Entropy of Endomorphisms of Lie Groups

We show, when $G$ is a nilpotent or reductive Lie group, that the entropy of any surjective endomorphism coincides with the entropy of its restriction to the toral component of the center of $G$. In particular, if $G$ is a semi-simple Lie group, the entropy of any surjective endomorphism always vanishes. Since every compact group is reductive, the formula for the entropy of a endomorphism of a compact group reduces to the formula for the entropy of an endomorphism of a torus.


Introduction
In [3] and [10], it is introduced a topological notion of entropy for proper maps on locally compact separable metrizable spaces. It is shown there that this topological entropy coincides with the supremum of the Kolmogorov-Sinai's entropies and also with the minimum of the Bowen's entropies. Using this variational principle, it is also shown in [10] that the topological entropy of a linear isomorphism of a finite dimensional vector space always vanishes. This shows that the Bowen's formula (see [2]) for the entropy of an endomorphism of a non-compact Lie Group gives just an upper bound for its topological entropy. At the end of [10], using again the variational principle, it is proved that the topological entropy of the endomorphism φ(z) = z 2 , where z ∈ C * , is equal to the topological entropy of its restriction to S 1 ⊂ C * . Using the same kind of reasoning presented there, one can show that the same result holds for any endomorphism φ(z) = z n , where n ∈ N.
These examples led us to the following conjecture. Since every connected abelian Lie group G is isomorphic to the product of a torus (the toral component of G) by a finite dimensional vector space and since the component of C * is just S 1 , the topological entropy of a proper endomorphism of a connected abelian Lie group G should be just the topological entropy of its restriction to the toral component of G.
More generally, for a connected Lie group G, we can consider the toral component of the identity component of the center of G, denoted by T (G) and called in the present paper just the toral component of G. From now on we will use the term entropy to refer to the topological entropy. In this paper, when G is a nilpotent or reductive Lie group, we show that the entropy of any surjective endomorphism coincides with the entropy of its restriction to T (G). Since every compact group is reductive, these results shed new light in the Bowen's formula even in the compact case. In fact, the formula for the entropy of a endomorphism of a compact group reduces to the formula for the entropy of an endomorphism of a torus (see [12]). In particular, if G is a compact semi-simple Lie group, the entropy of any surjective endomorphism always vanishes. One may wonder, for a general connected Lie group G, that the entropy of a surjective endomorphism coincides with the entropy of its restriction to T (G). At the end of this article, we present arguments that suggest that the general case is slightly different.
The paper is structured in sections corresponding to the classes of Lie groups for which we compute the topological entropy of their respective endomorphisms. But first, in a preliminary section, we collect some results used in the remaining sections. Since the concept of topological entropy given in [10] requires properness, we always start the remaining sections considering surjective endomorphisms and showing that these endomorphisms are in fact proper maps. In Section 3, we treat the abelian case and shows the above conjecture about entropy. Section 4 treats the nilpotent case. As in the abelian case, we conclude that the entropy of an endomorphism of G is the entropy of its restriction to T (G). In Section 5, the semi-simple case is considered. Using the relation in a linear semi-simple Lie group between endomorphisms, conjugations and linear maps, the recurrent set of a given conjugation is characterized. Then, the concept of Li-Yorke pairs is used to demonstrate that the entropy of endomorphisms of semi-simple Lie groups (even not linear ones) always vanishes. In particular, since T (G) is trivial in this case, the entropy of an endomorphism coincides with the entropy of its restriction to T (G). In Section 6, we compute the entropy of endomorphisms of reductive groups, by using the previous results for the abelian and semi-simple cases. And finally, in Section 7, we end the paper with some remarks about the general case.

Preliminaries
In this section, we collect some facts that are used in the next sections. In general, we just state the facts with references, but we do not present much details of the theory involved. From [10], Theorem 3.2, we have the following variational principle, where h (φ) is the topological entropy defined in [10]. That is, h (φ) is the supremum of the entropies taken over all admissible covers.
Proposition 2.1. Let X be a locally compact metrizable separable space and φ : X → X a proper map. Then As a consequence of Proposition 2.1 we present the following formula for the entropy of products, which is well known in the compact case.
Proposition 2.2. Let X and Y be locally compact metrizable separable spaces and φ : X → X, ψ : Y → Y proper maps. Then, Proof. From the variational principle, we have that On the other hand, by the variational principle there exist metrics d 1 and d 2 such that But Proposition 2.2.15 of [4] states that for the maximum distance d, Now, the variational principle leads us to The following proposition, which is used in section 6, also generalizes a simple result from the compact case (see Proposition 2.1 of [10]). Proposition 2.3. Let X and Y be locally compact metrizable separable spaces, and consider the diagram where φ, ψ and π are proper surjective maps. Then, The following result is proved in Corollary 16 of [2].

