The minimal Morse components of translations on flag manifolds are normally hyperbolic
TThe minimal Morse components of translationson flag manifolds are normally hyperbolic
Mauro Patr˜ao ∗ Lucas Seco † October 31, 2018
Abstract
Consider the iteration of an invertible matrix on the projectivespace: are the Morse components normally hyperbolic? As far as weknow, this was only stablished when the matrix is diagonalizable overthe complex numbers. In this article we prove that this is true in thefar more general context of an arbitrary element of a semisimple Liegroup acting on its generalized flag manifolds: the so called transla-tions on flag manifolds. This context encompasses the iteration of aninvertible non-diagonazible matrix on the real or complex projectivespace, the classical flag manifolds of real or complex nested subspacesand also symplectic grassmanians. Without these tools from Lie the-ory we do not know how to solve this problem even for the projectivespace.
AMS 2010 subject classification : Primary: 37D99, 53C30 Secondary: 37B35,22E46.
Key words: flag manifolds, normal hyperbolicity. ∗ Departamento de Matem´atica, Universidade de Bras´ılia. Campus Darcy Ribeiro.Bras´ılia, DF, Brasil. e-mail: [email protected] . Supported by CNPq grant n ◦ † Departamento de Matem´atica, Universidade de Bras´ılia. Campus Darcy Ribeiro.Bras´ılia, DF, Brasil. e-mail: [email protected] . a r X i v : . [ m a t h . D S ] M a y Introduction
Normal hyperbolicity of an invariant manifold is the natural generalization ofhyperbolicity of a fixed point, since it assures the existence of a linearizationin a neighborhood of the invariant manifold [15]. Consider the iteration ofan invertible matrix on the projective space. The simplest situation is whenthe matrix has eigenvalues of distinct absolute values. Then the matrixis diagonalizable and the corresponding eigendirections are clearly isolatedfixed points in the projective space. One can prove that they are hyperbolicfixed points and that the omega or alfa limit of any direction is one of theseeigendirections. Now if the matrix has eigenvalues with the same absolutevalue then the matrix is not necessarily diagonalizable and the directions ingeneralized eigenspaces of eigenvalues with the same absolute value give riseto a whole invariant manifold of directions. One can prove [7] that theseinvariant manifolds contain the alfa and omega limits of all directions andthat they are the components of a minimal Morse decomposition. TheseMorse components are, thus, the replacement of the eigendirections. So it isnatural to ask: are these Morse components normally hyperbolic?As far as we know, this normal hyperbolicity was only stablished whenthe matrix is diagonalizable over the complex numbers (with the possibil-ity of eigenvalues with the same absolute value) [6, 12, 7]. In terms of themultiplicative Jordan decomposition of the matrix, previous results were notable to deal with matrices with a non-trivial unipotent component. In thisarticle we prove that this normal hyperbolicity is true in the far more gen-eral context of an arbitrary element of a semisimple Lie group acting on itsgeneralized flag manifolds: the so called translations on flag manifolds. Thiscontext encompasses the iteration of an invertible non-diagonazible matrixon the real or complex projective space.Our approach uses techniques from Lie groups and generalized flag man-ifolds. In [7, 14] we generalized [6, 12] and, using the Jordan decomposition,we described the Morse components in the flag manifold and their corre-sponding stable manifolds as orbits of certain Lie groups acting on the flagmanifold. In this article we use the infinitesimal action of the Lie algebra tolift these orbits to the tangent bundle of the flag manifold and obtain natu-ral candidates for the stable and unstable bundles of each Morse component.Then, choosing an appropriate Riemannian metric which comes from the Liealgebra, we prove the normal hyperbolicity of each Morse component. With-out these tools from Lie theory we do not know how to solve this problem2ven for the projective space (see Example 1.3 and then Example 3.6).This broader context of generalized flag manifolds encompasses other in-teresting cases such as the classical flag manifolds of real or complex nestedsubspaces and also symplectic grassmanians, which were extensively studiedin the literature [2, 3, 4, 6, 7, 11, 12, 14, 16]. We remark that, in the widercontext of flows in flag bundles, it remains an open problem to know wetherthe minimal Morse components are always normally hyperbolic (see [14]).We end this introduction presenting two low dimensionsional examplesof non-diagonalizable matrices where the normal hyperbolicity can be easilyvisualized and a third four-dimensional example where the normal hyperbol-icity cannot be easily visualized.
