arXiv:0804.4244v1 [math.DS] 26 Apr 2008

Entropy and its variational principle for noncompact metric spaces

Mauro Patr?o a April 26, 2008

Abstract In the present paper, we introduce a natural extension of AKMtopological entropy for noncompact spaces and prove a variational principle which states that the topological entropy, the supremum of the measure theoretical entropies and the minimum of the metric theoretical entropies always coincide. We apply the variational principle to show that the topological entropy of automorphisms of simply connected nilpotent Lie groups always vanishes. This shows that the classical formula for the entropy of an automorphism of a noncompact Lie group is just an upper bound for its topological entropy.

AMS 2000 subject classi?cation: Primary: 37B40 37A35, Secondary: 22E25. Key words: Topological entropy, variational principle, non-compact metric spaces, automorphisms of Lie groups.

1

Introduction

Topological entropy was introduced in [1] by Adler, Konheim and MacAndrew (AKM) to study the dynamic of a continuous map T : X → X de?ned on a compact space X. They have also conjectured the well known variational principle which states that h(T ) = sup h? (T ),

?

where h(T ) is the topological entropy of T , h? (T ) is the ?-entropy of T and the supremum is taken over all T -invariant probabilities ? on X. 1

R. Bowen has extended in [2] the concept of topological entropy for noncompact metric spaces. This approach uses the concept of generator sets, which are de?ned by means of a given metric. For compact spaces this de?nition of entropy is, in fact, independent of the metric and, indeed, coincides with the AKM-topological entropy. However, for noncompact spaces Bowen’s entropy depends on the metric. In Lemma 1.5 of [4], it was proved the following variational principle for locally compact spaces sup h? (T ) = inf hd (T ),

? d

where hd (T ) is the d-entropy of T and the in?mum is taken over all metric d on X. In the present paper, we improve this result showing that the above in?mum is always attained at d, whenever d is a metric satisfying some special conditions. Theses metrics are called admissible metrics and always exist when X is a locally compact separable space. Moreover the supremum and the minimum coincide with a natural extension of AKM-topological entropy, introduced in the second section. In the third section, this variational principle is proven. In the last section, we apply the variational principle to determine the topological entropy of automorphisms of some Lie groups. First we consider linear isomorphisms of a ?nite dimensional vector space and characterize their recurrent sets in terms of their multiplicative Jordan decompositions. We show that the topological entropy of these linear isomorphisms always vanishes. This shows that the classical formula for its d-entropy, where d is the euclidian metric, is just an upper bound for its null topological entropy. Remember that this classical formula is given by hd (T ) =

|λ|>1

log |λ|,

where λ runs through the eigenvalues of the linear isomorphism T (cf. Theorem 8.14 in [7]). Thus it might be an interesting problem to calculate the topological entropy of a given automorphism φ of a general noncompact Lie group G, since the classical formula for its d-entropy, where d is some invariant metric, is as well just an upper bound for its topological entropy. We conclude this paper providing an answer for this problem when G is a simply connected nilpotent Lie group . We show that the topological entropy of its automorphisms also vanishes. 2

2

Topological entropy and admissible metrics

We start this section presenting an extension of the AKM-topological entropy introduced in [1]. Let X be a topological space and T : X → X be a proper map, i.e., T is a continuous map such that the pre-image by T of any compact set is compact. An admissible covering of X is an open and ?nite covering α of X such that, for each A ∈ α, the closure or the complement of A is compact. Given an admissible covering α of X, for every n ∈ N, we have that the set given by αn = {A0 ∩ T ?1 (A1 ) ∩ . . . ∩ T ?n (An ) : Ai ∈ α} is also an admissible covering of X, since T is a proper map. Given an admissible covering α of X, we denote by N(αn ) the smallest cardinality of all sub-coverings of αn . Exactly as in the compact case, it can be shown that the sequence log N(αn ) is subadditive, which implies the existence of the following limit 1 h(T, α) = lim log N(αn ). n→∞ n The topological entropy of the map T is thus de?ned as h(T ) = sup h(T, α),