Proposition 2.4 (Bowen's Formula).
If φ is an endomorphism of a Lie group G and d is a right invariant distance, then where λ are the eigenvalues of φ ′ , the differential of φ at the identity, counted with multiplicity.
The following result about compact principal bundles is proved in Theorem 19 of [2].
A tool we used in order to show that certain systems have zero entropy is the concept of Li-Yorke pairs. In this paper, we introduce a topological definition of a Li-Yorke pair, which is more adjusted to the non-compact case then the usual definition wich uses lim sup and lim inf of the distances between φ n (x) and φ n (y) (see page 2 of [1]). Definition 2.6 (Li-Yorke pair). Given a continuous function φ : We shall prove that positive entropy implies the existence of Li-Yorke pairs. A stronger version could be demonstrated along the lines of Theorem 2.3 in [1], but since we do not need that level of generality, we shall prove this simpler version.
Definition 2.7 (Pinsker σ-algebra and product measure). For a φ-invariant measure µ, the Pinsker σ-algebra P, is greatest sub-σ-algebra such that h µ| P (φ) = 0. The product of µ with itself over P, denoted by λ µ , is the measure on X × X such that, for any measurable sets A, B ⊂ X, [6]). Also, notice that if µ(A) = 0, then A ∈ P.
Proof. Since λ µ is ergodic and ∆ is φ × φ-invariant, it is enough to prove that λ µ (∆) = 1. Suppose on the contrary, that λ µ (∆) = 1. Then, for any measurable B ⊂ X, And since the conditional expectation of a non-negative function is non-negative, we have that Then, A ∈ P. Now, where the last equality is due to the fact that when restricted to A, E (1 B |P) = 0, and therefore, according to equation (1), Using those relations, since P is a sub-σ-algebra containing A, and containing also all measurable sets with null µ measure, we conclude that B ∈ P. But since B was arbitrary, this would imply that h µ (φ) = 0. The contradiction implies that λ µ (∆) = 1.
Proof. If (z, z) ∈ Z ×Z is not in the support of λ µ , then, there exists an open neighborhood U of (z, z), such that λ µ (U) = 0. In this case, there exists an open neighborhood A of z, such that z ∈ A × A ⊂ U. Then, But this contradicts the fact that z is in the support of µ. Proposition 2.10. Let X be a locally compact metrizable separable space and φ : X → X a proper map. If h (φ) > 0, then there exists a Li-Yorke pair for this system.
Proof. The fact that h (φ) > 0 implies, using Proposition 2.1, that there exists a φ-ergodic measure µ, such that Let Z = supp (µ) and W = supp (λ µ ). For each non-empty open set U of W in its relative topology, let The ergodicity of λ µ implies that λ µ (W U ) = 1. Therefore, if β is an enumerable basis for the topology of W , then λ µ (W β ) = 1 for Together with Lemma 2.8, this implies that In particular, for every point (a, b) ∈ W β , with a = b, there is a sequence n k such that (T n k a, T n k b) → (a, b). And also, by Lemma 2.9, if we pick c ∈ Z, then (c, c) ∈ W . And therefore, there is a sub-sequence m k such that (T m k a, T m k b) → (c, c). That is, any such pair (a, b) is a Li-Yorke pair.
The following lemma is a simple topological fact.

The Abelian Case
In this section, we determine the entropy of a surjective endomorphism φ of connected abelian Lie group G. In this case, the exponential map is a surjective group homomorphism with discrete kernel. Therefore, G can be identified with T (G) × R q , where T (G) is isomorphic to the p-dimensional torus R p /Z p and the group operation is addition. The endomorphisms of G can be identified with linear maps of the form where T : R p → R p is a linear map that leaves Z p invariant, and S : R q → R q is a linear isomorphism. Notice that for (x, y) ∈ T (G) × R q , the action of φ have the form φ n (x, y) = ( * , S n y).
But since S and S E coincide in cl (R(S)), we have that φ and T * 0 S E coincide in T (G)×cl (R(S)). And since S E has only eigenvalues with modulus 1, Bowen's formula and the variational principle gives that where d is an invariant distance. That is, the topological entropy of φ is just the topological entropy of φ restricted to its toral component.