Example 1.1
Let X = H + N in sl (3 , R ) , where its hyperbolic and nilpotentadditive Jordan components are given respectively by H = − N = and consider g t = exp( tX ) acting on the real projective plane P ( R ) , whichis a flag manifold of the simple Lie group G = SL(3 , R ) .The minimal Morse components are the projectivization of the eigenspacesof H : the repeller is the direction (0 : 0 : 1) and the attractor is the realprojective line ( ∗ : ∗ : 0) (a circle). Above we sketched the phase portrait ofthis flow and its linearization around the attractor, which is a linear flow onthe M¨obius strip over the unipotent flow on the projective line. Example 1.2
Now, let X = H + N in sl (2 , R ) × sl (2 , R ) , where its hyperbolicand nilpotent additive Jordan components are given respectively by H = (cid:18)(cid:18) − (cid:19) , (cid:18) (cid:19)(cid:19) N = (cid:18)(cid:18) (cid:19) , (cid:18) (cid:19)(cid:19) nd consider g t = exp( tX ) acting on the torus, which is a flag manifold ofthe semisimple Lie group G = SL(2 , R ) × SL(2 , R ) . Identifying the toruswith S × S , where each S is the projective line of R , g t acts on the firstcomponent by the exponential of ( − ) and acts on the second component bythe exponential of ( ) .The attractor (repeller) is the cartesian product of the attractor (repeller)in the first component by S : the attractor is the S “above” the torus, therepeller is the S “below” the torus. We sketched above the phase portraitof the flow on the torus and its linearization around the repeller, which is alinear flow on the cilinder over the unipotent flow on the projective line. Example 1.3
Consider the same matrix X of Example 1.1 but now as ele-ment of sl (3 , C ) and consider g t = exp( tX ) now acting in the complex projec-tive space P ( C ) , which is a four dimensional real flag manifold of the simpleLie group G = SL(3 , C ) . We have that g t = e t te t e t
00 0 e − t so that g t ( z : z : z ) = ( e t z + te t z : e t z : e − t z ) where z , z , z ∈ C are not simultaneously zero. If z (cid:54) = 0 then we canassume that z = 1 so that, dividing by e − t we have g t ( z : z : 1) = ( e t z + te t z : e t z : 1) → (0 : 0 : 1) when t → −∞ in R , and tends to ( ∗ : ∗ : 0) when t → + ∞ in R , where ∗ denotes arbitrary complex entries. If z = 0 then ( z : z : 0) (cid:55)→ z /z s a homeomorphism with the Riemann sphere which takes (1 : 0 : 0) to theinfinity ∞ of the complex plane, so that g t ( z : z : 0) = ( z + tz : z : 0) (cid:55)→ z z + t → ∞ = (1 : 0 : 0) when t → ±∞ in R .The minimal Morse components are the projectivization of the eigenspacesof H : the attractor is the complex projective line ( ∗ : ∗ : 0) = P ( C ) (asphere), the repeller is the direction (0 : 0 : 1) . Since P ( C ) is four dimen-sional, the stable bundle of the attractor should be a normal plane bundle overthe sphere (a complex line bundle over the complex projective line) which isnot easy to visualize. Thus, it is not clear if the attractor is normally hyper-bolic in this case. The rest of the paper is organized as follows. In the second section wepresent the preliminaries on dynamics, homogeneous spaces of Lie groups,semi-simple Lie groups and its flag manifolds, translations on flag manifolds.In the third section we prove our main result on normal hyperbolicity. Wefinish the article by revisiting Example 1.3.
We recall some concepts of topological dynamics (for more details, see [1]).Let φ : T × F → F be a continuous flow on a compact metric space ( F, d ),with discrete T = Z or continuous T = R time. Denote by ω ( x ), ω ∗ ( x ),respectively, the forward and backward omega limit sets of x . A Morsedecomposition of φ t , which is given by a finite collection of disjoint subsets {M , . . . , M n } of F such that(i) each M i is compact and φ t -invariant,(ii) for all x ∈ F we have ω ( x ) , ω ∗ ( x ) ⊂ (cid:83) i M i ,(iii) if ω ( x ) , ω ∗ ( x ) ⊂ M j then x ∈ M j .The minimal Morse decomposition is a Morse decomposition which is con-tained in every other Morse decomposition. Each set M i of a minimal Morse5ecomposition is called a minimal Morse component. The stable/unstableset of a morse component M i is the set of all points whose forward/backwardomega limit set is contained in M i .Now let φ be diffeomorphism on a Riemannian