On limit behavior of semigroup actions on noncompact spaces

Souza, Josiney

doi:10.1090/S0002-9939-2012-11248-1

On limit behavior of semigroup actions on noncompact spaces
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by Josiney A. Souza PDF

Proc. Amer. Math. Soc. 140 (2012), 3959-3972 Request permission

Abstract:

This paper is produced in response to the questioning of Morse decomposition for semigroup actions on noncompact spaces. We show how the limit behavior can be studied in arbitrary topological spaces by using powerful tools such as the Stone-Čech compactification and shadowing semigroups. We extend Conley’s characterization of chain recurrence in terms of attractors from the setting of flows on compact metric spaces to the setting of semigroup actions on any topological space.

References

Carlos José Braga Barros and Luiz A. B. San Martin, Chain control sets for semigroup actions, Mat. Apl. Comput. 15 (1996), no. 3, 257–276 (English, with English and Portuguese summaries). MR 1429552
Carlos J. Braga Barros and Luiz A. B. San Martin, Chain transitive sets for flows on flag bundles, Forum Math. 19 (2007), no. 1, 19–60. MR 2296065, DOI 10.1515/FORUM.2007.002
Carlos J. Braga Barros and Josiney A. Souza, Attractors and chain recurrence for semigroup actions, J. Dynam. Differential Equations 22 (2010), no. 4, 723–740. MR 2734477, DOI 10.1007/s10884-010-9164-3
Carlos J. Braga Barros and Josiney A. Souza, Finest Morse decompositions for semigroup actions on fiber bundles, J. Dynam. Differential Equations 22 (2010), no. 4, 741–760. MR 2734478, DOI 10.1007/s10884-010-9165-2
Braga Barros, C. J., Souza, J. A. and Reis, R. A. (2012). Dynamic Morse decompositions for semigroups of homeomorphisms and control systems. J. Dyn. Control Syst. 18, 1-19.
Sung Kyu Choi, Chin-Ku Chu, and Jong Suh Park, Chain recurrent sets for flows on non-compact spaces, J. Dynam. Differential Equations 14 (2002), no. 3, 597–611. MR 1917652, DOI 10.1023/A:1016339216210
Fritz Colonius and Wolfgang Kliemann, The dynamics of control, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2000. With an appendix by Lars Grüne. MR 1752730, DOI 10.1007/978-1-4612-1350-5
Mike Hurley, Chain recurrence and attraction in noncompact spaces, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 709–729. MR 1145617, DOI 10.1017/S014338570000643X
Mike Hurley, Noncompact chain recurrence and attraction, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1139–1148. MR 1098401, DOI 10.1090/S0002-9939-1992-1098401-X
Mike Hurley, Chain recurrence, semiflows, and gradients, J. Dynam. Differential Equations 7 (1995), no. 3, 437–456. MR 1348735, DOI 10.1007/BF02219371
Mike Hurley, Lyapunov functions and attractors in arbitrary metric spaces, Proc. Amer. Math. Soc. 126 (1998), no. 1, 245–256. MR 1458880, DOI 10.1090/S0002-9939-98-04500-6
Mauro Patrão, Morse decomposition of semiflows on topological spaces, J. Dynam. Differential Equations 19 (2007), no. 1, 181–198. MR 2279951, DOI 10.1007/s10884-006-9033-2
Mauro Patrão and Luiz A. B. San Martin, Semiflows on topological spaces: chain transitivity and semigroups, J. Dynam. Differential Equations 19 (2007), no. 1, 155–180. MR 2279950, DOI 10.1007/s10884-006-9032-3
Mauro Patrão and Luiz A. B. San Martin, Morse decomposition of semiflows on fiber bundles, Discrete Contin. Dyn. Syst. 17 (2007), no. 3, 561–587. MR 2276428, DOI 10.3934/dcds.2007.17.561
Verdi, M. A., Rocio, O. G. and San Martin, L. A. B. (2009). Semigroup Actions on Adjoint Orbits. To appear.
Stephen Willard, General topology, Dover Publications, Inc., Mineola, NY, 2004. Reprint of the 1970 original [Addison-Wesley, Reading, MA; MR0264581]. MR 2048350