Properness
We start proving that every surjective endomorphism φ of a connected nilpotent Lie group G is a proper map. Since the φ ′ is a surjective linear endomorphism of the Lie algebra, it is a linear isomorphism and thus a proper map. If the G is simply-connected nilpotent, we have φ is conjugated to its differential at the identity φ ′ and the following diagram commutes Thus φ is an automorphism and a proper map. We can reduce the general connected nilpotent case to the simply-connected one. But first we need the following lemmas. Proof. Let π : G → G be the universal covering of G. We have that G is isomorphic to G/ ker(π). Besides this, we have that Z( G) isomorphic to a finite dimensional vector space and that Therefore Z(G) 0 is isomorphic to Z( G)/ ker(π) and thus T (G) is isomorphic to T / ker(π), where T = π −1 (T (G))) is isomorphic to a vector subspace. On the other hand, we have that which shows that G/T (G) is simply connected, since G/ T is homeomorphic to the quotient of a vector space by a vector subspace. Proof. By Proposition 4.1, we have that G = G/T (G) is simply connected. Besides this, for any surjective endomorphism φ : G → G, there is a surjective endomorphism φ : G → G such that the following diagram commutes where π is the canonical projection. In fact, since φ (T (G)) is compact and abelian, it is necessarily contained in T (G). Thus we can define φ(π(x)) = π(φ(x)) = φ(x)T (G).
By the above discussion, since G is simply connected, we have that φ is an automorphism of G. Now, we claim that ker(φ) is a closed subset of the compact set T (G). In fact, we have that Thus we have that ker(φ) is compact and, by Lemma 2.11, we have that φ is a proper map.

Topological Entropy
In this subsection, we show that, as in the abelian case, the entropy of a surjective endomorphism of a nilpotent Lie group G coincides with the entropy of its restriction to the toral component of G. Proof. Let G = G/T (G), and denote by π : G → G/T (G) the canonical projection. By Proposition 4.1, we have that G is a simply-connected nilpotent Lie group and, as in the proof of Proposition 4.2, we can consider the induced endomorphism φ : G → G. We have that φ : G → G is conjugated to its differential at the unity φ ′ through the exponential map.
On the other hand, we have that φ ′ is a linear map and, by Proposition 4.2 in [10], that its recurrent set R( φ ′ ) is closed. We also know that there is a norm in g such that φ ′ | R( φ ′ ) is an isometry. In particular, for any closed ball B ⊂ g centered at 0, B ∩ R( φ ′ ) is compact and φ ′ -invariant. From the conjugation given by the exponential map, there is a distance in R( φ) such that any closed ball B ⊂ R( φ) centered at the unit is compact φ-invariant.
Let R = π −1 (R( φ)). Then, since R is closed, it follows that cl (R(φ)) ⊂ R. For any ε > 0, there exists an admissible covering This admissible cover can be chosen in a way that A 0 has compact complement, and A 1 , . . . , A k have compact closure, since, in a locally compact space, any admissible covering can be refined in this way. Let then B ⊂ R( φ) be a compact φ-invariant ball such that A 1 , . . . , A k all fall in B. Denoting K = π −1 (B), it follows that K is compact (since π is proper), and Therefore, On the other hand, applying Proposition 2.5 for the compact T (G)-principal bundle π| K : K → B, since φ is an isometry when restricted to B, we conclude that This way, And since ε was arbitrary, it follows that 5 The Semi-simple Case

Properness
We start proving that every surjective endomorphism φ of a connected semi-simple Lie group G is a proper map. In fact, we show that, in this case, every surjective endomorphism is an automorphism.
Proposition 5.1. Let φ : G → G be a surjective endomorphism of a semi-simple connected Lie group G. Then there is a k ∈ N such that φ k = C g for some g ∈ G, where C g is the conjugation by g. In particular, φ is an automorphism.
But since g is semi-simple, we know that there is a k ∈ N such that (φ ′ ) k is an internal endomorphism of g (see Theorem 5.4, page 423 of [7]). That is, there exists g ∈ G such that (φ ′ ) k = Ad(g) and hence Since G is generated by elements of the form exp X, it follows that φ k = C g . We have that φ is an automorphism, since C g is an automorphism.