α

where the supremum is taken over all admissible coverings α of X. We note that, when X is already compact, the above de?nition coincides with the AKM de?nition, since thus every continuous map is proper and every open and ?nite covering of X is admissible. The following result generalizes, for non-compact spaces, the relation of the topological entropies of two semi-conjugated maps. Proposition 2.1 Let T : X → X and S : Y → Y be two proper maps, where X and Y are topological spaces. If f : Y → X is a proper surjective map such that f ? S = T ? f , then we have that h(T ) ≤ h(S). Proof: Let α be an admissible covering of X. Since f is a proper map, it follows that f ?1 (α) = {f ?1 (A) : A ∈ α}

3

is an admissible covering of Y . We claim that if β is a subset of αn , then f ?1 (β) is a subset of f ?1 (α)n . In fact, B ∈ β if and only if B = A0 ∩ T ?1 (A1 ) ∩ . . . ∩ T ?n (An ) where Ai ∈ α, for each i ∈ {0, . . . , n}. Thus we have that f ?1 (B) = f ?1 (A0 ) ∩ f ?1 (T ?1 (A1 )) ∩ . . . ∩ f ?1 (T ?n (An )) = f ?1 (A0 ) ∩ S ?1 (f ?1 (A1 )) ∩ . . . ∩ S ?n (f ?1 (An )), where we used that f ?1 ? T ?i = S ?i ? T ?1 , since f ? S = T ? f . Thus it follows that f ?1 (β) ? f ?1 (α)n . Reciprocally, we claim that if γ is a subset of f ?1 (α)n , then γ = f ?1 (β), where β is some subset of αn . Proceeding analogously, if C ∈ γ, then C = f ?1 (A0 ) ∩ S ?1 (f ?1 (A1 )) ∩ . . . ∩ S ?n (f ?1 (An )) = f ?1 (A0 ∩ T ?1 (A1 ) ∩ . . . ∩ T ?n (An )), (1)

where Ai ∈ α, for each i ∈ {0, . . . , n}. Thus it follows that C = f ?1 (B), for some B ∈ αn , which implies that γ = f ?1 (β), where β is some subset of αn . If β is a sub-covering of αn , then f ?1 (β) is a sub-covering of f ?1 (α)n . Reciprocally, if γ is a sub-covering of f ?1 (α)n , then γ = f ?1 (β), where β is some sub-covering of αn . In fact, we have already known that γ = f ?1 (β), for some subset β of αn . We have to show just that β is a covering of X. But this is immediate, since γ = f ?1 (β) is a covering of Y , f is surjective and B = f (f ?1 (B)). Hence we have that N(αn ) = N(f ?1 (α)n ) and taking logarithms, dividing by n and taking limits, it follows that h(T, α) ≤ h(S, f ?1 (α)) ≤ h(S). Since α is an arbitrary admissible covering of X, we get that h(T ) ≤ h(S). Now we introduce the concept of entropy associated to some given metric in two di?erent ways. First we remember the classical de?nition, introduced in [2]. Given a metric space (X, d) and a continuous map T : X → X, we consider the metric given by dn (x, y) = max{d(T i (x), T i(y)) : 0 ≤ i ≤ n}, 4

which is equivalent to the original metric d. Given a subset Y ? X, a subset G is an (n, ε)-generator of Y if and only if, for every y ∈ Y , there exists x ∈ G such that dn (x, y) < ε. In other words, the collection of all ε-balls of dn centered at points of G is in fact a covering of Y . We denote by Gn (ε, Y ) the smallest cardinality of all (n, ε)-generators of Y and de?ne g(ε, Y ) = lim sup

n→∞

1 log Gn (ε, Y ). n

It can be proved that g(ε, Y ) is monotone with respect to ε, so we can de?ne hd (T, Y ) = lim g(ε, Y ).

ε↓0

The d-entropy of the map T is thus de?ned as hd (T ) = sup hd (T, K),

K

where the supremum is taken over all compact subsets K of X. The de?nition of d-entropy which we present in this paper is given by hd (T ) = sup hd (T, Y ),

Y

where now the supremum is taken over all subsets Y of X instead of just the compact ones. Note that here the metric d lies at the superscript. Since we have that hd (T, Y ) is monotone with respect to Y , it follows that hd (T ) = hd (T, X). We immediately notice that always hd (T ) ≤ hd (T ) and, when X is itself compact, we get the equality. Denoting Gn (ε, X) and g(ε, X), respectively, by Gn (ε) and g(ε), we get that hd (T ) = lim g(ε)

