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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) v.20 n.3 Antofagasta dic. 2001

http://dx.doi.org/10.4067/S0716-09172001000300002 

RELATIVE INVARIANCE FOR
MONOID ACTIONS

CARLOS BRAGA*
Universidade de Maringá, Brasil

Abstract

Let S be a topological monoid acting on the topological space M. Let J be a subset of M. Our purpose here is to study the subsets of M which correspond, under the action of S, to the relative (with respect to J) invariant control sets for control systems (see [4] section 3.3). The relation x ~ y if y Î cl(Sx) and x Î cl(Sy) is an equivalence relation and the classes with respect to this relation with nonempty interior in M are the control sets for the action of S. It is given conditions for the existence and uniqueness of relative invariant classes. As it was done for the control sets, we define an order in the classes and relate it to the relative invariant classes. We also show under certain condition that the relative invariant classes are relatively closed in J.

* Research partially supported by CAPES/PROCAD-Teoria de Lie e Aplicações, grant n0 00186/00-7.

References

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[5] Gottschalk, W.H. and Heldlund, G. A.: "Topological Dynamics''. American Mathematical Society, Providence, Rhode Island, 1955.         [ Links ]

[6] San Martin,L.A.B, and Tonelli, P.A.: Semigroup actions on homogeneous spaces. Semigroup Forum, vol. 50 (1995) 59-88.         [ Links ]

[7] San Martin,L.A.B.: "Control sets and semigroups in semi-simple Lie groups'' . In Semigroups in algebra, geometry and analysis. Gruyter Verlag, Berlin (1994).         [ Links ]

[8] San Martin L.A.B.: Order and domains of attraction of control sets in flag manifolds. Journal of Lie Theory , vol. 8, (1998) 111-128.         [ Links ]

[9] Ruppert, W.: "Compact Semitopological Semigroups: An Intrisic Theory''. Lecture Notes in Mathematics 1079, Springer Verlag, Berlin, Heidelberg, New York, Tokyo 1984.         [ Links ]

Received : April 2001.

Carlos Braga
Departamento de Matemática
Universidade Estadual de Maringá
Av. Colombo 5790
87020-900 Maringá-PR
Brazil
e-mail : cjbbarros@uem.br

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