manifold F and Dφ itsderivative. An invariant submanifold M ⊂ F is normally hyperbolic if thetangent bundle of F over M has invariant vector subbundles V + and V − and positive constants c and λ < µ such that(i) T F | M = T M ⊕ V − ⊕ V + (ii) | Dφ n v | ≤ ce − λn | v | for all v ∈ V − and n ≥ | Dφ n v | ≤ ce λn | v | for all v ∈ V + and n ≤ | Dφ n v | ≤ ce µ | n | | v | for all v ∈ T M and n ∈ Z in this case, V − is said to be the stable bundle and V + the unstable bundleof M . If φ t is a differentiable flow on F , t ∈ R , an invariant submani-fold M is normally hyperbolic if its is normally hyperbolic for the time onediffeomorphism φ . For the theory of Lie groups and its homogeneous spaces we refer to Helgason[9] and for the theory of principal bundles we refer to Steenrod [17]. Let G be a real Lie group with Lie algebra g where g ∈ G acts on X ∈ g by theadjoint action gX = Ad( g ) X . We have that Ad(exp( X )) = e ad( X ) whereexp : g → G is the exponential of G , Ad and ad are, respectively, the adjointrepresentation of G and g .Let a Lie group G act on a manifold F on the left by the differentiablemap G × F → F , ( g, x ) (cid:55)→ gx . Fix a point x ∈ F . The isotropy subgroup G x is the set of all g ∈ G such that gx = x . We say that the action is transitiveor, equivalently, that F is a homogeneous space of G , if F equals the orbit Gx of x (and hence the orbit of every point of F ). In this case, the map G → F g (cid:55)→ gx is a submersion onto F which is a differentiable locally trivial principal fiberbundle with structure group the isotropy subgroup G x . Quotienting by G x we get the diffeomorphism G/G x ∼ −→ F gG x (cid:55)→ gx L is a Lie subgroup of G , the orbit Lx is the set of all hx , h ∈ L . Therestriction of the principal fiber bundle G → F to L gives the submersiononto the orbit Lx L → Lx l (cid:55)→ lx which is a differentiable locally trivial principal fiber bundle with structuregroup L x = L ∩ G x . If Lx is an embedded submanifold of F then aroundevery point in Lx there exists a differentiable local section from Lx to L thatis a restriction of a local section from F to G of the principal fiber bundle G → F .Since the map G → F is a submersion, the derivative of the map g (cid:55)→ gx on the identity gives the infinitesimal action of g , more precisely, a surjectivelinear map g → T F x Y (cid:55)→ Y · x whose kernel is the isotropy subalgebra g x , the Lie algebra of G x . The deriva-tive of the map g : F → F , x (cid:55)→ gx , gives the action of G on tangent vectors gv = Dg ( v ), v ∈ T F , which is related to the infinitesimal action by g ( Y · x ) = gY · gx For a subset q ⊂ g , denote by q · x the set of all tangent vectors Y · x , Y ∈ q .It follows that T F gx = g ( g · x ). In particular, for l ∈ L , the tangent space ofthe orbit Lx at lx is given by l ( l · x ) ⊂ T F lx , where l ⊂ g is the Lie algebraof L . Thus, the tangent bundle of the orbit is given by T ( Lx ) = L ( l · x )Let E be another manifold with a differentiable action of G , a map f : F → E is said to be G -equivariant if f ( gx ) = gf ( x ). Such a G -equivariant map isautomatically differentiable. For the theory of real semisimple Lie groups and their flag manifolds we referto Duistermat-Kolk-Varadarajan [6], Helgason [9], Knapp [13] and Warner[18]. Let G be a connected real Lie group with semi-simple Lie algebra g .Fix a Cartan decomposition g = k ⊕ s and denote by (cid:104)· , ·(cid:105) the associatedCartan inner product. Let K be the connected subgroup with Lie algebra k ,it is a maximal compact subgroup of G . Since ad( X ) is anti-symmetric for7 ∈ k , the Cartan inner product is K -invariant. Since ad( X ) is symmetric for X ∈ s , a maximal abelian subspace a ⊂ s can be simultaneously diagonalizedso that g splits as an orthogonal sum of g α = { X ∈ g : ad( H ) X = α ( H ) X, ∀ H ∈ a } where α ∈ a ∗ (the dual of a ). We have that g = m ⊕ a , where m is thecentralizer of a in k . A root is a functional α (cid:54) = 0 such that its root space g α (cid:54) =0, denote the set of roots by Π. We thus have the root space decompositionof g , given by the orthogonal sum g = m ⊕ a ⊕ (cid:88) α ∈ Π g α Fix a Weyl chamber a + ⊂ a and let Π + be the corresponding positiveroots, Π − = − Π + the negative roots and Σ the set of simple