Chaos, attraction, and control for semigroup actions. (English) Zbl 1457.37008
Summary: This manuscript studies sensitivity and chaos for semigroup actions on completely regular spaces. The main results explain how the notions of attraction and control play a fundamental role in the investigation of Auslander-Yorke and Li-Yorke chaos. A general type of non-chaotic semigroup action is exhibited and a criterium for chaos in control systems is presented.
MSC:
| 37B02 | Dynamics in general topological spaces |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 20M30 | Representation of semigroups; actions of semigroups on sets |
| 16W22 | Actions of groups and semigroups; invariant theory (associative rings and algebras) |
References:
| [1] | Alves, RWM; Rocha, VHL; Souza, JA, A characterization of completely regular spaces, Int. J. Math., 26, 1550032 (2015) · Zbl 1314.54016 · doi:10.1142/S0129167X15500329 |
| [2] | Alves, RWM; Souza, JA, Cantor-Kuratowski theorem in uniformizable spaces, Topol. Appl., 252, 158-168 (2019) · Zbl 1407.54016 · doi:10.1016/j.topol.2018.11.017 |
| [3] | Braga Barros, CJ; Reis, RA, On the number of control sets on compact homogeneous spaces, Port. Math., 60, 359-371 (2003) · Zbl 1079.22004 |
| [4] | Braga Barros, CJ; Rocha, VHL; Souza, JA, Lyapunov stability for semigroup actions, Semigroup Forum, 88, 227-249 (2014) · Zbl 1353.37029 · doi:10.1007/s00233-013-9527-2 |
| [5] | Braga Barros, CJ; Rocha, VHL; Souza, JA, Lyapunov stability and attraction under equivariant maps, Can. J. Math., 67, 1247-1269 (2015) · Zbl 1357.37024 · doi:10.4153/CJM-2015-007-7 |
| [6] | Braga Barros, CJ; San Martin, LAB, On the number of control sets on projective spaces, Syst. Control Lett., 29, 21-26 (1996) · Zbl 0883.93016 · doi:10.1016/0167-6911(96)00047-3 |
| [7] | Ceccherini-Silberstein, T.; Coornaert, M., Sensitivity and Devaney’s chaos in uniform spaces, J. Dyn. Control Syst., 19, 568-579 (2013) · Zbl 1338.37015 · doi:10.1007/s10883-013-9182-7 |
| [8] | Colonius, F.; Kliemann, W., Some aspects of control systems as dynamical systems, J. Dyn. Differ. Equ., 5, 469-494 (1993) · Zbl 0784.34050 · doi:10.1007/BF01053532 |
| [9] | Colonius, F.; Kliemann, W., The Dynamics of Control (2000), Boston: Birkhäuser, Boston · Zbl 0718.34091 |
| [10] | Devaney, R., An Introduction to Chaotic Dynamical Systems (1989), Redwood: Addison-Wesley, Redwood · Zbl 0695.58002 |
| [11] | Huang, W.; Ye, X., Devaney’s chaos or 2-scattering implies Li-Yorke chaos, Topol. Appl., 117, 259-272 (2002) · Zbl 0997.54061 · doi:10.1016/S0166-8641(01)00025-6 |
| [12] | Kliemann, W., Recurrence and invariant measures for degenerate diffusions, Ann. Probab., 15, 690-707 (1987) · Zbl 0625.60091 · doi:10.1214/aop/1176992166 |
| [13] | Kontorovich, E.; Megrelishvili, M., A note on sensitivity of semigroup actions, Semigroup Forum, 76, 133-141 (2008) · Zbl 1161.47029 · doi:10.1007/s00233-007-9033-5 |
| [14] | Li, T.; Yorke, J., Period 3 implies chaos, Am. Math. Mon., 82, 985-992 (1975) · Zbl 0351.92021 · doi:10.1080/00029890.1975.11994008 |
| [15] | Polo, F., Sensitive dependence on initial conditions and chaotic group actions, Proc. Am. Math. Soc., 138, 2815-2826 (2010) · Zbl 1200.28017 · doi:10.1090/S0002-9939-10-10286-X |
| [16] | Rybak, OV, Li-Yorke sensitivity for semigroup actions, Ukr. Math. J., 65, 752-759 (2013) · Zbl 1345.37007 · doi:10.1007/s11253-013-0811-9 |
| [17] | San Martin, LAB, Invariant control sets on flag manifolds, Math. Control Signals Syst., 6, 41-61 (1993) · Zbl 0780.93024 · doi:10.1007/BF01213469 |
| [18] | San Martin, LAB; Tonelli, PA, Semigroup actions on homogeneous spaces, Semigroup Forum, 50, 59-88 (1995) · Zbl 0826.22007 · doi:10.1007/BF02573505 |
| [19] | Schneider, FM; Kerkhoff, S.; Behrisch, M.; Siegmund, S., Chaotic actions of topological semigroups, Semigroup Forum, 87, 590-598 (2013) · Zbl 1339.37028 · doi:10.1007/s00233-013-9517-4 |
| [20] | Souza, JA, Lebesgue covering lemma on nonmetric spaces, Int. J. Math., 24, 1350018 (2013) · Zbl 1266.54069 · doi:10.1142/S0129167X13500183 |
| [21] | Souza, JA, Recurrence theorem for semigroup actions, Semigroup Forum, 83, 351-370 (2011) · Zbl 1261.54028 · doi:10.1007/s00233-011-9334-6 |
| [22] | Souza, JA; Tozatti, HVM; Rocha, VHL, On stability and controllability for semigroup actions, Topol. Methods Nonlinear Anal., 48, 1-29 (2016) · Zbl 1362.37046 · doi:10.12775/TMNA.2016.039 |
| [23] | Wang, H.; Long, X.; Fu, H., Sensitivity and chaos of semigroup actions, Semigroup Forum, 84, 81-89 (2012) · Zbl 1267.37010 · doi:10.1007/s00233-011-9335-5 |
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