Jordan decomposition and dynamics
on flag manifolds
Let be a semisimple Lie algebra and . In this article, we relate the Jordan decomposition of (or ) with the dynamics induced on generalized flag manifolds by the right invariant continuous-time flow generated by (or the discrete-time flow generated by ). 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 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 , which can be regarded as an extension of the dynamics generated by an element . In this context, we generalize Floquet theory and extend the previous results to this case.
AMS 2000 subject classification: Primary: 37B35, 22E46, 37C20, Secondary: 37B30, 37B40.
Key words: Jordan decomposition, recurrence, Morse decomposition, generalized flag manifolds, structural stability, Conley index, Floquet theory.
Let be a linear group acting differentially on a manifold and its Lie algebra. We denote by , or , the right invariant continuous-time flow generated by or the discrete-time flow generated by . More precisely, when , we have that and, when , we have that is the -iterate of . When we just write . Throughout this paper, we call a linear flow. It induces a differentiable flow on given by , where and or . We call these flows linearly induced flows.
Take , where is a semisimple Lie algebra. In  it is considered a continuous-time flow generated by real semisimple element acting on the flag manifolds of : they show it is a Morse-Bott gradient flow, describe its fixed point set and their stable sets. In  it is analized a continuous-time flow generated by an element which is the sum of two commuting elements of , one of which induces a gradient vector field and the other generates a one-parameter group of isometries. In the context of , the articles [1, 2] study the discrete-time flow generated by an arbitrary element : they characterize the structurally stable ones.
In this article, we study the dynamics of linearly induced flows , for both continuous and discrete times, acting on a generalized flag manifold of . This context includes, for example, volume preserving and Hamiltonian linearly induced flows acting, respectively, on Grassmanian manifolds and on Grassmanian of isotropic subspaces, such as the Grassmanian of the Lagrangian subspaces. This dynamics is related with the Jordan decomposition of the flow , which is defined in terms of the Jordan decomposition of or (see Section 3). We also consider the dynamical aspects of linear differential equations with periodic coefficients in , which can be regarded as an extension of the dynamics generated by an element . In what follows we describe the structure of this article. We note that we recover, in this setting, the results of [14, 13] about flows in flag bundles with chain recurrent compact Hausdorff base.
In the preliminaries we recall some notions of Conley theory and of semisimple Lie theory, proving some useful results.
In Section 3, we recall the Jordan decomposition in and and show that the flow can be written as a product of commuting flows , where , , are linear flows in which are called, respectively, the elliptic, hyperbolic and unipotent components of . We finish this section with a result about the good behavior of the Jordan decomposition under a certain representation of which is related to a natural immersion of a flag manifold into a projective space.
Section 4 is made up of various subsections. In the first one, we look at the linearly induced flow of on a flag manifold as the restriction of a linearly induced flow on a projective space. Using the results of the appendix about dynamics on projective spaces and the results of  about the action of a real semisimple element on the flag manifolds, we generalize these results characterizing the recurrent and chain recurrent sets, the finest Morse decomposition, including its stable sets, in terms of the fixed points of the Jordan components. For example, we get the following result.
The recurrent and chain recurrent sets of in the flag manifold are given, respectively, by
where and are the fixed points of these flows in .
As a byproduct, we show that the entropy of these flows always vanishes. In Section 4.2 we define the conformal flows as the ones whose unipotent part in the Jordan decomposition is trivial, this is the kind of linear flow considered in . For these flows, we compute the Conley indexes of all Morse components. We note that we can compute the Conley index of the attractor for every flow , with no restrictions. We then introduce the regular flows, which are a particular case of the conformal flows. We show that they are dense in or , which implies Theorem 8.1 of  about the density of continuous-time conformal flows. Using this and the previous results we obtain the next result which generalizes results of [1, 2] obtained in the context of discrete-time flows generated by an arbitrary element .
The following conditions are equivalent:
is Morse-Smale and
is structurally stable.
Finally, we consider the dynamical aspects of linear differential equations with periodic coefficients in , which can be regarded as an extension of the dynamics generated by an element . In this context, we generalize Floquet theory and then extend the previous results to this case.
2.1 Flows on topological spaces
Let be a continuous flow on a compact metric space , with discrete or continuous time. For a -invariant set , we define its stable and unstable sets respectively as
where , are the limit sets of . We denote by the set of all recurrent points, that is
and by the set of all fixed points, that is
A linear flow on a vector bundle is called normally hyperbolic if can be written as a Whitney sum of their stable and unstable set and there exist a norm in and constants such that , when is in the stable set, and , when is in the unstable set. We say that a -invariant set is normally hyperbolic if there exists a neighborhood of where the flow is conjugated to a normally hyperbolic linear flow restricted to some neighborhood of the null section.
