daniel Glasscock, Andreas Koutsogiannis, Florian Karl Richter. Multiplicative combinatorial properties of return time sets in minimal dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5891-5921. doi: 10.3934/dcds.2019258

[12]

José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781

[13]

Víctor Jiménez López, Gabriel Soler López. A topological characterization of ω-limit sets for continuous flows on the projective plane. Conference Publications, 2001, 2001 (Special) : 254-258. doi: 10.3934/proc.2001.2001.254

[14]

Qiuxia Liu, Peidong Liu. Topological stability of hyperbolic sets of flows under random perturbations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 117-127. doi: 10.3934/dcdsb.2010.13.117

[15]

Silvére Gangloff, Alonso Herrera, Cristobal Rojas, Mathieu Sablik. Computability of topological entropy: From general systems to transformations on Cantor sets and the interval. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4259-4286. doi: 10.3934/dcds.2020180

[16]

João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465

[17]

Chris Good, Sergio Macías. What is topological about topological dynamics?. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1007-1031. doi: 10.3934/dcds.2018043

[18]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817

[19]

Woochul Jung, Keonhee Lee, Carlos Morales, Jumi Oh. Rigidity of random group actions. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6845-6854. doi: 10.3934/dcds.2020130

[20]

Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (236)
  • HTML views (229)
  • Cited by (0)

Other articles
by authors

[Back to Top]