ε↓0

1 log Gn (ε). n→∞ n Our ?rst result is a generalization of the well know result which states that, if (X, d) is compact, then the topological and the metric entropies coincide. In the non-compact case, this remains true but we need to ask for some regularity in the metric. Let (X, d) be a metric space. The metric d is admissible if the following conditions are veri?ed: g(ε) = lim sup 5

and that

(1) If αδ = {B(x1 , δ), . . . , B(xk , δ)} is a covering of X, for every δ ∈ (a, b), where 0 < a < b, then there exists δε ∈ (a, b) such that αδε is admissible. (2) Every admissible covering of X has a Lebesgue number. We observe that, if (X, d) is compact, then d is automatically admissible. Proposition 2.2 Let T : X → X be a proper map, where (X, d) is a metric space. If d is an admissible metric, then h(T ) = hd (T ). Proof: First we claim that, for any admissible covering α of X, it follows that g(|α|) ≤ h(T, α), where |α| is the maximum of the diameters of A ∈ α. In fact, the elements of αn are given by A0 ∩T ?1 (A1 )∩. . .∩T ?n (An ), where Ai ∈ α. Let β be a sub-covering of αn . For each B ∈ β, take an x ∈ B and consider G the set of all such points. We claim that G is an (n, |α|)-generator set of X. In fact, let y ∈ X and take some A0 ∩T ?1 (A1 ) ∩. . .∩T ?n (An ) ∈ β containing y. Taking x ∈ G such that x ∈ A0 ∩ T ?1 (A1 ) ∩ . . . ∩ T ?n (An ), we have that d(T i(x), T i (y)) < |α|, for every i ∈ {0, 1, . . . , n}, since T i (x), T i (y) ∈ Ai . Hence, for each sub-covering β of αn , there exists an (n, |α|)-generator set of X such that Gn (|α|) ≤ #G ≤ #β. Thus we get that Gn (|α|) ≤ N(αn ). Taking logarithms, dividing by n and taking limits, it follows as claimed that g(|α|) ≤ h(T, α). Now we claim that, for all ε > 0, there exists δε ∈ (ε/2, ε) such that h(T, αδε ) ≤ g(ε/2), where αδε is an admissible covering of balls with radius equals to δε . In fact, if G = {x1 , . . . , xk } is an (n, ε/2)-generator of X, for every δ ∈ (ε/2, ε), we have that βδ = {B(xi , δ) ∩ . . . ∩ T ?n (B(T n (xi ), δ)) : xi ∈ G} is a covering of X. To see this, given x ∈ X, there exists xi ∈ G such that dn (x, xi ) < ε/2 < δ and thus we have that x ∈ B(xi , δ) ∩ . . . ∩ T ?n (B(T n (xi ), δ)). This implies that αδ = {B(T l (xi ), δ) : xi ∈ G, 0 ≤ l ≤ n} is a ?nite covering of X, for every δ ∈ (ε/2, ε). Since d is an admissible metric, there exists δε ∈ (ε/2, ε) such that αδε is an admissible covering of 6

balls with radius equals to δε . Furthermore we have that βδε is a sub-covering n of αδε . Hence, for each (n, ε/2)-generator G of X, there exist δε ∈ (ε/2, ε) n and a sub-covering βδε of αδε such that N(αδε ) ≤ #βδε ≤ #G. Thus it follows that N(αδε ) ≤ Gn (ε/2). Taking logarithms, dividing by n and taking limits, we get as claimed that there exists δε ∈ (ε/2, ε) such that h(T, αδε ) ≤ g(ε/2). Since |αδε | ≤ 2δε ≤ 2ε, we have that g(2ε) ≤ g(|αδε |) ≤ h(T, αδε ) ≤ g(ε/2). Taking limits with ε ↓ 0, it follows that hd (T ) = lim h(T, αδε ) = sup h(T, αδε ).

