The dynamic of a Lie group endomorphism

Víctor Ayala; Heriberto Román-Flores; Adriano Da Silva

doi:10.1515/math-2017-0120

Article Open Access

The dynamic of a Lie group endomorphism

Víctor Ayala , Heriberto Román-Flores and Adriano Da Silva

Published/Copyright: December 29, 2017

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$Open Mathematics$

From the journal Open Mathematics Volume 15 Issue 1

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.

Keywords: Lie group; Dynamic; Endomorphism

MSC 2010: 20K30; 22E15

1 Introduction

In [1] was shown that associated to a given continuous flow of automorphisms on a connected Lie group G there are dynamical subgroups of G that are intrinsically connected with the behavior of the flow. 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 flow of automorphisms. In the present paper we extend such results by showing that for any G-endomorphism, one can also define 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 2 and trace 0.

2 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 field and ϕ :𝔤→ 𝔤 an endomorphism. For any eigenvalue α of ϕ let us consider its generalized eigenspace given by

gα={X∈g;(ϕ−α)nX=0,forsomen≥1}.

If β is also an eigenvalue of ϕ then

[gα,gβ]⊂gαβ, (1)

where 𝔤_αβ = {0} 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 Z₁,… Z_r ∈ 𝔤_λ such that

ϕ(Zj)=λZj+Zj−1,j=1,…r with Z0=0.

To prove the proposition it is enough to show the following:

if {X1,…,Xn}⊂gα and {Y1,…,Ym}⊂gβ

are linearly independent sets, hence

[Xi,Yj]⊂gαβ,i=1,…,n;j=1,…,m.

The proof is done by induction on the sum i+j. In fact, since

ϕ[Xi,Yj]=[ϕXi,ϕYj]=[αXi+Xi−1,βYj+Yj−1]=αβ[Xi,Yj]+α[Xi,Yj−1]+β[Xi−1,Yj]+[Xi−1,Yj−1]

we obtain

(ϕ−αβ)[Xi,Yj]=α[Xi,Yj−1]+β[Xi−1,Yj]+[Xi−1,Yj−1]. (2)

If i = j = 1 we get (ϕ − αβ)[X₁,Y₁] = 0 which implies [X₁,Y₁] ∈ 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)[Xi,Yj]∈ker(ϕ−αβ)n for some n≥1.

Consequently,

(ϕ−αβ)n+1[Xi,Yj]=0

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

ϕ=ϕSϕN=ϕNϕS

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 field of the scalars is algebraically closed. To prove that ϕ^S is an automorphism, it is enough to show that

ϕS([X,Y])=[ϕS(X),ϕS(Y)]

for every couple of basis elements.

Since 𝔤 is decomposed in generalized eigenspaces of ϕ it is enough to show that ϕ^S satisfies the property of automorphisms for X ∈ 𝔤_α, 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

ϕS([X,Y])=α⋅β[X,Y] and [ϕS(X),ϕS(Y)]=[αX,βY]=α⋅β[X,Y]

showing that ϕ^S is in fact an automorphism. Therefore,

ϕN=ϕS−1ϕ

is also an automorphism ending the proof. □

Let 𝔤 be a Lie algebra over a closed field. 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 define the following subsets of 𝔤 where α is an arbitrary ϕ-eigenvalue

gϕ=⨁α̸=0gα,kϕ=ker⁡(ϕd)g+=⨁|α|>1gα,g0=⨁α:|α|=1gαg−=⨁0<|α|<1gα,g+,0=g+⊕g0 and g−,0=g−⊕g0.

Also, we denote by 𝔤_ϕ = 𝔤⁺⊕ 𝔤⁰⊕ 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 defined. In fact, let us denote by 𝔤 the complexification of 𝔤. By considering the 𝔤-endomorphism ϕ induced by ϕ we can define the subalgebras

g¯ϕ¯,k¯ϕ¯,g¯∗, where ∗ =+,0,−.

