The dynamic of a Lie group endomorphism

Abstract For a given endomorphism φ on a connected Lie group G this paper studies several subgroups of G that are intrinsically connected with the dynamic behavior of φ.


Introduction
In [1] was shown that associated to a given continuous ow of automorphisms on a connected Lie group G there are dynamical subgroups of G that are intrinsically connected with the behavior of the ow. The author shows there that only by looking at such subgroups one can get information about the controllability of any control system whose drift generates a 1-parameter ow of automorphisms. In the present paper we extend such results by showing that for any G-endomorphism, one can also de ne such subgroups and they still share many of the properties of the continuous case.
On the other hand, we use the results of this article to study the notion of entropy in our forthcoming paper "Topological entropy of Lie group automorphisms".
The paper is structured as follows. In Section 2 we introduce the subalgebras induced by an arbitrary endomorphism ϕ on the Lie algebra g. Then, we show that g decomposes in a dynamical way. In Section 3 we prove that the g-decompositions can be carried on to a connected Lie group. And the associated endomorphism φ of G allows us to associate to φ subgroups that contains most of its dynamical behavior. In the sequence, we establish the main properties of such subgroups. At the end we show an example on the Euclidean Lie group R d and on Sl(n, R), the group of real matrices of order and trace .

Lie algebra endomorphisms
The aim of this section is to introduce the Lie subalgebras induced by a g-endomorphism and show their main properties. For general facts on Lie algebras we use the reference [2].
Let g be a Lie algebra of dimension d and assume that ϕ : g → g is an endomorphism of g. That is, ϕ is a linear map satisfying ϕ[X, Y] = [ϕX, ϕY] for any X, Y ∈ g.
Proposition 2.1. Let g be a Lie algebra over a closed eld and ϕ : g → g an endomorphism. For any eigenvalue α of ϕ let us consider its generalized eigenspace given by If β is also an eigenvalue of ϕ then where g αβ = { } if αβ is not an eigenvalue of ϕ.
Proof. In order to decomposes the ϕ-eigenspace g λ in its Jordan components, we consider r linear independent vectors Z , . . . Zr ∈ g λ such that To prove the proposition it is enough to show the following: if {X , . . . , Xn} ⊂ gα and {Y , . . . , Ym} ⊂ g β are linearly independent sets, hence The proof is done by induction on the sum i + j. In fact, since If i = j = we get (ϕ − αβ)[X , Y ] = which implies [X , Y ] ∈ g αβ . Let us assume that the result holds for i + j < n and let i + j = n. By the induction hypothesis, every term in the right-side of equation (2) is in g αβ which implies (ϕ − αβ)[X i , Y j ] ∈ ker (ϕ − αβ) n for some n ≥ . Consequently, showing that [X i , Y j ] ∈ g αβ and concluding the proof.
In the sequel we prove a primary decomposition for any g-automorphism.
Proposition 2.2. Let ϕ be an automorphism of g and consider its Jordan decomposition with ϕ S semisimple and ϕ N unipotent. Then ϕ S and ϕ N are also g-automorphisms.
Proof. Without lost of generality we can assume that the eld of the scalars is algebraically closed. To prove that ϕ S is an automorphism, it is enough to show that for every couple of basis elements.
Since g is decomposed in generalized eigenspaces of ϕ it is enough to show that ϕ S satis es the property of automorphisms for X ∈ gα, Y ∈ g β and α, β eigenvalues of ϕ. From Proposition 1, [X, Y] ∈ g αβ . On the other hand, since the eigenspaces of ϕ and ϕ S coincide, we get showing that ϕ S is in fact an automorphism. Therefore, is also an automorphism ending the proof.
Let g be a Lie algebra over a closed eld. Proposition 2.2 allows to associate to any endomorphism ϕ of g several Lie subalgebras that are intrinsically connected with its dynamics. In fact, let us de ne the following subsets of g where α is an arbitrary ϕ-eigenvalue gα, g +, = g + ⊕ g and g −, = g − ⊕ g . Also, we denote by g ϕ = g + ⊕ g ⊕ g − and g = g ϕ ⊕ k ϕ . By the property (1) is easy to see that all these subspaces are in fact Lie subalgebras. Moreover, g + and g − are nilpotent. If g is a real Lie algebra, the algebras above are well de ned. In fact, let us denote byḡ the complexi cation of g. By considering theḡ-endomorphismφ induced by ϕ we can de ne the subalgebras gφ,kφ,ḡ * , where * = +, , −.
Moreover, since all the mentionedḡ-subalgebras are invariant by complex conjugation, they are also the complexi cation of the following ϕ-invariant subalgebras of g g ϕ =ḡφ ∩ g, k ϕ =kφ ∩ g, and g * =ḡ * ∩ g with g + and g − nilpotent Lie subalgebras. We should notice that the equality k ϕ = ker(ϕ d ) is still true.

