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Open Math. 2017; 15: 1477–1486
Open Mathematics
Research Article
Víctor Ayala*, Heriberto Román-Flores, and Adriano Da Silva
The dynamic of a Lie group endomorphism
https://doi.org/10.1515/math-2017-0120 Received April 10, 2017; accepted August 17, 2017.
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: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 Rd
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.
*Corresponding Author: Víctor Ayala: Universidad de Tarapacá, Instituto de Alta Investigación, Casilla 7D, Arica. Sede
Esmeralda Iquique, Chile, E-mail: [email protected]
Heriberto Román-Flores: Universidad de Tarapacá, Instituto de Alta Investigación, Casilla 7D, Arica, Chile,
E-mail: [email protected]
Adriano Da Silva: Instituto de Matemática, Universidade Estadual de Campinas, Cx. Postal 6065, 13.081-970 Campinas-SP,
Brasil, E-mail: [email protected]
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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, for some n ≥ 1}.
If β is also an eigenvalue of ϕ then
[gα, gβ]⊂gαβ, (1)
where gαβ={0}if αβ is not an eigenvalue of ϕ.
Proof. In order to decomposes the ϕ-eigenspace gλin its Jordan components, we consider r linear
indepen-dent vectors Z1, . . . Zr∈gλsuch 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]∈gαβ. 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 gαβ
which implies
(ϕ − αβ)[Xi, Yj]∈ker (ϕ − αβ)n for some n ≥ 1.
Consequently,
(ϕ − αβ)n+1[Xi, Yj] = 0
showing that [Xi, Yj]∈gαβand 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 ϕSsemisimple and ϕNunipotent. Then ϕSand ϕNare also g-automorphisms.
Proof. Without lost of generality we can assume that the field of the scalars is algebraically closed. To prove that ϕS
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 ϕS
satisfies the property of automorphisms for X∈gα, Y∈gβand α, β eigenvalues of ϕ. From Proposition 1, [X, Y]∈gαβ. On the other
hand, since the eigenspaces of ϕ and ϕS
coincide, we get
The dynamic of a Lie group endomorphism | 1479
showing that ϕS
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= M α̸=0 gα, kϕ= ker(ϕd) g+=M |α|>1 gα, g0= M α:|α|=1 gα g− = M 0<|α|<1 gα, g+,0= g+⊕g0 and g−,0= g−⊕g0.
Also, we denote by gϕ= g+⊕g0⊕g−and g = gϕ⊕kϕ. 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 ¯ϕinduced by ϕ we can define the subalgebras
¯gϕ¯, ¯kϕ¯, ¯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= ¯gϕ¯ ∩g, kϕ= ¯kϕ¯ ∩g, and g*= ¯g*∩g
with g+and g−nilpotent Lie subalgebras. We should notice that the equality k
ϕ= ker(ϕd) is still true.
Remark 2.3. In the real or complex case the restriction of ϕ|gϕ is an automorphism of gϕ. Furthermore, the
restriction of ϕ to the Lie subalgebras g+, g0and g−satisfies the inequalities
|ϕm(X)| ≥ cµ−m|X| for any X∈g+and m∈N,
and
|ϕm(Y)| ≤ c−1µm|Y| for any Y∈g−and m∈N,
for some c≥ 1 and µ∈(0, 1).
Furthermore, for all a > 0 and Z∈g0it 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 ϕi: gi→ gian endomorphism of the Lie algebra giover a closed field. Assume
that f : g1→ g2is a surjective linear map such that f◦ϕ1= ϕ2◦f . Hence, it holds f(gϕ1) = gϕ2, f(kϕ1) = kϕ2,
f(g+1) = g+2, f (g01) = g02 and f(g−1) = g−2.
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Proof. Let α be an eigenvalue of ϕ1 and X ∈ gα. 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 (g+1)⊂g+2, f (g01)⊂g20 and f (g−1)⊂g−2.
Since for i = 1, 2, gi= gϕi⊕kϕiand f is a surjective linear map, we must have
f(gϕ1) = gϕ2and f (kϕ1) = kϕ2. By the restriction of f to gϕ1we 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 gibe real algebras and ϕi : gi → gian endomorphism. If f : g1 → g2is 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 gi, i = 1, 2 and the complex extensions of
ϕ1, ϕ2and 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)meX| ≤ c−1µm|X|, for any m∈Z+, X∈g
we say that φ is contracting. On the other hand, if
|(dφ)meX| ≥ cµ−m|X|, for any m∈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,
The dynamic of a Lie group endomorphism | 1481 Proof. If H and K are closed subgroups then H∩Kis 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∩Kis 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⊂kand 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∈eU⊂H
, y∈eV ⊂K
and g∈W∩H, we get K3y= x−1g∈H⇒y∈W∩(H∩K) ={e}.