Moreover, since all the mentioned 𝔤-subalgebras are invariant by complex conjugation, they are also the complexification of the following ϕ-invariant subalgebras of 𝔤

gϕ=g¯ϕ¯∩g,kϕ=k¯ϕ¯∩g, and g∗=g¯∗∩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⁺, 𝔤⁰ and 𝔤⁻ satisfies the inequalities

|ϕm(X)| ≥cμ−m|X|foranyX∈g+andm∈N,

and

|ϕm(Y)| ≤c−1μm|Y|foranyY∈g−andm∈N,

for some c≥ 1 and μ ∈ (0,1).

Furthermore, for all a > 0 and Z ∈ 𝔤⁰ it holds that

|ϕm(Z)|μa|m|→0 as m→±∞.

In the sequel we prove that any linear map commuting with two endomorphisms preserves their associated decompositions.

Proposition 2.4

For i = 1,2, let ϕ_i:𝔤_i→ 𝔤_i an endomorphism of the Lie algebra 𝔤_i over a closed field. Assume that f:𝔤₁→ 𝔤₂ is a surjective linear map such that f∘ ϕ₁ = ϕ₂∘ f. Hence, it holds

f(gϕ1)=gϕ2,f(kϕ1)=kϕ2,f(g1+)=g2+,f(g10)=g20andf(g1−)=g2−.

Proof

Let α be an eigenvalue of ϕ₁ and X ∈ 𝔤_α. There exists n≥ 1 such that (ϕ₁−α)ⁿX = 0. By the commutating property, we get

(ϕ2−α)nf(X)=f(ϕ1−α)nX=f(0)=0.

Consequently, f((𝔤₁)_α)⊂ (𝔤₂)_α, where (𝔤₂)_α = {0} if α is not an eigenvalue of ϕ₂. In particular we obtain

f(gϕ1)⊂gϕ2,f(kϕ1)⊂kϕ2,f(g1+)⊂g2+,f(g10)⊂g20 and f(g1−)⊂g2−.

Since for i = 1,2, 𝔤_i = 𝔤_{ϕ_i}⊕ 𝔨_{ϕ_i} and f is a surjective linear map, we must have

f(gϕ1)=gϕ2 and f(kϕ1)=kϕ2.

By the restriction of f to 𝔤_ϕ₁ we recover all the equalities ending the proof. □

Proposition 2.4 is still true for the real case.

Corollary 2.5

For i = 1,2, let 𝔤_i be real algebras and ϕ_i: 𝔤_i→ 𝔤_i an endomorphism. If f: 𝔤₁→ 𝔤₂ is a surjective linear map such that f∘ ϕ₁ = ϕ₂∘ f, the same equalities as in Proposition 2.4 hold.

Proof

The proof follows by considering the complexification of 𝔤_i, i = 1,2 and the complex extensions of ϕ₁, ϕ₂ and f. Then, we apply Proposition 2.4. □

3 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,

φ(gh)=φ(g)φ(h) for any g,h∈G.

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.

Definition 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≥ 1 and μ ∈ (0,1) such that

|(dφ)emX| ≤c−1μm|X|,foranym∈Z+,X∈g

we say that φ is contracting. On the other hand, if

|(dφ)emX| ≥cμ−m|X|,foranym∈Z+,X∈g

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,

HandKareclosed⇔H∩Kisadiscretesubgroup.

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) = 0. 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

0∈U⊂h, 0∈V⊂k and e∈W⊂G

such that the map

f:U×V→W defined by f(X,Y)=eXeY

is a diffeomorphism.

Without loss of generality, we can assume that W is small enough in order to obtain

W∩(H∩K)={e}.

In particular, if g = xy where x ∈ e^U⊂ H, y ∈ e^V⊂ K and g ∈ W ∩ H, we get

K∋y=x−1g∈H⇒y∈W∩(H∩K)={e}.

Thus, H ∩ W = e^U = f(U × {0}). Therefore, H ∩ W is closed in W since U × {0} is closed in U × V. As a consequence

H∩W=cl(H)∩W.

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. □

Definition 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, G_φ, K_φ, G⁺, G⁰, G⁻, G^+,0 and G^−,0 associated with the Lie subalgebras 𝔤_ϕ, 𝔨_ϕ, 𝔤⁺, 𝔤⁰, 𝔤⁻, 𝔤^+,0 and 𝔤^−,0, respectively.