We recall here the definitions and results related to the concept of chain recurrence and chain transitivity introduced in  (see also ). Take , and . A -chain from to is a sequence of points and a sequence of times such that and , for all .
Given a subset we write for the set of all such that there is a -chain from a point to . Also we put
If , we write
Also, for we write and define the relation if , which is transitive, closed and invariant by , i.e., we have if , for all . For every the set is invariant as well.
Define the relation if and . Then is said to be chain recurrent if . We denote by the set of all chain recurrent points. It is easy to see that the restriction of to is an equivalence relation. An equivalence class of is called a chain transitive component or a chain component, for short.
Now we prove two results which will be used further on.
Let be a flow of such that is an isometry for all . Then, for each and each , there exists a sequence such that .
Proof: By the compactness of , we have that the sequence has a convergent subsequence. Thus, given and , there exist such that and
Hence there exists a sequence such that .
Let , be commuting flows of , . Assume that is an isometry for all and that for each there exists such that the omega and alpha limits of by are precisely . Then the composition is a chain recurrent flow.
Proof: Fix . Given and we will construct an -chain from to . By the assumption on , there exists and such that
for all . Taking , it follows that the points and times define an -chain of , since
Now, since the isometry is recurrent (see Lemma 2.1), there exists such that . Thus the points and times define an -chain of . In fact, using the commutativity of and and using that is an isometry, we have
by the above construction. Finally, using again the commutativity of and we have that
by the choice of .
Now we relate Morse decompositions to chain transitivity. First let us recall that a finite collection of disjoint subsets defines a Morse decomposition when
each is compact and -invariant,
for all we have ,
if then .
Each set of a Morse decomposition is called a Morse component. If is a Morse decomposition of , then it is immediate that decomposes as the disjoint union of stable sets .
The finest Morse decomposition is a Morse decomposition which is contained in every other Morse decomposition. The existence of a finest Morse decomposition of a flow is equivalent to the finiteness of the number of chain components (see , Theorem 3.15). In this case, each Morse component is a chain transitive component and vice-versa. We say that the flow is normally hyperbolic if there exists the finest Morse decomposition and their Morse components are normally hyperbolic.
2.2 Semi-simple Lie theory
For the theory of semisimple Lie groups and their flag manifolds we refer to Duistermat-Kolk-Varadarajan , Helgason  and Warner . To set notation let be a semisimple Lie algebra and acting in canonically. We identify throughout the Lie algebra of with , that is, we write to mean , where . Thus, for and , it follows that . Note that if is a connected Lie group with Lie algebra , then . It follows that the adjoint action of in is the canonical action of in .
Fix a Cartan involution of with Cartan decomposition . The form , where is the Cartan-Killing form of , is an inner product.
Fix a maximal abelian subspace and a Weyl chamber . We let be the set of roots of , the positive roots corresponding to , the set of simple roots in and the negative roots. The Iwasawa decomposition of the Lie algebra reads with where is the root space associated to . As to the global decompositions of the group we write and with , , and .
The Weyl group associated to is the finite group generated by the reflections over the root hyperplanes in , . For each and we define , for all . We have that and that this is a transitive action of on . The maximal involution of is the (only) element of which is such that .
Given a subset of simple roots , let
and put . Let also
and put . The subset singles out the subgroup of the Weyl group which acts trivially on .
The standard parabolic subalgebra of type with respect to chamber is defined by
Let the dimension of and denote the grassmanian of -dimensional subspaces of by . The flag manifold of type is the orbit , with base point , which identifies with the homogeneous space . Since the center of normalizes , the flag manifold depends only on the Lie algebra of . The empty set gives the maximal flag manifold with basepoint .
For we denote by , , , the centralizer of in , , , respectively, i.e, the elements in those groups which fix . Note that centralizes if and only if it commutes with . In fact, this follow from and the injectivity of . When we put
An element induces a vector field on a flag manifold with flow . This is a gradient vector field with respect to a given Riemannian metric on (see , Section 3). The connected sets of fixed point of this flow are given by
so that they are in bijection with the cosets in . Each -fixed point connected set has stable manifold given by
whose union gives the Bruhat decomposition of :
The unstable manifold is
We note that both and are open and dense in . Since the centralizer of leaves invariant and normalizes both and , it follows and are -invariant. We note that these fixed points and (un)stable sets remain the same if is replaced by some such that .
