Conley Decompositions and Actions on Flag Manifolds

This work is concerned with the action of a semisimple element g of the semisimple connected Lie group Sl(d;R) on ag manifolds or F. These manifolds are obtained as left cosets of a parabolic group T. We consider minimal sets of g on F , assigning to each minimal set M an index K(M) which appears if one consider an special decomposition of the vector space R d related to g called the Conley Decomposition. there is a continuous curve : 0;1] ! F joining M and N and such that each (t) be almost periodic if and only if K(M) = K(N). We give an application of the theorem in control theory. 1. Introduction We present a brief resum ee on ag manifolds to x some notations. This approach follows closely that of the paper by Guivarch and Raugi GR89]. The details can be found there and in the book by Rohlin and Fuchs RF81], chapter 3.