The subgroups G⁺, G⁰ 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.

Proposition 3.4

It holds:

All the dynamical subgroups are φ-invariant
There exists a natural number d such that the subgroup K_φ = ker(φ^d)₀ is normal. Moreover,
G=GφKφandGφ=Im(φd)
The restriction of φ is expanding on G⁺ and contracting on G⁻
If G_φ is a solvable Lie group it holds that
Gφ=G+,0G−=G−,0G+=G+G0G− (3)
If G_φ is semisimple and G⁰ is compact, then G_φ = G⁰. Therefore, if G is any connected Lie group such that G⁰ is compact, then G_φ has also the decomposition (3).

Proof

It is well known that the following diagram is commutative,
g⟶ϕ=(dφ)ehexpg↓↪↓exphG⟶φH_¯

Since the Lie subgroups are connected, their φ-invariance follows directly from the ϕ-invariance of their own Lie algebras.
K_φ and ker (φ^d)₀ are connected Lie subgroups with the same Lie algebra
kφ=ker⁡(ϕd).

So, the desired equality follows.

Moreover, since ker (φ^d) is a normal subgroup of G, its connected component of the identity K_φ is also normal. In particular, the product G_φK_φ is a connected subgroup of G with Lie algebra 𝔤_ϕ⊕ 𝔨_ϕ = 𝔤. Therefore, by uniqueness we get G = G_φK_φ. From this G-decomposition and the φ-invariance of G_φ we obtain
Im(φd)=φd(G)=φd(Gφ)φd(Kφ)⊂Gφ.

On the other hand, since ϕ restricted to 𝔤_ϕ is an automorphism, it turns out that
eX=eϕdϕ|gϕ−d(X)=φdeϕ|gϕ−d(X)∈Im(φd), for all X∈gϕ.

Consequently, G_φ⊂ Im(φ^d) which concludes the proof
Follows directly by the definition of G⁺ and G⁻ and by Remark 2.3.
For the decomposition G = G_φK_φ one can easily show that
G+,0=G+G0=G0G+.

Thus, G^−,0 = G⁻G⁰ = G⁰G⁻. Hence, in order to prove the result it is enough to show that
Gφ=G+,0G−.

We prove it by induction on the dimension of G_φ.
1. If dim(G_φ) = 1 the group G_φ is Abelian and the result is certainly true
2. Let us assume that the result holds for any endomorphism φ such that G_φ is solvable with dim(G_φ) < n.
3. Consider a φ-endomorphism of G with G_φ solvable and dim(G_φ) = n.
The assumption of G_φ solvable implies that there exists a nontrivial closed normal Lie subgroup B_φ of G_φ which is Abelian and φ-invariant, (see for instance the proof in Proposition 2.9 of [1]). By considering the homogeneous space H_φ = G_φ/B_φ we obtain a connected solvable Lie group H_φ such that
dim⁡(Hφ)=dim⁡(Gφ)−dim⁡(Bφ)<n.

Moreover, the canonical projection π :G_φ → H_φ induces on H_φ a well-defined surjective endomophism φ͠ given by φ͠(π(g)) = π (φ (g)).

By the induction hypothesis we obtain H_φ = H^+,0H⁻. By taking derivative
ϕ~∘(dπ)e=(dπ)e∘ϕ.

Therefore, Proposition 2.4 and the fact that all the subgroups are connected give us
π(G+,0)=H+,0 and π(G−)=H−.

Consequently,
Hφ=π(G+,0G−) and so Gφ=G+,0G−Bφ.

The Lie subgroup B_φ is Abelian, hence B_φ = B^+,0B⁻ with B^+,0⊂ G^+,0 and B⁻⊂ G⁻. But, B is also normal, so
G=G+,0G−Bφ=G+,0BφG−=G+,0B+,0B−G−=G+,0G−

which ends the proof of item 4.
Let us start by proving that the second assertion is implied by the first one. We know that R_φ is φ-invariant. As before, we obtain an induced surjective endomorphism φ͠ on G_φ/R_φ such that
Gφ/Rφ0=π(G0), where π:Gφ→Gφ/Rφ

is the canonical projection.