Remark 2.3.
In the real or complex case the restriction of ϕ|g ϕ is an automorphism of g ϕ . Furthermore, the restriction of ϕ to the Lie subalgebras g + , g and g − satis es the inequalities |ϕ m (X)| ≥ cµ −m |X| for any X ∈ g + and m ∈ N, for some c ≥ and µ ∈ ( , ).
Furthermore, for all a > and Z ∈ g it holds that In the sequel we prove that any linear map commuting with two endomorphisms preserves their associated decompositions.
Proposition 2.4. For i = , , let ϕ i : g i → g i an endomorphism of the Lie algebra g i over a closed eld. Assume Proof. Let α be an eigenvalue of ϕ and X ∈ gα. There exists n ≥ such that (ϕ − α) n X = . By the commutating property, we get Since for i = , , g i = g ϕi ⊕ k ϕi and f is a surjective linear map, we must have By the restriction of f to g ϕ we recover all the equalities ending the proof. Proposition 2.4 is still true for the real case.
Corollary 2.5. For i = , , let g i be real algebras and ϕ i :

Lie group endomorphisms
In the sequel all the Lie groups considered are real. For given Lie groups G, H a continuous map φ : G → H is said to be a homomorphism if it preserves the group structure. That is, If G = H such map is said to be an endomorphism of G.
Our aim here is to show that associated with any endomorphism of a connected Lie group G there are connected Lie subgroups which contain most of the dynamic information of the endomorphism. Throughout the paper we always assume the Lie groups and their subgroups are connected.
De nition 3.1. Let G, H be Lie groups with Lie algebras g, h, respectively, and φ : G → H an homomorphism. If there are constants c ≥ and µ ∈ ( , ) such that the homomorphism φ is said to be expanding.
Next, we characterize same general topological property of Lie subgroups that will be needed in the next sections.
Lemma 3.2. Let G be a Lie group with Lie algebra g and, H and K Lie subgroups of G with Lie algebras h and k, respectively such that h ⊕ k = g. Then, H and K are closed ⇔ H ∩ K is a discrete subgroup.
Proof. If H and K are closed subgroups then H ∩ K is also a closed Lie subgroup. As g decomposes into a direct sum of the corresponding Lie subalgebras, it follows that dim(H ∩ K) = . Hence, the result follows.
Reciprocally, assume that H ∩ K is a discrete subgroup of G. By Proposition 6.7 of [3] and also by the hypothesis on h and k, there exist open neighborhoods U, V and W with is a di eomorphism. Without loss of generality, we can assume that W is small enough in order to obtain In particular, if g = xy where x ∈ e U ⊂ H, y ∈ e V ⊂ K and g ∈ W ∩ H, we get Hence, H has nonemtpy interior in cl(H) which only happens if H = cl(H), showing that H is in fact a closed subgroup of G. Analogously, it is possible to prove that K is a closed subgroup of G as stated.

De nition 3.3. Let φ be an endomorphism of a Lie group G. A Lie subgroup H ⊂ G is said to be φ-invariant if φ(H) ⊂ H.
If H ⊂ G is a φ-invariant Lie subgroup, the restriction φ| H is an endomorphism of H in the induced topology. Let us consider a Lie group G and φ : G → G a continuous endomorphism. In order to avoid cumbersome notations, from here we write ϕ = (dφ)e. The dynamical subgroups of G induced by φ are the Lie subgroups, Gφ, Kφ, G + , G , G − , G +, and G −, associated with the Lie subalgebras g ϕ , k ϕ , g + , g , g − , g +, and g −, , respectively. The subgroups G + , G and G − are called the unstable, central and stable subgroups of φ in G, respectively. The following result sets the main properties of these subgroups.

If Gφ is semisimple and G is compact, then Gφ = G . Therefore, if G is any connected Lie group such that G is compact, then Gφ has also the decomposition (3).
Proof. 1. It is well known that the following diagram is commutative, Since the Lie subgroups are connected, their φ-invariance follows directly from the ϕ-invariance of their own Lie algebras. 2. Kφ and ker(φ d ) are connected Lie subgroups with the same Lie algebra kφ = ker(ϕ d ).
So, the desired equality follows. Moreover, since ker(φ d ) is a normal subgroup of G, its connected component of the identity Kφ is also normal. In particular, the product GφKφ is a connected subgroup of G with Lie algebra g ϕ ⊕ k ϕ = g. Therefore, by uniqueness we get G = GφKφ. From this G-decomposition and the φ-invariance of Gφ we obtain On the other hand, since ϕ restricted to g ϕ is an automorphism, it turns out that Consequently, Gφ ⊂ Im(φ d ) which concludes the proof 3. Follows directly by the de nition of G + and G − and by Remark 2.3.

For the decomposition G = GφKφ one can easily show that
Thus, G −, = G − G = G G − . Hence, in order to prove the result it is enough to show that We prove it by induction on the dimension of Gφ. i) If dim(Gφ) = the group Gφ is Abelian and the result is certainly true ii) Let us assume that the result holds for any endomorphism φ such that Gφ is solvable with dim(Gφ) < n. iii) Consider a φ-endomorphism of G with Gφ solvable and dim(Gφ) = n.
The assumption of Gφ solvable implies that there exists a nontrivial closed normal Lie subgroup Bφ of Gφ which is Abelian and φ-invariant, (see for instance the proof in Proposition 2.9 of [1]). By considering the homogeneous space Hφ = Gφ/Bφ we obtain a connected solvable Lie group Hφ such that dim(Hφ) = dim(Gφ) − dim(Bφ) < n.
By the induction hypothesis we obtain Hφ = H +, H − . By taking derivative 5. Let us start by proving that the second assertion is implied by the rst one. We know that Rφ is φinvariant. As before, we obtain an induced surjective endomorphism φ on Gφ/Rφ such that Gφ/Rφ = π(G ), where π : Gφ → Gφ/Rφ is the canonical projection.
But, Gφ/Rφ is semisimple and π(G ) is compact, therefore Moreover, Rφ is a solvable Lie subgroup which by item 4. decomposes as Rφ = R +, R − . Finally, as stated. Now, assume that Gφ is semisimple and G is a compact subgroup. Since ϕ|g ϕ is an automorphism, Theorem 5.4 of [4] implies that there exists k ∈ N such that ϕ| k g ϕ = Ad(g) for some g ∈ Gφ. It follows that g + Ad(g) = g + , g Ad(g) = g and g − Ad(g) = g − . Now, because Gφ is semisimple, there exists an Iwasawa decomposition Gφ = KAN and elements a ∈ A, u ∈ K and n ∈ N such that Ad(g) = Ad(u)Ad(a)Ad(n) with Ad(a) hyperbolic, Ad(n) unipotent and Ad(u) elliptic commutating matrices (see Chapter IX, Lemma 7.1 of [4]). Therefore, i) g + = g + Ad(g) is the sum of eigenspaces with positive eigenvalues of Ad(a) ii) g − = g − Ad(g) is the sum of eigenspaces with negative eigenvalues of Ad(a), and iii) g = g Ad(g) = ker(Ad(a)). Furthermore, the subgroup A is a simply connected Abelian Lie group and A ⊂ G . By the compactness hypothesis of G we must have a = e. So, g + = g − = { } implying that G = G as stated.
These facts bring topological consequences on the induced subgroups.
Proposition 3.6. Suppose that φ restricted to Gφ is an automorphism in the induced topology of G. Then, The dynamical subgroups induced by φ are closed in G 3. For n ≥ d, ker(φ n ) = Kφ. In particular, ker(φ n ) is connected.
Because z ∈ G , equation (6) implies that in the last inequality, each term on the right hand goes to zero as n → +∞. Therefore, ϱ(y, e) = ⇒ G −, ∩ G + = {e} as desired.
By the assumption, φ| Gφ is an automorphism. From that we get Gφ ∩ Kφ = {e}. Then, Proposition 3.2 implies that Gφ is closed in G. Using again Proposition 3.2 and item 1., we also obtain that G + , G , G − , G +, and G −, are closed subgroups of Gφ. As a consequence, they are also closed subgroups of G.
In the sequel we prove that some strong topological property of G are also maintained by φ. Proof. By Proposition III.3.17 of [5] both, the subgroup ker(φ d ) and the quotient G/ ker(φ d ) are simply connected, for any n ≥ d . Since the application is a covering map, Proposition 6.12 of [6] implies that Kφ = ker(φ d ).
Moreover, from the decomposition G = GφKφ we obtain that φ d : G → Gφ is a surjective continuous homomorphism. Thus, by the canonical isomorphism theorem it follows that Gφ and G/ ker φ d are isomorphic, showing in particular that Gφ is simply connected. Knowing that ϕ restricted to g ϕ is an automorphism and Gφ is simply connected, we must have that φ restricted to Gφ is an automorphism, ending the proof.

