Jordan decomposition and dynamics on flag manifolds

Abstract

Let g be a real semisimple Lie algebra and G = Int(g). In this article, we relate the Jordan decomposition of X ∈ g (or g ∈ G) with the dynamics induced on generalized flag manifolds by the right invariant continuous-time flow generated by X (or the discrete-time flow generated by g). We characterize the recurrent set and the finest Morse decomposition (including its stable sets) of these flows and show that its entropy always vanishes. We characterize the structurally stable ones and compute the Conley index of the attractor Morse component. When the nilpotent part of X is trivial, we compute the Conley indexes of all Morse components. Finally, we consider the dynamical aspects of linear differential equations with periodic coefficients in g, which can be regarded as an extension of the dynamics generated by an element X ∈ g. In this context, we generalize Floquet theory and extend our previous results to this case.

A method to find generators of a semi-simple Lie group via the topology of its flag manifolds

“…Moreover, H has a unique attractor fixed point set, say att (H ), that has an open and dense stable manifold σ (H ); cf. [5,7]. This means that if x ∈ σ (H ) then its ω-limit set ω(x) is contained in att (H ).…”

“…It is known that H is a gradient vector field with respect to some Riemmannian metric on F ; cf. [5, Proposition 3.3 (ii)] and [7].…”

“…[5, Corollary 3.5] and [7]. Here Z H = {g ∈ G : Ad(g)H = H } is the centralizer of H in G and K H = Z H ∩ K is the centralizer in K .…”

The Minimal Morse Components of Translations on Flag Manifolds are Normally Hyperbolic

“…This broader context of generalized flag manifolds encompasses other interesting cases such as the classical flag manifolds of real or complex nested subspaces and also symplectic grassmanians, which were extensively studied in the literature [2][3][4]6,7,11,12,14,16]. We remark that, in the wider context of flows in flag bundles, it remains an open problem to know wether the minimal Morse components are always normally hyperbolic (see [14]).…”

A note on the jordan decomposition

“…As a byproduct we obtain that a semisimple linear Lie group is the connected component of the identity of an algebraic group and hence closed. Our interest in this subject arose in the article [1], where we related the Jordan decomposition with the dynamics on the flag manifold.…”