But, G_φ/R_φ is semisimple and π(G⁰) is compact, therefore
π(G0)=Gφ/Rφ0=Gφ/Rφ hence Gφ=G0Rφ.

Moreover, R_φ is a solvable Lie subgroup which by item 4. decomposes as R_φ = R^+,0R⁻. Finally,
Gφ=G0Rφ=G0R+,0R−⊂G+,0G−⊂Gφ

as stated.

Now, assume that G_φ is semisimple and G⁰ 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|gϕk=Ad(g) for some g ∈ G_φ. It follows that
gAd(g)+=g+,gAd(g)0=g0 and gAd(g)−=g−.

Now, because G_φ is semisimple, there exists an Iwasawa decomposition G_φ = 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,
1. g+=gAd(g)+ is the sum of eigenspaces with positive eigenvalues of Ad(a)
2. g−=gAd(g)− is the sum of eigenspaces with negative eigenvalues of Ad(a), and
3. g0=gAd(g)0 = ker(Ad(a)).
Furthermore, the subgroup A is a simply connected Abelian Lie group and A ⊂ G⁰. By the compactness hypothesis of G⁰ we must have a = e. So, 𝔤⁺ = 𝔤⁻ = {0} implying that G⁰ = G as stated. □

Definition 3.5

Let φ be an endomorphism of the Lie group G. We say that φ decomposes G if G_φ satisfy (3), i.e.,

Gφ=G+,0G−=G−,0G+=G+G0G−.

Let us assume that φ restricted to G_φ is in fact an automorphism. From 2.3 we get that for any right (left) invariant Riemannian metric ϱ

ϱ(φn(x),e)≤c−1μnϱ(x,e), for any x∈G−,n∈N, and (4)

ϱ(φn(y),e)≥cμ−nϱ(y,e), for any y∈G+,n∈N. (5)

Moreover, for any a > 0,

ϱ(φn(z),e)μa|n|→0,n→±∞ for any z∈G0. (6)

These facts bring topological consequences on the induced subgroups.

Proposition 3.6

Suppose that φ restricted to G_φ is an automorphism in the induced topology of G. Then,

G^+,0 ∩ G⁻ = G⁺ ∩ G⁻ = G⁰ ∩ G⁻ = G^−,0 ∩ G⁺ = G⁺ ∩ G⁰ = {e}
The dynamical subgroups induced by φ are closed in G
For n≥ d, ker (φⁿ) = K_φ. In particular, ker(φⁿ) is connected.

Proof

Since other cases are analogous, we just show G^−,0 ∩ G⁺ = {e}

Let y ∈ G^−,0 ∩ G⁺, x ∈ G⁻ and z ∈ G⁰ such that y = xz. The right invariance of the metric gives

ϱ(φn(y),e)=ϱ(φn(x)φn(z),e)≤ϱ(φn(x),e)+ϱ(φn(z),e).

Since y ∈ G⁺ and x ∈ G⁻, from (5) and (4), it follows that

cμ−nϱ(y,e)≤ϱ(φn(z),e)+c−1μnϱ(x,e).

Hence,

ϱ(y,e)≤c−1ϱ(φn(z),e)μn+c−2μ2nϱ(x,e).

Because z ∈ G⁰, equation (6) implies that in the last inequality, each term on the right hand goes to zero as n → + ∞. Therefore,

ϱ(y,e)=0⇒G−,0∩G+={e}

as desired.

2. For n ∈ ℕ we know that

Gφ∩ker⁡(φn)=ker⁡(φ|Gφn). (7)

By the assumption, φ|_{G_φ} is an automorphism. From that we get G_φ ∩ K_φ = {e}. Then, Proposition 3.2 implies that G_φ is closed in G. Using again Proposition 3.2 and item 1., we also obtain that G⁺,G⁰,G⁻,G^+,0 and G^−,0 are closed subgroups of G_φ. As a consequence, they are also closed subgroups of G.

3. Let n≥ d, x ∈ ker (φⁿ) and consider the decomposition x = gk with g ∈ G_φ and k ∈ K_φ given by item 2. of Proposition 3.4. Hence,

Gφ∋g=xk−1∈ker⁡(φn)Kφ⊂ker⁡(φn).