Corollary 3.8. Let G be a simply connected Lie group. Then, any subgroup induced by an endomorphism φ of G is closed.
The next result shows that the unstable/stable subgroup of a compact φ-invariant subgroup of Gφ is contained in its center. This implies the decomposition of the group when Gφ is compact. If Gφ is compact, we get G ′ φ ⊂ G and so Gφ = Z Gφ G .
Since Z Gφ is solvable subgroup, item 4. of Proposition 3.4 implies that Z Gφ is contained in G +, G − which gives us the desired conclusion.
For the special case of solvable Lie groups more is true. In fact, Theorem 3.10. Let G be a solvable Lie group and φ an endomorphism of G. If φ| Gφ is an automorphism, then any xed point of φ is contained in G .
Proof. Since φ| Gφ is an automorphism we know that Gφ ∩ Kφ = {e}. Therefore, the decomposition of x ∈ G as x = gk with g ∈ Gφ and k ∈ Kφ is unique. Thus, x = gk is a xed point of φ if and only if g and k are xed points of φ. Since φ d (k) = e we must have k = e. So, we only have to analyze the case where g ∈ Gφ is a xed point. By Proposition 3.4 item 4., we know that g = g g g with g ∈ G + , g ∈ G and g ∈ G − Moreover, by Proposition 3.6 item 1. and the φ-invariance of the subgroups it turns out that g is a xed point of φ if and only if g i is a xed point of φ for i = , , . However, since g ∈ G + , from the equation (5) we obtain ϱ(g , e) = ϱ(φ n (g ), e) ≥ cµ −n ϱ(g , e), for any n ∈ N which happens if and only if g = e.
In the same way, by using the fact g ∈ G − is a xed point and the equation (4), we get that g = e showing that x = g ∈ G as we stand.

Examples
Example 3.11. Take G = R d , A ∈ gl(d) a d × d matrix and the endomorphism φ A of G given by φ A (x) = Ax. In this case, the subgroups induced by φ A are given by sums of the eigenspaces of A.