Luiz A. B. San Martin

Semigroup actions on homogeneous spaces

LetG be a connected semi-simple Lie group with finite center andS⊄G a subsemigroup with interior points. LetG/L be a homogeneous space. There is a natural action ofS onG/L. The relationx≤y ify ∈Sx, x, y ∈G/L, is transitive but not reflexive nor symmetric. Roughly, a control set is a subsetD ⊄G/L, inside of which reflexivity and symmetry for ≤ hold. Control sets are studied inG/L whenL is the minimal parabolic subgroup. They are characterized by means of the Weyl chambers inG meeting intS. Thus, for eachw ∈W, the Weyl group ofG, there is a control setDw.D1 is the only invariant control set, and the subsetW(S)={w:Dw=D1} turns out to be a subgroup. The control sets are determined byW(S)/W. The following consequences are derived: i)S=G ifS is transitive onG/H, i.e.Sx=G/H for allx ∈G/H. HereH is a non discrete closed subgroup different fromG andG is simple. ii)S is neither left nor right reversible ifS #G iii)S is maximal only if it is the semigroup of compressions of a subset of some minimal flag manifold.

Invariant control sets on flag manifolds

LetG be a semisimple Lie group and letS ⊂G be a subsemigroup with nonempty interior. In this paper we study invariant control sets for the action ofS on homogeneous spaces ofG. These sets on the boundary manifolds of the group are characterized in terms of the semisimple elements contained in intS. From this characterization a result on controllability of control systems on semisimple Lie groups is derived. Invariant control sets for the action of S on the boundaries of larger groups¯G withG ⊂¯G are also studied. This latter case includes the action ofS on the projective space and on the flag manifolds.

Chain transitive sets for flows on flag bundles

Abstract We study the chain transitive sets and Morse decompositions of flows on fiber bundles whose fibers are compact homogeneous spaces of Lie groups. The emphasis is put on generalized flag manifolds of semi-simple (and reductive) Lie groups. In this case an algebraic description of the chain transitive sets is given. Our approach consists in shadowing the flow by semigroups of homeomorphisms to take advantage of the good properties of the semigroup actions on flag manifolds. The description of the chain components in the flag bundles generalizes a theorem of Selgrade for projective bundles with an independent proof.

Invariant almost Hermitian structures on flag manifolds

Abstract Let G be a complex semi-simple Lie group and form its maximal flag manifold F =G/P=U/T where P is a minimal parabolic (Borel) subgroup, U a compact real form and T=U∩P a maximal torus of U. We study U-invariant almost Hermitian structures on F . The (1,2)-symplectic (or quasi-Kahler) structures are naturally related to the affine Weyl groups. A special form for them, involving abelian ideals of a Borel subalgebra, is derived. From the (1,2)-symplectic structures a classification of the whole set of invariant structures is provided showing, in particular, that nearly Kahler invariant structures are Kahler, except in the A2 case.

A Multiplicative Ergodic Theorem for Rotation Numbers

Given a vector fieldX on a Riemannian manifoldM of dimension at least 2 whose flow leaves a probability measureμ invariant, the multiplicative ergodic theorem tells us thatμ-a.s. every tangent vector possesses a Lyapunov exponent (exponential growth rate) that is equal to one of finitely many basic exponents corresponding toX andμ. We prove that, in the case of a simple Lyapunov spectrum, every tangent planeμ-a.s. possesses a rotation number that is equal to one of finitely many basic rotation numbers corresponding toX andμ. Rotation in a plane is measured as the time average of the infinitesimal changes of the angle between a frame moved by the linearized flow and the same frame parallel-transported by a (canonical) connection.

On Global Controllability of Discrete-Time Control Systems*

Let xn+1=exp(A+uB)xn be a discrete-time control system which is obtained by discretizing a continuous-time bilinear system. This paper gives sufficient conditions on the matricesA andB for global controllability of this system. These conditions are similar to those of Jurdjevic and Kupka [JK1] for the continuous-time case.

Morse and Lyapunov spectra and dynamics on flag bundles

This paper studies characteristic exponents of flows in relation with the dynamics of flows on flag bundles. The starting point is a flow on a principal bundle with semi-simple group G. Projection against the Iwasawa decomposition G = KAN defines an additive cocycle over the flow with values in a = logA. Its Lyapunov exponents (limits along trajectories) and Morse exponents (limits along chains) are studied. It is proved a symmetric property of these spectral sets, namely invariance under the Weyl group. It is proved also that these sets are located in certain Weyl chambers, defined from the dynamics on the associated flag bundles. As a special case linear flows on vector bundles are considered.

Conley indexes and stable sets for flows on flag bundles

Consider a continuous flow of automorphisms of a G-principal bundle which is chain transitive on its compact Hausdorff base. Here G is a connected non-compact semi-simple Lie group with finite centre. The finest Morse decomposition of the induced flows on the associated flag bundles were obtained in previous articles. Here we describe the stable sets of these Morse components and, under an additional assumption, their Conley indexes.

Nonexistence of invariant semigroups in affine symmetric spaces

Abstract. Let

Nonreversibility of subsemigroups of semi-simple Lie groups

\left( G,L,\tau \right)