Topological Entropy
In this subsection, we use the previous result in order to show that surjective endomorphisms of connected semi-simple Lie groups have zero entropy. Proposition 5.1, shows that some iteration of a surjective endomorphism φ of a semi-simple Lie group G is in fact a conjugation by some element of G. Thus we first consider the dynamics of conjugations. For some g ∈ G, we denote by G g the centralizer of g in G, which is the set of fixed points of the conjugation C g .
Lemma 5.2. Let G be a connected linear semi-simple Lie group, and C g : G → G the conjugation by g ∈ G. Then, where g = ehu is the multiplicative Jordan decomposition of g. In particular, C g restricted to its recurrent set is an isometry for some distance.
Proof. Notice that C g = Ad(g)| G , where Ad is the adjoint representation of Gl(d), with G ≤ Gl(d). Also, Lemma 3.6 in [11] shows that is the Jordan decomposition for Ad(g).
Since the recurrent set behaves well with respect to restrictions, we know from Proposition 2.12 that The last claim follows, since C g coincides in R(C g ) with the elliptic linear isomorphism Ad(e).
Since G is connected and semi-simple, there is k > 0 such that φ k = C g , for some g ∈ G. Therefore, it is enough to prove that h (C g ) = 0. From proposition 2.10, we know that, if h (C g ) > 0, there exists a Li-Yorke pair for C g , that is, two distinct elements a, b ∈ G, such that (C n k g (a), C n k g (b)) → (a, b) and (C m k g (a), C m k g (b)) → (c, c), for some c ∈ G. Consider C Ad(g) , and notice that Ad •C g = C Ad(g) • Ad. Now, since a, b ∈ R(C g ), we also have that Ad(a), Ad(b) ∈ R(C Ad(g) ). But C Ad(g) | R(C Ad(g) ) = C Ad(e) | R(C Ad(g) ) is an isometry for some distance in Ad(G). This way, the fact that C m k Ad(g) (Ad(a)), C m k Ad(g) (Ad(b)) converges to (Ad(c), Ad(c)) implies that Ad(a) = Ad(b).
So, we know that a = wu and b = wv for some w ∈ G and u, v ∈ Z(G). We also have that But this means that u = v. And then a = b, contradicting the fact that they are a Li-Yorke pair.
6 The Reductive Case

Properness
Let G be a connected reductive Lie group. It will be useful to consider the surjective group homomorphism where Z(G) 0 is the identity component of its center, and G ′ = [G, G] is the derived group which is connected and semi-simple. Also, G and Z(G) 0 × G ′ have the same Lie algebra z × g ′ , where g is the Lie algebra of G, z is the Lie algebra of Z(G) 0 and g ′ = [g, g] is the Lie algebra of G ′ .
Lemma 6.1. Let φ : G → G be a surjective endomorphism of a connected reductive Lie group G. Then, φ induces the surjective homomorphism since φ(Z(G) 0 ) is a connected subgroup of Z(G) containing the identity, and therefore is a subset of Z(G) 0 . It happens that φ(G ′ ) ⊂ G ′ also holds. But this implies that φ ′ (z) = z and φ ′ (g ′ ) = g ′ . And because both Z(G) 0 and G ′ are connected, we have that φ| Z(G) 0 : Z(G) 0 → Z(G) 0 and φ| G ′ : G ′ → G ′ are surjective. The commutativity of the above diagram is an immediate consequence of the fact that φ is an homomorphism. Proposition 6.2. Let φ : G → G be a surjective endomorphism of a connected reductive Lie group G. If π is a proper map, then φ is a proper map.
Proof. First observe that the endomorphism φ presented in Lemma 6.1 is proper, since it is the product of two proper maps. In fact, we have that φ| Z(G) 0 and φ| G ′ are proper endomorphisms, respectively, by Propositions 4.2 and 5.1. Considering the diagram in Lemma 6.1, we have that, if K ⊂ G is compact, then φ −1 (K) = π • φ −1 • π −1 (K) is also compact, since φ and π are proper maps.

Topological Entropy
We start computing the topological entropy of the endomorphism φ. Proof. Using Proposition 2.2, we have that The result follows, since, from the abelian and semi-simple cases, we know that h φ| Z(G) 0 = h φ| T (G) and that h (φ| G ′ ) = 0. Proof. Since G is a universal covering and since the Lie algebras of G and Z(G) 0 × G ′ coincide, the homomorphism π is a conjugation between φ and φ.
Now we consider the case where G is not homeomorphic to Z(G) 0 × G ′ . Proposition 6.5. Let φ : G → G be a surjective endomorphism of a reductive connected Lie group G. If the projection π : Z(G) 0 × G ′ → G is proper, then h (φ) = h φ| T (G) .
Proof. Consider the endomorphismφ from Lemma 6.1. Now, use Proposition 2.3 and Proposition 6.3 to conclude that which is well defined, since φ(R) = R, we would have that Thus we get the above first formula, since G/R is a connected semi-simple Lie group. Now, considering R ′ , the derived subgroup of R, and the diagram which is well defined, since φ(R ′ ) = R ′ , we would have that Since R ′ is connected nilpotent and R/R ′ is connected abelian, putting all together, we would have that We would conclude that the formula of the topological entropy of a surjective proper endomorphism of a connected Lie group would reduce to the formula of the topological entropy of a surjective endomorphism of a torus.