We note that, since the spectrum of in is , it follows that the spectrum of in is .
We conclude with a useful lemma about the decomposition semisimple elements. We say that is semisimple if is diagonalizable over and that is semisimple if is diagonalizable over .
We have that
If is semisimple, then there exists an Cartan decomposition such that where and .
If is semisimple, then there exists an Cartan decomposition such that , where , and .
Proof: For item (i), since is semisimple, there exists a Cartan subalgebra such that (see the proof of Proposition 126.96.36.199, p.105 of ). By Proposition 188.8.131.52, p.89 of , there exists a Cartan involution such that is -invariant. Thus we have that
Writing , with and , we have that and commute, since is abelian.
For item (ii), since is semisimple, there exists a Cartan subgroup such that (since the centralizer of in contains a Cartan subalgebra, see the proof of Proposition 184.108.40.206, p.120 of ). Denote by the associated Cartan subalgebra. By Proposition 220.127.116.11, p.89 of , there exists a Cartan involution such that is -invariant. Thus, by Proposition 18.104.22.168, p.109 of , we have that
Writing , with and , where . Since centralizes , it
follows that and commute, showing that .
3 Jordan decomposition
In this section we recall the additive and the multiplicative Jordan decompositions. Let be a finite dimensional vector space.
If , then we can write , where is semisimple with imaginary eigenvalues, is diagonalizable in with real eigenvalues and is nilpotent. The linear maps , and commute, are unique and called, respectively, the elliptic, the hyperbolic, and the nilpotent components of the additive Jordan decomposition of (see Section ? of [humphreys]).
If , then we can write , where is an isometry relative to some appropriate inner product, is diagonalizable in with positive eigenvalues and is the exponential of a nilpotent linear map. The linear maps , and commute, are unique and called, respectively, the elliptic, the hyperbolic and the unipotent components of the multiplicative Jordan decomposition of (see Lemma IX.7.1 p.430 of ). We denote by the matrix given by the logarithm of the diagonal elements of in the Jordan basis. Writing , , , in the Jordan basis, we see that they commute with .
Take a semisimple Lie algebra. We say that , where , is the Jordan decomposition of in if is the additive Jordan decomposition of in . In this case, , and commute, are unique and called, respectively, the elliptic, the hyperbolic, and the nilpotent components of .
We note that the conjugate of a Jordan decomposition is the Jordan decomposition of the conjugate. Now we prove the following useful result.
Let , where is a semisimple Lie algebra. Then we have that
For each , there exists the Jordan decomposition . Furthermore, there exists an Iwasawa decomposition such that and .
For each , its multiplicative Jordan components lie in . Moreover, there exist a unique such that and an Iwasawa decomposition such that and .
Proof: For item (i), by Proposition 22.214.171.124, p.104 of , there exists a unique decomposition , where commute, is semisimple and is nilpotent. By Lemma 2.3, there exists an Cartan decomposition such that , where and . This is the additive Jordan decomposition of in , since is -anti-symmetric and is -symmetric. It remains to show that and commute with . We first note that is invertible and that commutes with if and only if commutes with . In fact, we have that
It follows that commutes with . In order to show that and commute with , we write
By the uniqueness of the additive Jordan decomposition in , we have that and . Since , we can choose an Iwasawa decomposition such that and .
For item (ii), by Proposition 126.96.36.199, p.120 of , there
exists a unique decomposition , where commute, is semisimple and is the
exponential of a nilpotent linear map. By Lemma
2.3, there exists an Cartan decomposition
such that , where , and . This is
the multiplicative Jordan decomposition of , since is a
-isometry and is -positive. In
order to show that and commute with , one
can proceed as in Lemma IX.7.1 p.431 of . By the
uniqueness of the multiplicative Jordan decomposition in ,
it follows that , and , showing
that the multiplicative Jordan components of lie in . By the
proof of Lemma IX.7.3 item (i) p.431 of , we have
that lies in the Lie algebra of and thus there exists a
unique such that , since is
injective. Since both and commute with , it follows that and can be
diagonalized in the same basis. Since and using the injectivity of , it follows
that . Thus we can choose an Iwasawa
decomposition such that and .
Let be a linear group. Now we define the Jordan decomposition of a linear flow in , . If then , and we can use the additive Jordan decomposition to write , where , and . If we can use the multiplicative Jordan decomposition to write for each . It follows that in both cases the linear flows , , , commute.