Thus, H∩W = eU = 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⊂Gis a φ-invariant Lie subgroup, the restriction φ|His 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+, G0, G−, G+,0 and G−,0 associated with the Lie subalgebras gϕ, kϕ, 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:
1. All the dynamical subgroups are φ-invariant
2. There exists a natural number d such that the subgroup Kφ= ker(φd)0is normal. Moreover,
G= GφKφ and Gφ= Im(φd)
3. The restriction of φ is expanding on G+and contracting on G− 4. If Gφis a solvable Lie group it holds that
Gφ= G+,0G−= G−,0G+= G+G0G− (3)
5. If Gφis semisimple and G0is compact, then Gφ= G0. Therefore, if G is any connected Lie group such that
G0is compact, then Gφhas also the decomposition (3).
Proof. 1. It is well known that the following diagram is commutative,
g ϕ= (dφ)e −→ h expg↓ ,→ ↓exph G φ −→ H
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Since the Lie subgroups are connected, their φ-invariance follows directly from the ϕ-invariance of their own Lie algebras.
2. Kφand ker(φd)0are connected Lie subgroups with the same Lie algebra
kφ= ker(ϕd).
So, the desired equality follows. Moreover, since ker(φd
) is a normal subgroup of G, its connected component of the identity Kφis also
normal. In particular, the product GφKφis a connected subgroup of G with Lie algebra gϕ⊕kϕ= g. Therefore,
by uniqueness we get G = GφKφ. From this G-decomposition and the φ-invariance of Gφwe obtain
Im(φd) = φd(G) = φd(Gφ)φd(Kφ)⊂Gφ.
On the other hand, since ϕ restricted to gϕis an automorphism, it turns out that
eX= eϕ d ϕ|−dgϕ(X) = φd eϕ|−dgϕ(X)
∈Im(φd), for all X∈gϕ.
Consequently, Gφ⊂Im(φd) which concludes the proof
3. Follows directly by the definition of G+and G−and by Remark 2.3. 4. For the decomposition G = GφKφone 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 Gφ.
i) If dim(Gφ) = 1 the group Gφis Abelian and the result is certainly true
ii) Let us assume that the result holds for any endomorphism φ such that Gφis solvable with dim(Gφ) < n.
iii) Consider a φ-endomorphism of G with Gφsolvable and dim(Gφ) = n.
The assumption of Gφsolvable implies that there exists a nontrivial closed normal Lie subgroup Bφof
Gφwhich is Abelian and φ-invariant, (see for instance the proof in Proposition 2.9 of [1]). By considering the
homogeneous space Hφ= Gφ/Bφwe obtain a connected solvable Lie group Hφsuch that
dim(Hφ) = dim(Gφ) − dim(Bφ) < n.
Moreover, the canonical projection π : Gφ → Hφinduces on Hφa well-defined surjective endomophismφe given byφe(π(g)) = π(φ(g)).
By the induction hypothesis we obtain Hφ= H+,0H−. By taking derivative
e
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+,0and π(G−) = H−. Consequently,
Hφ= π(G+,0G−) and so Gφ= G+,0G−Bφ.
The Lie subgroup Bφis Abelian, hence Bφ= B+,0B−with B+,0⊂G+,0and B−⊂G−. But, B is also normal, so
G= G+,0G−Bφ= G+,0BφG−= G+,0B+,0B−G−= G+,0G−
The dynamic of a Lie group endomorphism | 1483
5. Let us start by proving that the second assertion is implied by the first one. We know that Rφis
φ-invariant. As before, we obtain an induced surjective endomorphismφeon Gφ/Rφsuch that
Gφ/Rφ
0
= π(G0), where π : Gφ→ Gφ/Rφ
is the canonical projection.
But, Gφ/Rφis semisimple and π(G0) is compact, therefore
π(G0) = Gφ/Rφ
0
= Gφ/Rφhence Gφ= G0Rφ.
Moreover, Rφis a solvable Lie subgroup which by item 4. decomposes as Rφ= R+,0R−. Finally,
Gφ= G0Rφ= G0R+,0R−⊂G+,0G−⊂Gφ
as stated.