Therefore, (7) implies x = k ∈ K_φ, concluding the proof. □

In the sequel we prove that some strong topological property of G are also maintained by φ.

Proposition 3.7

Let φ be an endomorphism of a simply connected Lie group G. Then, G_φ and K_φ are simply connected. Moreover, the restriction of φ to G_φ is an automorphism.

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

G/Kφ→G/ker⁡(φd)

is a covering map, Proposition 6.12 of [6] implies that K_φ = ker (φ^d).

Moreover, from the decomposition G = G_φK_φ we obtain that φ^d:G → G_φ is a surjective continuous homomorphism. Thus, by the canonical isomorphism theorem it follows that G_φ and G/kerφd are isomorphic, showing in particular that G_φ is simply connected.

Knowing that ϕ restricted to 𝔤_ϕ is an automorphism and G_φ is simply connected, we must have that φ restricted to G_φ 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 G_φ is contained in its center. This implies the decomposition of the group when G_φ is compact.

Theorem 3.9

Let G be a Lie group and φ an endomorphism of G. If H⊂ G_φ is a φ-invariant compact subgroup, then H⁺,H⁻⊂ Z_H the center of H. In particular, if G_φ is compact G is decomposable.

Proof

Since H is a compact subgroup it is in particular reducible and so H = Z_HH^′, where H^′ is the derivated subgroup. Since both, H^′ and Z_H are φ-invariant subgroups and H^′ is semisimple, item 5. of Proposition 3.4 implies that (H^′)⁰ and by the φ-invariance, H⁺ and H⁻ are subsets of Z_H.

If G_φ is compact, we get

Gφ′⊂G0 and so Gφ=ZGφG0.

Since Z_{G_φ} is solvable subgroup, item 4. of Proposition 3.4 implies that Z_{G_φ} is contained in G^+,0G⁻ 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 |_{G_φ} is an automorphism, then any fixed point of φ is contained in G⁰.

Proof

Since φ|_{G_φ} is an automorphism we know that G_φ ∩ K_φ = {e}. Therefore, the decomposition of x ∈ G as x = gk with g ∈ G_φ and k ∈ K_φ is unique. Thus, x = gk is a fixed point of φ if and only if g and k are fixed points of φ. Since φ^d(k) = e we must have k = e. So, we only have to analyze the case where g ∈G_φ is a fixed point.

By Proposition 3.4 item 4., we know that

g=g1g2g3 with g1∈G+, g2∈G0 and g3∈G−

Moreover, by Proposition 3.6 item 1. and the φ-invariance of the subgroups it turns out that g is a fixed point of φ if and only if g_i is a fixed point of φ for i = 1,2,3. However, since g₁ ∈ G⁺, from the equation (5) we obtain

ϱ(g1,e)=ϱ(φn(g1),e)≥cμ−nϱ(g1,e), for any n∈N

which happens if and only if g₁ = e.

In the same way, by using the fact g₃ ∈ G⁻ is a fixed point and the equation (4), we get that g₂ = e showing that x = g₂ ∈ G⁰ 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.

Example 3.12

Consider G = Sl(n,ℝ), the group of the invertible matrices with determinant equals to one. If A = diag(a₁ > … > a_d) is a matrix with trace equal to zero we can induce the automorphism φ_A: G → G defined by

φA(B)=eA B e−A

where e^A is the exponential of the square matrix A.

An easy calculation shows that in this case

G+={B∈G:B is upper triangular with 1's in the main diagonal},G−={B∈G:B is lower triangular with 1's in the main diagonal}

and

G0={B∈G;B is diagonal}.

Acknowledgement

The first author was supported by Proyecto Fondecyt n^o 1150292, Conicyt. The second author was supported by Proyecto Fondecyt n^o 1151159, Conicyt, Chile. The third author was supported by Fapesp grant n 2016/11135-2.

The first and third author would like to thank the Centro de Estudios Científicos, CECs in Valdivia, Chile, through Prof. Jorge Zanelli, to provide an excellent work environment for developing part of this article.

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Received: 2017-4-10

Accepted: 2017-8-17

Published Online: 2017-12-29

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