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