I. Random Dynamical Systems and Their Generators.- 1. Basic Definitions. Invariant Measures.- 2. Generation.- II. Multiplicative Ergodic Theory.- 3. The Multiplicative Ergodic Theorem in Euclidean Space.- 4. The Multiplicative Ergodic Theorem on Bundles and Manifolds.- 5. The MET for Related Linear and Affine RDS.- 6. RDS on Homogeneous Spaces of the General Linear Group.- III. Smooth Random Dynamical Systems.- 7. Invariant Manifolds.- 8. Normal Forms.- 9. Bifurcation Theory.- IV. Appendices.- Appendix A. Measurable Dynamical Systems.- A.1 Ergodic Theory.- A.2 Stochastic Processes and Dynamical Systems.- A.3 Stationary Processes.- A.4 Markov Processes.- Appendix B. Smooth Dynamical Systems.- B.1 Two-Parameter Flows on a Manifold.- B.4 Autonomous Case: Dynamical Systems.- B.5 Vector Fields and Flows on Manifolds.- References.

Random Dynamical Systems

This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

Set-valued analysis

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The Foundations Of Differential Geometry

Harmonic analysis on semi‐simple lie groups—i

I. Introduction: Transcending Nationalism: The European Community 1958-2000

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.

Semigroup actions on homogeneous spaces

This paper studies semiflows on topological spaces. A concept of chain recurrence, based on families of coverings, is introduced and related to Morse decomposition. The chain transitive components are studied via semigroup theory by the introduction of the shadowing semigroups associated to a semiflow.

Semiflows on Topological Spaces: Chain Transitivity and Semigroups ⁄

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.

Invariant control sets on flag manifolds

Invariant almost Hermitian structures on flag manifolds

Linear control systems on Lie groups were introduced by Markus [3] and also studied by Ayala and Tirao in [1]. For this class of control systems we establish controllability results in the compact case and also in the semi-simple non-compact case. Wealso show that the forward orbit from the identity is not in general a semigroup.

Luiz A. B. San Martin

Papers

Semigroup actions on homogeneous spaces

Semiflows on Topological Spaces: Chain Transitivity and Semigroups ⁄

Invariant control sets on flag manifolds

Invariant almost Hermitian structures on flag manifolds

Controllability properties of a class of control systems on lie groups