roots. Fix asubset of simple roots Θ ⊂ Σ and consider the nilpotent subalgebras n ± Θ = (cid:88) α ∈ Π ± −(cid:104) Θ (cid:105) g α and n ± (Θ) = (cid:88) α ∈ Π ± ∩(cid:104) Θ (cid:105) g α where (cid:104) Θ (cid:105) is the set of roots given by linear combinations of roots in Θ. Let n ± = n ± ∅ , then n ± = n ± Θ ⊕ n ± (Θ). The minimal parabolic subalgebra is givenby p = m ⊕ a ⊕ n + and the standard parabolic subalgebra p Θ of type Θ ⊂ Σis given by p Θ = n − (Θ) ⊕ p so by the root space decomposition we have the orthogonal sum g = n − Θ ⊕ p Θ Let p the dimension of p Θ and denote the grassmanian of p -dimensionalsubspaces of g by Gr p ( g ). The flag manifold of type Θ is the orbit F Θ = G p Θ ⊂ Gr p ( g )with base point b Θ = p Θ whose isotropy subalgebra is p Θ itself and isotropysubgroup is the parabolic subgroup P Θ . It follows that F Θ has dimensiondim( n − Θ ) and that G/P Θ ∼ −→ F Θ gP Θ (cid:55)→ gb Θ
8s a G -equivariant diffeomorphism. We also have that K acts transitively in F Θ with isotropy subgroup K Θ = K ∩ P Θ so that K/K Θ ∼ −→ F Θ , kK Θ (cid:55)→ kb Θ ,is a K -equivariant diffeomorphism.The Weyl group W is the finite group generated by the reflections overthe root hyperplanes α = 0 in a , α ∈ Π. W acts on a by isometries and canbe alternatively be given as W = M ∗ /M where M ∗ and M are the normalizerand the centralizer of a in K , respectively. An element w of the Weyl group W can act in g by taking a representative in M ∗ . This action centralizes a ,normalizes m , permutes the roots Π and thus permutes the root spaces g α ,where w g α = g wα does not depend on the representative chosen in M ∗ . Wethus have the basepoint wb Θ = w p Θ whose isotropy subalgebra w p Θ has theorthogonal complement w n − Θ in g , that is g = w n − Θ ⊕ w p Θ For the description of the flow h t = exp( tH ), t ∈ R , induced by H ∈ cl a + on the flag manifold F Θ see ([6], Section 3). Its connected set of fixed pointsis labeled by w ∈ W , each one given by the orbitfix Θ ( H, w ) = G H wb Θ = K H wb Θ , which is an embedded submanifold of F Θ , where G H and K H denote the cen-tralizer of H respectively in G and K and fix Θ ( H, w ). Consider the nilpotentsubalgebras n ± H = (cid:88) ± α ( H ) > g α given by the the sum of the positive/negative eigenspaces of ad( H ) in g .Since G H leaves invariant each eigenspace of ad( H ) it follows that n ± H is G H -invariant. Let N ± H be the corresponding connected Lie subgroups, then N ± H fix Θ ( H, w )is an embedded submanifold of F Θ which is the unstable/stable manifold offix Θ ( H, w ). Here we collect some previous results about the dynamics of a flow g t oftranslations of a real semisimple Lie group G acting on its flag manifolds F Θ .9he flow g t is either given by the iteration of some g ∈ G , for t ∈ Z , or byexp( tX ), for t ∈ R , where X ∈ g , and g t acts on F Θ by left translations.Since G acts on its flag manifolds by the adjoint action we will assume that G is a linear Lie group, and thus g is linear Lie algebra.The usual additive Jordan decomposition writes a matrix as a commut-ing sum of a semisimple and a nilpotent matrix and we can decompose thesemisimple part further as the commuting sum of its imaginary and its realpart, where each part commutes with the nilpotent part and the matrix isdiagonalizable over the complex numbers iff its nilpotent part is zero. Thisgeneralizes to a multiplicative Jordan decomposition of the flow g t in thesemisimplie Lie group G (see Section 2.3 of [7]), providing us with a commu-tative decomposition g t = e t h t u t where there exist a Cartan decomposition of g with a corresponding maximalcompact subgroup K and a Weyl chamber a + such that the elliptic compo-nent e t lies in K , the hyperbolic component is such that h t = exp( tH ),where H ∈ cl a + , and the unipotent component is such that u t = exp( tN ),with N ∈ g nilpotent. Furthermore, we have that h t , e t and u t lie in G H ,the centralizer of H in G , and that g t is diagonalizable iff its unipotent partis u t = 1. We have that the hyperbolic component H dictates the minimalMorse components (see Proposition 5.1 and Theorem 5.2 of [7]). Proposition 2.1