Now, assume that Gφis semisimple and G0 is a compact subgroup. Since ϕ|gϕ is an automorphism,
Theorem 5.4 of [4] implies that there exists k∈N such that ϕ|k
gϕ = Ad(g) for some g∈Gφ. It follows that
g+Ad(g)= g+, g0Ad(g)= g0 and g−Ad(g)= g−.
Now, because Gφis semisimple, there exists an Iwasawa decomposition Gφ = KAN and elements a ∈ A,
u∈Kand n∈Nsuch 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) g0= g0Ad(g)= ker(Ad(a)).
Furthermore, the subgroup A is a simply connected Abelian Lie group and A⊂G0. By the compactness hypothesis of G0we 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 Gφsatisfy (3),
i.e.,
Gφ= G+,0G−= G−,0G+= G+G0G−.
Let us assume that φ restricted to Gφis 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 Gφis an automorphism in the induced topology of G. Then,
1. G+,0∩G−= G+∩G−= G0∩G−= G−,0∩G+= G+∩G0={e}
2. The dynamical subgroups induced by φ are closed in G 3. For n≥ d, ker(φn
) = Kφ. In particular,ker(φn) is connected.
Proof. Since other cases are analogous, we just show G−,0∩G+={e}
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Let y∈G−,0∩G+, x∈G−and z∈G0such 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|nGφ). (7)
By the assumption, φ|Gφis an automorphism. From that we get Gφ∩Kφ={e}. Then, Proposition 3.2 implies
that Gφis closed in G. Using again Proposition 3.2 and item 1., we also obtain that G+, G0, G−, G+,0and G−,0
are closed subgroups of Gφ. 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∈Gφand k∈Kφgiven by item 2.
of Proposition 3.4. Hence,
Gφ3g= xk−1∈ker(φn)Kφ⊂ker(φn).
Therefore, (7) implies x = k∈Kφ, 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, Gφand Kφare simply
connected. Moreover, the restriction of φ to Gφis an automorphism.
Proof. By Proposition III.3.17 of [5] both, the subgroup ker(φd
) and the quotient G/ ker(φd
) 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 Kφ= ker(φd).
Moreover, from the decomposition G = GφKφwe obtain that φd : G → Gφis a surjective continuous
homomorphism. Thus, by the canonical isomorphism theorem it follows that Gφ and G/ ker
φd are
isomorphic, showing in particular that Gφis simply connected.
Knowing that ϕ restricted to gϕis an automorphism and Gφis simply connected, we must have that φ
restricted to Gφis 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 Gφis compact.
Theorem 3.9. Let G be a Lie group and φ an endomorphism of G. If H⊂Gφis a φ-invariant compact subgroup,
then H+, H−⊂Z
Hthe center of H. In particular, if Gφis compact G is decomposable.
Proof. Since H is a compact subgroup it is in particular reducible and so H = ZHH′, where H′is the derivated
subgroup. Since both, H′and ZHare φ-invariant subgroups and H′is semisimple, item 5. of Proposition 3.4
implies that (H′)0and by the φ-invariance, H+and H−are subsets of Z
The dynamic of a Lie group endomorphism | 1485
If Gφis compact, we get
G′φ⊂G0and so Gφ= ZGφG 0
.
Since ZGφis solvable subgroup, item 4. of Proposition 3.4 implies that ZGφis 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|Gφis an automorphism, then
any fixed point of φ is contained in G0.
Proof. Since φ|Gφis an automorphism we know that Gφ∩Kφ ={e}. Therefore, the decomposition of x∈G
as x = gk with g∈Gφand k∈Kφis unique. Thus, x = gk is a fixed point of φ if and only if g and k are fixed
points of φ. Since φd
(k) = e we must have k = e. So, we only have to analyze the case where g∈Gφis a fixed
point.
By Proposition 3.4 item 4., we know that
g= g1g2g3with g1∈G+, g2∈G0and 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 giis 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∈G0as we stand.
Examples
Example 3.11. Take G = Rd
, A∈gl(d) a d × d matrix and the endomorphism φAof G given by φA(x) = Ax. In
this case, the subgroups induced by φAare 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 > . . . > ad) is a matrix with trace equal to zero we can induce the automorphism φA : G → G
defined by
φA(B) = eABe−A
whereeAis 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 no
1150292, Conicyt. The second author was supported by Proyecto Fondecyt no
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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