ε↓0 ε>0

In order to complete the proof, it remains to show that the above supremum is equal to h(T ). For any admissible covering α of X, take ε a Lebesgue n number of this covering. We claim that N(αn ) ≤ N(αδε ), where αδε is given n above. In fact, since every element of αδε if given by B0 ∩ . . . ∩ T ?n (Bn ), where {B0 , . . . , Bn } are balls of radius δε < ε, there exist {A0 , . . . , An } ? α such that Bi ? Ai , for all i ∈ {0, . . . , n}. Thus we have that B0 ∩ . . . ∩ T ?n (Bn ) ? A0 ∩ . . . ∩ T ?n (An ),

n showing that, for each sub-covering β of αδε , there exists a sub-covering γ of αn such that N(αn ) ≤ #γ ≤ #β. Hence we get as claimed that n N(αn ) ≤ N(αδε ). Taking logarithms, dividing by n and taking limits, it follows that h(T, α) ≤ h(T, αδε ) ≤ sup h(T, αδε ), ε>0

which shows that h(T ) = sup h(T, αδε ),

ε>0

completing the proof. From now one we assume that X is a locally compact space. Thus we have associated to X its one point compacti?cation, which we will denote by X. We have that X is de?ned as the disjoint union of X with {∞}, where ∞ is some point not in X called the point at the in?nity. The topology in X consists by the former open sets in X and by the sets A ∪ {∞}, where 7

the the complement of A in X is compact. If T : X → X is a proper map, de?ning T : X → X by T (x) = T (x), x = ∞ , ∞, x=∞

we have that T is also proper map, called the extension of T to X. To see this, we only need to verify that T is continuous at ∞. If A ∪ {∞} is a neighborhood of ∞, then the complement of A is compact and we have that T ?1 (A ∪ {∞}) = T ?1 (A) ∪ {∞} is also a neighborhood of ∞, since T is proper and thus the complement of T ?1 (A) is also compact. The following result shows that the restriction to X of any metric on X is always admissible and that their respective entropies coincide. Proposition 2.3 Let T : X → X be a proper map, where X is a locally compact separable space. Let d be the metric given by the restriction to X of some metric d on X, the one point compacti?cation of X. Then it follows that d is an admissible metric and that hd (T ) = hd T , where T is the extension of T to X. In particular, we get that h(T ) = h T . Proof: First we show that Gn (ε) is ?nite for every ε > 0. Let G = {x1 , . . . , xk } ? X be an (n, ε/2)-generator set of X. By the density of X in X, it follows that there exist {x1 , . . . , xk } ? X, such that dn (xi , xi ) < ε/2. If x ∈ X ? X, we have that dn (x, xi ) < ε/2, for some xi ∈ G. Hence it follows that dn (x, xi ) ≤ dn (x, xi ) + dn (xi , xi ) < ε/2 + ε/2 = ε, showing that G = {x1 , . . . , xk } is an (n, ε)-generator set of X. If we choose G such that #G = Gn (ε/2), then we get as claimed that Gn (ε) ≤ Gn (ε/2) < ∞. 8

e

Now assume that αδ = {B(x1 , δ), . . . , B(xk , δ)} is a covering of X, for every δ ∈ (a, b), where 0 < a < b. Since, for each ?xed δ, the number of balls are ?nite, it follows that there exists δε ∈ (a, b) such that ∞ ∈ S(xi , δε ), for / every i ∈ {1, . . . , k}, where S(x, r) is the sphere in X of radius r centered in x. Denoting by B(x, r) the open ball in X of radius r centered in x, it remains two alternatives: 1) the point ∞ is inside B(xi , δε ) or 2) the point ∞ is not in the closure of B(xi , δε ). In the ?rst case, the complement of B(xi , δε ) in X is equal to the complement of B(xi , δε ) in X, which is compact. In the second alternative, there exist a open neighborhood U of ∞ which has empty intersection with B(xi , δε ). Thus B(xi , δε ) = B(xi , δε ) is in the complement of U in X, which is compact. This shows that the closure or the complement of B(xi , δε ) in X is compact, showing that αδε is already admissible. We show now that every admissible covering of X has a Lebesgue number. If α is admissible covering of X, there exists an open covering α of X such that A ∈ α if and only if there exists A ∈ α, with A = A ∩ X. In fact, if α is an admissible covering of X, there exists at least one A ∈ α with compact complement in X and such that its closure in X is not compact. If we de?ne A = A ∪ {∞}, we have that A is an open neighborhood of {∞} in X. Thus de?ning α = A ∪ {B ∈ α : B = A}, it follows as claimed that α is an open covering of X such that A ∈ α if and only if there exists A ∈ α, with A = A ∩ X. If ε is a Lebesgue number for α, then we claim that ε is a also a Lebesgue number for α. To see this, if B(x, ε) is a ball in X, there exists A ∈ α such that B(x, ε) ? A. Thus it follows that B(x, ε) ? A, where A = A ∩ X ∈ α, completing the proof that d is an admissible metric. e Finally we show that hd (T ) = hd T . Let G = {x1 , . . . , xk } be an (n, ε/2)-generator set of X. By the density of X in X, if x ∈ X, there exists x ∈ X such that dn (x, x) < ε/2, since dn is topologically equivalent to d. Thus we have that dn (x, xi ) < ε/2, for some xi ∈ G. Hence it follows that dn (x, xi ) ≤ dn (x, x) + dn (x, xi ) < ε/2 + ε/2 = ε, showing that G = {x1 , . . . , xk } is an (n, ε)-generator set of X. Choosing G such that #G = Gn (ε/2), we get that Gn (ε) ≤ Gn (ε/2). Since we have also shown above that Gn (ε) ≤ Gn (ε/2), taking logarithms, dividing by n and 9