The minimal Morse components of g t on F Θ are givenby fix Θ ( H, w ) , w ∈ W , and their unstable/stable manifolds are given by N ± H fix Θ ( H, w ) . First we construct an appropriate Riemannian metric of F Θ . Fix the Cartaninner product (cid:104)· , ·(cid:105) in g and recall that it is K -invariant. Let g x be theisotropy subalgebra at x ∈ F Θ , then g = g ⊥ x ⊕ g x where ⊥ denotes orthogonal complement with respect to the Cartan innerproduct. Let k ∈ K , since the isotropy subalgebra satisfies g kx = k g x , by the K -invariance of the Cartan inner product we have that g ⊥ kx = ( k g x ) ⊥ = k ( g ⊥ x )10ote that g ⊥ x → T ( F Θ ) x X (cid:55)→ X · x (1)is a linear isomorphism. Define the inner product in T ( F Θ ) x (cid:104) X · x, Y · x (cid:105) x = (cid:104) X, Y (cid:105) where
X, Y ∈ g ⊥ x Proposition 3.1
We have that (cid:104)· , ·(cid:105) x defines a K -invariant Riemannianmetric of F Θ such that the map (1) is an isometry. Furthermore, for Y ∈ g we have | Y · x | x ≤ | Y | with equality iff Y ∈ g ⊥ x . Proof:
Let X ∈ g ⊥ x , then k ( X · x ) = kX · kx , where kX ∈ k ( g ⊥ x ) = g ⊥ kx .The same holds for kY ∈ g ⊥ kx , thus by the K -invariance of the Cartan innerproduct we have (cid:104) k ( X · x ) , k ( Y · x ) (cid:105) kx = (cid:104) kX · kx, kY · kx (cid:105) kx = (cid:104) kX, kY (cid:105) = (cid:104) X, Y (cid:105) = (cid:104) X · x, Y · x (cid:105) x To prove the smoothness of this metric, consider the local charts ψ s of T M constructed from a local section s : U ⊂ F Θ → K of the projection K → F Θ , k (cid:55)→ kb Θ , as follows ψ s : U × g ⊥ b Θ → T M ( x, Y ) (cid:55)→ s ( x )( Y · b Θ )Since s ( x ) ∈ K and s ( x ) b Θ = x , it follows that s ( x ) maps g ⊥ b Θ to g ⊥ x . By the K -invariance it follows that (cid:104) ψ s ( x, X ) , ψ s ( x, Y ) (cid:105) x = (cid:104) X, Y (cid:105) which proves thesmoothness.For the last property, write Y = Y + Y according to g = g ⊥ x ⊕ g x . Then Y · x = Y · x , and thus | Y · x | x = | Y | ≤ | Y | with equality iff Y = 0 iff Y = Y ∈ g ⊥ x .This construction of an invariant metric is related to reductive homoge-nous spaces of K (see [9]) but as a model for the tangent space, instead of asubspace of the Lie algebra of K as usual, here we use a subspace of the Liealgebra of G : this will be more appropriate to the study of G -action in whatfollows. 11ow we recall the candidates for the stable and unstable vector subbun-dles presented in [14] which complement the tangent bundle of each Morsecomponent M = fix Θ ( H, w ). The tangent bundle of M = G H wb Θ inside T F Θ is given by (see Section 2.2) T M = G H ( g H · wb Θ ) ⊂ T F Θ Recall the orthogonal decomposition g = g H ⊕ n − H ⊕ n + H where each g H , n ± H is G H -invariant. Define V ± = G H ( n ± H · wb Θ ) ⊂ T F Θ Proposition 3.2
We have that V ± and T M are G H -invariant vector sub-bundles of T F Θ over M and we have the orthogonal Whitney sum T F Θ | M = T M ⊕ V − ⊕ V + In particular V − ⊕ V + is the normal subbundle of T M . Furthermore, v ∈ T M x or v ∈ V ± x can be written uniquely as Y · x , for Y ∈ g H ∩ g ⊥ x or Y ∈ n ± H ∩ g ⊥ x respectively. Proof:
The G H -invariance is immediate from the definitions. To prove that V − is a subbundle, first note that, since M = K H wb Θ and n − H is K H invariant,it follows that V − = K H ( n − H · wb Θ ). By the orthogonal decomposition g = w n − Θ ⊕ w p Θ and by the root space decomposition, we have that n − H = ( n − H ∩ w n − Θ ) ⊕ ( n − H ∩ w p Θ ) (2)Since we have the isotropy subalgebra g wb Θ = w p Θ , it follows that g ⊥ wb Θ = w n − Θ . Let x ∈ M . We have that x = kwb Θ , k ∈ K H , so that k n − H = n − H , kw p Θ = k g wb Θ = g x and kw n − Θ = k g ⊥ wb Θ = g ⊥ x
12t follows that n − H = ( n − H ∩ g ⊥ x ) ⊕ ( n − H ∩ g x ) (3)so that the map ( n − H ∩ g ⊥ x ) → V − x , Y (cid:55)→ Y · x , is a linear isomorphism.Now let us prove local triviality. Since M = K H wb Θ is an embeddedsubmanifold of F Θ , there exists a differentiable local section (cid:101) s : (cid:101) U → K H of the projection K H → M , l (cid:55)→ lwb Θ , on a neighbourhood (cid:101) U of M , suchthat (cid:101) s is the restriction of a local section s : U → K of the projection K → F Θ , k (cid:55)→ kwb Θ , on neighbourhood U of F Θ . Consider the local chart ψ s ( x, Y ) = s ( x )( Y · wb Θ ) of T ( F Θ ) as in the previous proof. It follows that ψ s restricted to (cid:101) U × ( n − H ∩ g ⊥ wb Θ ) is a local chart of V − given by ψ (cid:101) s : ( x, Y ) (cid:55)→ (cid:101) s ( x )( Y · wb Θ )since (cid:101) s ( x ) ∈ K H and (cid:101) s ( x ) wb Θ = x implies that (cid:101) s ( x ) maps n − H ∩ g ⊥ wb Θ to n − H ∩ g ⊥ x . This shows that V − is a vector subbundle.Since equations (2) and (3) also hold for n + H , the same arguments holdsfor V + , showing that it is also a vector subbundle. Indeed, equations (2) and(3) also hold for g + H , so it follows that g ⊥ x = ( g H ∩ g ⊥ x ) ⊕ ( n − H ∩ g ⊥ x ) ⊕ ( n + H ∩ g ⊥ x )is an orthogonal sum and that the maps( g H ∩ g ⊥ x ) → T M x , ( n ± H ∩ g ⊥ x ) → V ± x given by Y (cid:55)→ Y · x are linear isomorphisms. In particular, since the map (1)is an isometry, it follows that T F Θ | M = T M ⊕ V − ⊕ V + is an orthogonalWhitney sum.These constructions of vector bundles are related to associated bundlesof the principal bundle K H → K H wb Θ (see [17]) but here we constructedthem as subbundles of T M : this will be more appropriate to the study ofthe G -action in what follows.In order to prove normal hyperbolicity we need the following lemma. Lemma 3.3
Let H (cid:54) = 0 . We have that | h t Y | ≤ e − µt | Y | , for Y ∈ n − H , t ≥ nd | h t Y | ≤ e µt | Y | , for Y ∈ n H , t ≤ where µ = min { α ( H ) : α ( H ) > , α ∈ Π } Proof:
For Y ∈ n ± H , we have that h t Y = e t ad( H ) Y , where e t ad( H ) is (cid:104)· , ·(cid:105) -symmetric with eigenvalues in n ± H given by { e ± α ( H ) t : α ( H ) > , α ∈ Π } since ad( H ) is (cid:104)· , ·(cid:105) -symmetric with eigenvalues in n ± H given by {± α ( H ) : α ( H ) > , α ∈ Π } Writing Y as the orthogonal sum of eigenvectors Y = (cid:80) α Y α , we have that | h t Y | = | (cid:88) α e ± α ( H ) t Y α | ≤ (cid:88) α e ± α ( H ) t | Y α | For t > Y ∈ n − H , we have that | h t Y | ≤ e − µt (cid:88) α | Y α | = e − µt | Y | since e − α ( H ) t < e − µt , for all α ∈ Π with α ( H ) >
0. For t < Y ∈ n H ,we have that | h t Y | ≤ e µt (cid:88) α | Y α | = e µt | Y | since e α ( H ) t < e µt , for all α ∈ Π with α ( H ) > Theorem 3.4
Each Morse component fix Θ ( H, w ) is normally hyperbolic. Proof:
By the Jordan decomposition of g t we have the following commuta-tive decompostition g t = e t h t u t where h t = exp( tH ), with H ∈ g hyperbolic, u t = exp( tN ), with N ∈ g nilpotent, and e t , u t ∈ G H , the centralizer of H in G . Furthermore, we canassume that H ∈ cl a + and that e t ∈ K H , the centralizer of H in K .14y Proposition 3.2 a tangent vector v ∈ V ± x can be written as v = Y · x ,for Y ∈ n ± H ∩ g ⊥ x . By Proposition 3.1 we have that | v | = | Y | and also that | g t v | = | g t Y · g t x | ≤ | g t Y | Since g t ∈ G H implies g t Y ∈ n ± H , it is enough to show that the inequalitieshold for g t restricted to n ± H . This follows from standard linear algebra andwe will sketch the argument here for the readers’ convenience. By Lemma3.3, there exists µ > | h t X | ≤ e − µt | X | , for t ≥ X ∈ n − H .Since we can assume that e t ∈ K H , it follows that | g t Y | = | h t u t Y | ≤ e − µt | u t Y | Since u t = exp( tN ), for some nilpotent N ∈ g , we have that u t Y = e t ad( N ) Y .By the triangle inequality, we have that | u t Y | = | e t ad( N ) Y | ≤ (cid:88) k ≥ t k k ! (cid:107) ad( N ) k (cid:107)| Y | = p ( t ) | Y | where (cid:107) · (cid:107) is the operator norm induced by the norm | · | in g and p ( t ) is apolynomial, since ad( N ) is nilpotent. Thus, for v ∈ V − , we have that | g t v | ≤ e − µt p ( t ) | v | , t > V + is completely analogous and we get | g t v | ≤ e µt p ( t ) | v | , t < T M , note that for x ∈ M = fix Θ ( H, w ) we have g t x = e t u t x , thus g t acts as e t u t in T M . By Proposition 3.2 a tangent vector v ∈ T M x canbe written as v = Y · x , for Y ∈ g H ∩ g ⊥ x . By Proposition 3.1 we have that | v | = | Y | and also that | g t v | = | e t u t Y · e t u t x | ≤ | e t u t Y | = | u t Y | ≤ p ( t ) | Y | = p ( t ) | v | where we used that e t ∈ K H and the same inequality for | u t Y | as above.Since e − µ t p ( t ) → t → + ∞ , it is bounded by c for t ≥