Now take , where we identify the Lie algebra of with (see Section 2.2). Let , for all . By Lemma 3.1, each Jordan component of also lies in , for all . If then this is immediate. When , then , where . Thus we can use the Jordan decomposition to write , where , and . In both continuous and discrete time cases, we also have that each Jordan components of the flow lie in , where is given by Lemma 3.1, when .
Let be a finite dimensional representation, where is its infinitesimal representation. When , it is immediate that . When , we have that , for . Denoting , it follows also that .
Now we consider the behavior of the Jordan decomposition with respect to the canonical representation of the general linear group in , where be a finite dimensional vector space, given by
For the the canonical representation of the general linear group in , we have that
Take . If is elliptic (resp. hyperbolic, nilpotent), then is elliptic (resp. hyperbolic, nilpotent). In particular, if is the additive Jordan decomposition of , then is the additive Jordan decomposition of .
Take . If is elliptic (resp. hyperbolic, unipotent), then is elliptic (resp. hyperbolic, unipotent). In particular, if is the Jordan decomposition of , then is the Jordan decomposition of .
If is the Jordan decomposition of , then is the Jordan decomposition of .
Proof: First observe that the complexification of wedge product of is equal to the wedge product of the complexification of , that is . In fact, it is immediate that and that both have the same dimension. Note also that
We claim that
where such that . For this is immediate. By induction on
Since it follows that .
For item (i), taking nilpotent then there exists such that . From the above claim, it follows that . In fact,
where such that . Thus, for each there exists such that . Therefore , which implies that , for all . Now taking elliptic, there exists a -basis of such that , where is purely imaginary. Then
is a -basis of such that
This implies that is elliptic, since is purely imaginary. The hyperbolic case is analogous.
For item (ii), taking unipotent, then with nilpotent so that is unipotent, by using item (i). Now taking is elliptic, there exists a -basis of such that , where with . Then
is a -basis of such that
This implies that is elliptic, since . The hyperbolic case is analogous.
Item (iii) follows immediately from the previous items.
We recall the well known Plücker embedding, which is given by
where is a basis of . This embedding has the following equivariance property
where is the canonical representation presented in Lemma 3.2. If is a linearly induced flow it follows that
4 Dynamics in flag manifolds
In this section, we relate the Jordan decomposition of in to the dynamics of the induced linear flow on the flag manifolds of , where is a semisimple Lie algebra. The main results of the section deals with the characterization of the recurrent set and the finest Morse decomposition in terms of the fixed points of the Jordan components.
Recall that, as seen in Section 3, when then each multiplicative Jordan component of lies in , where is such that . Furthermore, there exists a Weyl chamber such that .
Let . It follows that , and induce flows in the flag manifold . If , we know that , so we can restrict the Plücker embedding (see Section 3) to and get an embedding , where . Since is -invariant, we have the following equivariance property
where is the restriction to of the canonical representation presented in Lemma 3.2.
4.1 Recurrence, chain recurrence and entropy
The next proposition shows that the fixed points of the hyperbolic part of are Morse components for the flow in the flag manifold. This result is proved by using Proposition A.5, which is in fact a particular case when the flag manifold is the projective space.
Let be a flow on . The set
is a Morse decomposition for . Furthermore, the stable and unstable sets of are given by
Proof: Since and since , it follows that is -invariant.
Now we show that is the stable set of . By the Bruhat decomposition of , it is enough to show that is contained in the stable set of . Let , then , where and . Then
where , since . Now we show that . This follows by Lemma A.3, since the spectral radius of the restriction of to is smaller than 1. In fact, by the Jordan decomposition, is given by the greatest eigenvalue of its hyperbolic component, which is given by the restriction of to . These eigenvalues are given by , where with , so that . Now if then , so that lies in the closed subset .
For the unstable set we proceed analogously. It follows that
contains all the alpha and omega limit sets. In order to show that the set is a Morse decomposition for it is enough to prove that if , , then . First recall that
where is the Plücker embedding. By hypothesis , are contained in the connected set of , so they lie in the same connected component of which is given by an eigenspace of . Using Lemmas 3.2 and A.4, it follows that , which shows that . Then there exists such that . By the invariance of , we get that
showing that . The proof for the unstable
set is completely analogous.
We note that and are open and dense (see Section 2.2) so that and are, respectively, the only attractor and repeller which are thus denoted by and . Using the previous result, we obtain the desired characterization of the finest Morse decomposition.
Let be a flow on and