taking limits, we get that g(ε) ≤ g(ε/2) Therefore it follows that hd (T ) = lim g(4ε) ≤ lim g(2ε) = hd T

ε↓0 ε↓0 e e

and

g(ε) ≤ g(ε/2).

≤ lim g(ε) = hd (T ),

ε↓0

showing that hd (T ) = hd T . The last statement now follows applying Proposition 2.2.

3

The variational principle

In this section, we present a full extension for non-compact sets of the well known variational principle involving entropies. We start with a proper map T : X → X and remember the concept of entropy associated to some given T -invariant probability on X. A T -invariant probability on X is a Borel measure ? such that ?(X) = 1 and ?(T ?1(A)) = ?(A), for all Borel subsets A ? X. We denote by PT (X) the set of all T -invariant probability on X. A Borel partition A of X is a partition of X where all of its elements are Borel sets. Given a ?nite Borel partition A of X, for every n ∈ N, we have that the set given by An = {A0 ∩ T ?1 (A1 ) ∩ . . . ∩ T ?n (An ) : Ai ∈ A} is also a ?nite Borel partition of X, since T is continuous. For a given ?nite Borel partition A of X, we de?ne its associated n-entropy as H(An ) =

B∈An

φ(?(B)),

where φ : [0, 1] → R is the continuous function given by φ(x) = ?x log(x), x ∈ (0, 1] . 0, x=0

It can be shown that the sequence H(An ) is subadditive, which implies the existence of the following limit h(T, A) = lim 10 1 H(An ). n→∞ n

The ?-entropy of the map T is thus de?ned as h? (T ) = sup h(T, A),

A

where the supremum is taken over all ?nite Borel partition A of X. The next result was ?rst proved in the Lemma 1.5 of [4], where they used the fact that the supremum of the measure entropies taken over all invariant probabilities is equals to that one just taken over all ergodic invariant probabilities. We present here an elementary proof of the quoted result which dos not use this fact. In the following PT (X) denotes the set of all T -invariant probabilities on X. Lemma 3.1 Let T : X → X be a proper map and T : X → X be its extension in the one point compacti?cation X of the locally compact separable space X. Then it follows that sup h? (T ) = sup h? T , e

? ? e

where the suprema are taken, respectively, over PT (X) and over PT (X). e Proof: If ? ∈ PT (X), de?nig ? A h? (T ) = h? T , showing that e sup h? (T ) ≤ sup h? T . e

? ? e

= ? A ∩ X , it is immediate that

? ∈ PT (X), since X and {∞} are T -invariant sets. It is also immediate that e

Now let ? ∈ PT (X). If ?(∞) = 1, it is immediate that h? T e e

= 0, since

X and {∞} are T -invariant sets. Thus we can assume that ?(∞) = c < 1. It is also immediate that ?= We claim that h? T e

1 1?c

?|B(X) ∈ PT (X).

≤ h? (T ). If A = {A1 , . . . , Al } is a measurable par-

tition of X, de?ning Ai = Ai ∩ X, it follows that A = {A1 , . . . , Al } is a 11

measurable partition of X. We have that, B ∈ An if and only if there exists B ∈ An , with B = B ∩ X. This follows, because T ?j Ai ∩ X = T ?j (Ai ) for any j ∈ {0, . . . , n} and any i ∈ {1, . . . , l}, since X and {∞} are T invariant sets. It follows that H(An ) =

e e B∈An

φ ? B

= φ ? B∞

+

e e B=B∞

φ ? B

,

where ∞ ∈ B∞ . Since ? B = (1 ? c)?(B), for each B = B∞ , it follows that φ(a?(B)), H(An) = b +

B∈An

where a = 1 ? c and b = φ ? B∞ H(An ) = b ?

B∈An

? φ(a?(B)). Hence

a?(B) log(a?(B)) ?(B)(log(a) + log(?(B))

B∈An

= b?a = b ? a log(a)

?(B)

B∈An

+a

B∈An

φ(?(B))

= b + φ(a) + aH(An ). It follows that H(An ) ≤ d + H(An ), since a = 1 ? c ≤ 1 and b + φ(a) ≤ d = 2 max{φ(x) : x ∈ [0, 1]}. Dividing by n and taking limits, we get that h? T , A ≤ h? (T, A) ≤ h? (T ). e Since A is arbitrary, we have that h? T e ≤ h? (T ) ≤ sup h? (T ).

?

12

Since ? ∈ PT (X) is also arbitrary, it follows that e sup h? T e

? e

≤ sup h? (T ),

?

completing the proof. Now we prove the variational principle for entropies in a locally compact separable space. In Lemma 1.5 of [4], it was proved that sup h? (T ) = inf hd (T ).

? d

Here we improve this result showing that the above in?mum is always attained at d, whenever d is an admissible metric. Moreover the supremum and the minimum coincide with the topological entropy introduced in the Section 2. Theorem 3.2 Let T : X → X be a continuous map, where X is a locally compact separable space. Then it follows that sup h? (T ) = h(T ) = min hd (T ),

? d

where the minimum is attained whenever d is an admissible metric. Proof: The the variational principle for compact spaces states that h T = sup h? T . e

? e

By Proposition 1.4 of [4], we have that sup h? (T ) ≤ inf hd (T ).

? d

Applying Lemma 3.1, it follows that h T = sup h? (T ) ≤ inf hd (T ).

? d

Applying Propositions 2.2 and 2.3, we get that hd (T ) ≤ hd (T ) = h(T ) = sup h? (T ) ≤ inf hd (T ),

? d

where, in the ?rst two terms, d is any admissible metric.

13

4

Topological entropy of automorphisms

In this section, we compute the topological entropy for automorphisms of simply connected nilpotent Lie groups. We start with linear isomorphisms of a ?nite dimensional vector space. For this, we need to determine the recurrent set of a linear isomorphism in terms of its multiplicative Jordan decomposition (see Lema 7.1, page 430, of [5]). If T : V → V is a linear isomorphism, where V is a ?nite dimensional vector space, then we can write T = TH TE TU , where TH : V → V is diagonalizable in V with positive eigenvalues, TE : V → V is an isometry relative to some appropriate norm and TU : V → V is a linear isomorphism which can be decomposed into the sum of the identity map plus some nilpotent linear map. The linear maps TH , TE and TU commute, are unique and called, respectively, the hyperbolic, the elliptic and the unipotent components of the multiplicative Jordan decomposition of T . In the next result, we prove that the recurrent set R(T ) of T is given as the intersection of the ?xed points of the hyperbolic and unipotent components. Using this characterization, we also show that the topological entropy of T always vanishes. Proposition 4.1 Let T : V → V be a linear isomorphism, where V is a ?nite dimensional vector space. Then the recurrent set of T is given by R(T ) = ?x (TH ) ∩ ?x (TU ) . Furthermore, it follows that h(T ) = 0. Proof: Let PT : PV → PV be the map induced by T in the projective space of V and, for a subspace W ? V , denote by PW its projection in PV . By Proposition 2 of [3], we have that R(PT ) = ?x (PTH ) ∩ ?x (PTU ) . By linearity, it is immediate that P R(T ) ? R(PT ), where A denotes the linear subspace generated by A ? V . Thus it follows that R(T ) ? eig(TH ) ∩ eig(TU ), where eig(S) denotes the union of the eigenspaces of a linear map S : V → V . Since TU is unipotent, all of its eigenvalues are equals to one, implying that eig(TU ) = ?x(TU ). Hence R(T ) ? eig(TH ) ∩ ?x(TU ). Now let v ∈ R(T ) 14

be such that TH v = λv, for some λ > 0. Since the multiplicative Jordan decomposition is commutative, it follows that

n n n n |T n v| = |TE TH TU v| = |TH v| = λn |v|,

where we used the fact that TE is an isometry relative to some appropriate norm | · |. This shows that λ is equals to one and thus that R(T ) is contained in ?x(TH )∩?x(TU ). Since ?x(TH )∩?x(TU ) is invariant by TE , we can consider the restriction of T to this set, which is just the restriction of TE . Thus we get that R(T ) = ?x(TH ) ∩ ?x(TU ), since the restriction of TE to ?x(TH ) ∩ ?x(TU ) is an isometry whose orbits have compact closure. Now to show that h(T ) = 0, we ?rst note that, by the Poincar? recurrence e theorem, for every ? ∈ PT (X), we have that h? (T ) = h? T |R(T ) , since R(T ) is a closed set. By the variational principle, it follows that h(T ) = h T |R(T ) ≤ hd T |R(T ) , for every metric d. Since T |R(T ) = TE |R(T ) is an isometry relative to some appropriate metric d, we get that hd T |R(T ) = 0, completing the proof. We note that the classical formula for the d-entropy of a linear isomorphism, where d is the euclidian metric, is just an upper bound for its null topological entropy. Thus it might be an interesting problem to calculate the topological entropy of a given automorphism φ of a noncompact Lie group G, since the classical formula for its d-entropy, where d is some invariant metric, is as well just an upper bound for its topological entropy. Remember that the classical formula is given by hd (φ) =

|λ|>1

log |λ|,

where λ runs through the eigenvalues of d1 φ : g → g, the di?erential of the automorphism φ at the identity element of G. The following theorem gives an answer for the above problem when G is a simply connected nilpotent Lie group. Theorem 4.2 Let φ : G → G be an automorphism, where G is a simply connected nilpotent Lie group. Then it follows that h(φ) = 0.

15

Proof: If G is a connected and simply connected nilpotent Lie group, we have that the exponential map is a di?eomorphism between g and G (see Theorem 1.127, page 107, of [6]). Since φ(exp(X)) = exp(d1 φX), we get that φ and d1 φ are conjugated maps. By Proposition 2.1, it follows that h(φ) = h(d1 φ) = 0. We note that, if the fundamental group of G is not trivial, it is possible the existence of an automorphism with positive topological entropy, even when G is abelian. In fact, the map φ(z) = z 2 is an automorphism of the abelian Lie group S 1 and it is well known that it has topological entropy h(φ) = log(2) > 0. We have that the canonical homomorphism π : R → S 1 , given by π(x) = eix , is a semi-conjugation between φ and the automorphism φ of the universal covering R of S 1 , given by φ(x) = 2x. Although h φ = 0, we can not apply Proposition 2.1 to conclude that h(φ) = 0, since the canonical homomorphism π is a proper map if and only if the fundamental group of G is ?nite.

References

[1] R. Adler, A. Konheim and H. MacAndrew : Topological entropy. Trans. Americ. Math Soc. 114 (1965), 309-319. [2] R. Bowen : Topological entropy for noncompact sets. Trans. Americ. Math Soc. 184 (1973), 125-136. [3] T. Ferraiol, M. Patr?o, L. Santos and L. Seco : Multiplicative Jordan a decomposition and dynamics on ?ag manifolds. preprint (2008). [4] M. Handel and B. Kitchens: Metrics and entropy for non-compact spaces. Isr. J. Math., 91 (1995), 253-271. [5] Helgason, S. Di?erential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York, (1978). [6] Knapp, A. W. : Lie Groups Beyond an Introduction, Progress in Mathematics, v. 140, Birkh¨user, Boston (2004). a [7] P. Walters : An introduction to ergodic theory. GTM, Springer-Verlag, Berlin, 1981. 16