0, so thate − µt p ( t ) = e − µ t (cid:16) e − µ t p ( t ) (cid:17) ≤ c e − µ t , t ≥ µt p ( t ) = e µ t (cid:16) e µ t p ( t ) (cid:17) ≤ c e µ t , t ≤ − µ | t | p ( t ) → t → ±∞ , it is bounded by c for t ∈ T , sothat p ( t ) = e µ | t | (cid:0) e − µ | t | p ( t ) (cid:1) ≤ c e µ | t | , t ∈ T Conditions (ii), (iii) and (iv) of normal hyperbolicity then follows by choosing λ = µ and c = max { c , c , c } .It follows that V ± are the unstable/stable bundle of g t . By the mainresult of [15], we obtain a linearization of this flow in a neighborhood of eachminimal Morse component M = fix Θ ( H, w ). Corollary 3.5
Let V = V − ⊕ V + . There exists a differentiable map V → F Θ which takes the null section to fix Θ ( H, w ) and such that: ( i ) Its restriction to some neighborhood of the null section V inside V isa g t -equivariant diffeomorphism onto some neighborhood of fix Θ ( H, w ) inside F Θ . ( ii ) Its restrictions to V ± are g t -equivariant diffeomorphisms, respectively,onto the unstable/stable manifolds N ± H fix Θ ( H, w ) . Proof:
It is enough to note that the action of g t on V is given by the restric-tion of the differential of the action of g t on F Θ and also that the equivarianceproperty is equivalent to the conjugation property of [15].We remark that, in the wider context of flows in flag bundles, it remainsan open problem to know wether the minimal Morse components are alwaysnormally hyperbolic. In [14] our main tool requires a G H -equivariant lin-earization of the flow generated by H around a connected component of itsfixed point set on the flag manifold which, unfortunately, we were only ableto construct in some situations. It would be nice if the linearization we getin the previous result could be made G H -equivariant so that it could be usedto provide the linearization of the flows on flag bundles.We end this article by revisiting Example 1.3 of the introduction.16 xample 3.6 Given a non-null vector v ∈ C n , denote its correspondingdirection by [ v ] = C v . Then the complex projective space P ( C n ) is the set ofsuch directions and the action of an invertible n × n matrix g on P ( C n ) isgiven by g [ v ] = [ gv ] . If v = ( z , . . . , z n ) we denote its direction by [ v ] = ( z : . . . : z n ) . The canonical complex line bundle γ ( C n ) over P ( C n ) is the vectorbundle given by γ ( C n ) = { ( x, v ) : v ∈ x } ⊂ P ( C n ) × C n .On P ( C ) , consider the flow g t = exp( tX ) of translations given in Ex-ample 1.3, where the matrix X is an element of sl (3 , C ) and has hyperboliccomponent H = (cid:16) − (cid:17) We have that P ( C ) is a flag manifold of the simple real Lie group G =SL(3 , C ) of complex matrices of determinant one, whose Lie algebra is thereal Lie algebra g = sl (3 , C ) of complex traceless matrices. In order to seethis, let k = su (3) be the subalgebra of anti-hermitian matrices and let s bethe subspace of hermitian matrices in g . This gives a Cartan decomposition g = k ⊕ s whose corresponding Cartan inner product is proportional to theusual hermitian inner product on complex matrices, more precisely we have (cid:104) X, Y (cid:105) = 6 tr( XY † ) , where Y † is the conjugate transpose of Y . The subgroup K = SU(3) of unitary matrices k of determinant one, that is, kk † = I and det( k ) = 1 , has Lie algebra k and is a maximal compact subgroup of G . Thesubspace a ⊂ s of real diagonal matrices is a maximal abelian subalgebra of s , the corresponding root spaces are given by C E ij , i (cid:54) = j , where E ij is theelementary matrix with entry in row i column j and entries elsewhere.An element H = diag( λ , λ , λ ) ∈ a is such that ad( H ) has eigenvalue α ij ( H ) = λ i − λ j in C E ij . Fix the Weyl chamber given by the subset a + ⊂ a of diagonal matrices with decreasing diagonal entries. The positive root spacesare C E , C E , C E , so that n + = (cid:110)(cid:16) ∗ ∗∗ (cid:17)(cid:111) and n − = (cid:110)(cid:16) ∗∗ ∗ (cid:17)(cid:111) where ∗ denotes arbitrary complex entries. Since the centralizer of a in k is given by m = √− a , we have that m ⊕ a = d the subspace of complextraceless diagonal matrices. It follows that the minimal parabolic subalgebra p = d ⊕ n + is the set of upper triangular matrices in g . Fix the simple root Θ = { α } and thus the corresponding negative root space C E . It followsthat p Θ = C E ⊕ p so that its corresponding parabolic subgroup P Θ and rthogonal complement n − Θ are given by P Θ = (cid:110)(cid:16) ∗ ∗ ∗∗ ∗ (cid:126) ∗ (cid:17) ∈ G (cid:111) and n − Θ = (cid:110)(cid:16) ∗∗ (cid:17)(cid:111) Note that P Θ is precisely the isotropy subgroup of the direction (1 : 0 : 0) in P ( C ) . It follows that P ( C ) is the flag manifold of G of type Θ , moreprecisely, the map F Θ → P ( C ) gb Θ (cid:55)→ g (1 : 0 : 0) , g ∈ G is a G -equivariant diffeomorphism.To describe the dynamics first note that, since the adjoint action of G in g is by conjugation, by the block form of H we have that g ∈ G centralizes H iff it leaves invariant its eigenspaces. Thus we have G H = (cid:110)(cid:16) ∗ ∗∗ ∗ ∗ (cid:17) ∈ G (cid:111) and K H = (cid:26)(cid:16) a − bb a (cid:17) : a, b ∈ C | a | + | b | = 1 (cid:27) (cid:39) SU(2)
Then note that the positive eigenspaces of ad( H ) are C E and C E so that n + H = (cid:110)(cid:16) ∗∗ (cid:17)(cid:111) and n − H = (cid:110)(cid:16) ∗ ∗ (cid:17)(cid:111) The attractor is given by w = 1 so that it is M = G H (1 : 0 : 0)= ( ∗ : ∗ : 0) = P ( C ) Put b Θ = (1 : 0 : 0) , the stable bundle of the attractor is the normal bundle V − = G H ( n − H · b Θ ) so that we have the orthogonal Whitney sum T P ( C ) | P ( C ) = T P ( C ) ⊕ V − We claim that V − is the canonical complex line bundle γ = γ ( C ) over thecomplex projective line P ( C ) , in particular V − is a non-trivial vector bundle.Indeed, we have that γ = (cid:26) (( a : b ) , c ( a, b )) : a, b, c ∈ C | a | + | b | = 1 (cid:27) ⊂ P ( C ) × C By the proof of Proposition 3.2, we have that V − = K H ( n − H ∩ n − Θ · b Θ ) , where n − H ∩ n − Θ = (cid:110)(cid:16) ∗ ∗ (cid:17)(cid:111) ∩ (cid:110)(cid:16) ∗∗ (cid:17)(cid:111) = (cid:110)(cid:16) ∗ (cid:17)(cid:111) = C Y − here we let Y − = E . Define the K H -equivariant map V − → γ k ( yY − · b Θ ) (cid:55)→ ( k (1 : 0) , yk (1 , where k ∈ K H and y ∈ C . By fixing k , we see clearly that it is an isomor-phism between fibers thus, in order to show it is a bundle homeomorphism,we must only show it is well defined. If k ( yY − · b Θ ) = k (cid:48) ( y (cid:48) Y − · b Θ ) then kb Θ = k (cid:48) b Θ so that k (cid:48) = kl where l ∈ K H ∩ P Θ = (cid:26)(cid:16) c c (cid:17) : c ∈ C | c | = 1 (cid:27) (cid:39) S Thus yY − = l ( y (cid:48) Y − ) = y (cid:48) lY − , where the action of l on Y − is by conjugation lY − l − = (cid:16) c c (cid:17) (cid:16) (cid:17) (cid:16) c c (cid:17) = (cid:16) c (cid:17) = cY − It follows that yY − = y (cid:48) cY − , so that y (cid:48) = yc , since cc = 1 . We also havethat k (cid:48) = kl = (cid:16) a − bb a (cid:17) (cid:16) c c (cid:17) = (cid:16) ca − cbcb ca (cid:17) Thus k ( yY − · b Θ ) (cid:55)→ ( k (1 : 0) , yk (1 , a : b ) , y ( a, b )) and k (cid:48) ( y (cid:48) Y − · b Θ ) (cid:55)→ (( ca : cb ) , yc ( ca, cb )) = (( a : b ) , y ( a, b )) which shows the map is well defined and thus gives a vector bundle isomor-phism from V − to γ . Since V − is a vector bundle over a sphere, this fact canalso be seen by computing its clutching function over the equator: it turnsout to be homotopic to z (cid:55)→ z , | z | = 1 , which is the clutching function of thecanonical complex line bundle over P ( C ) (see [8], Proposition 1.11). Thereal counterpart of this is Example 1.1, where the stable bundle of the attrac-tor is the infinite M¨oebius strip, which is the canonical real line bundle overthe real projective line.Note that the K -invariant metric restricted to V − is proportional to thenatural metric of γ ⊂ P ( C ) × C , given by the hermitian inner product in thesecond factor. Nevertheless, the above vector bundle isomorphism V − (cid:39) γ isonly K H -equivariant. Since g t ∈ G H it follows that, even for this example in P ( C ) , γ cannot be used directly to study the normal dynamics of g t over theattractor: the invariant metric and the subbundle V − constructed with toolsfrom Lie theory are much more appropriate for this. eferences [1] J. Alongi and G. Nelson: Recurrence and Topology, AMS GraduateStudies in Mathematics, vol. 85, Providence (2007).[2] G. Ammar and C. Martin: The Geometry of Matrix Eigenvalues Meth-ods , Acta Appl. Math. (1986), 239-278.[3] V. Ayala, F. Colonius, W. Kliemann: Dynamical characterization of theLyapunov form of matrices , Linear Algebra and its Applications, (2005), 272-290.[4] S. Batterson:
Structurally Stable Grassmann Transformations , Trans.Amer. Math. Soc., (1977), 385-404.[5] C. Conley: