arX

iv:1

007.

0533

v1 [

mat

h.G

R]

4 J

ul 2

010

Entropy on abelian groups

Dikran Dikranjan

dikran.dikranjan@dimi.uniud.it

Dipartimento di Matematica e Informatica,

Universita di Udine,

Via delle Scienze, 206 - 33100 Udine, Italy

Anna Giordano Bruno

anna.giordanobruno@math.unipd.it

Dipartimento di Matematica Pura e Applicata,

Universita di Padova,

Via Trieste, 63 - 35121 Padova

Abstract

We extend to endomorphism of arbitrary abelian groups the definition of the algebraic entropy h given byPeters for automorphisms and we study the properties of h. In particular, we prove the Addition Theorem forh and we obtain a Uniqueness Theorem for h in the category of all abelian groups and their endomorphisms.The third of our main results is the Bridge Theorem connecting the algebraic entropy and the topologicalentropy by the Pontryagin duality.

1 Introduction

Inspired by the notion of entropy invented by Clausius in thermodynamics in the fifties of the nineteenthcentury, Shannon introduced the notion of entropy in Information Theory by the end of the forties of thelast century. A couple of years later, Kolmogorov and Sinai introduced the notion of (measure) entropy inergodic theory. By appropriate modification of their definition, Adler, Konheim, and McAndrew [1] obtainedthe notion of topological entropy htop(ψ) of a continuous self-map ψ : X → X of a compact topological spaceX (see Section 7 for the definition).

The compact groups and their continuous endomorphisms provide an instance where both the measure andthe topological entropy can be applied. Indeed, every compact group admits a translation invariant (Haar)measure. Moreover, as noticed by Halmos [9] a continuous endomorphism ψ : K → K of a compact group Kis measure preserving if and only if ψ is surjective. It was established by Berg that for a surjective continuousendomorphism ψ : K → K of a compact group K the measure entropy and the topological entropy coincide.As far as non-surjective endomorphisms ψ : K → K are concerned, the measure entropy of ψ is not defined (asψ is not measure preserving), while htop(ψ) still makes sense. However, as observed in [26], the restriction of ψto the subgroup Im∞ψ =

⋂n∈N ψ

n(K) is surjective and htop(ψ) = htop(ψ ↾Im∞ψ) (see Lemma 7.6). This allowsus to say that in this sense the topological entropy and the measure entropy of the continuous endomorphismsof compact groups coincide.

Using ideas briefly sketched in [1], Weiss [30] developed the definition of algebraic entropy for endomor-phisms of abelian groups as follows. Let G be an abelian group and let φ : G→ G be an endomorphism of G.For a finite subgroup F of G and n ∈ N, let Tn(φ, F ) = F + φ(F ) + . . .+ φn−1(F ) be the n-th φ-trajectory ofF (while T (φ, F ) =

∑n∈N φ

n(F ) is the φ-trajectory of F ). Moreover, the limit (which is the algebraic entropyof φ with respect to F )

H(φ, F ) = limn→∞

log |Tn(φ, F )|

n(1.1)

exists. The algebraic entropy of φ is

ent(φ) = sup{H(φ, F ) : F ≤ G finite}. (1.2)

According to the main result of Weiss [30], the topological entropy of a continuous endomorphism ψ : K →

K of a profinite abelian group coincides with the algebraic entropy of the adjoint map ψ : K → K of ψ, whereK is the Pontryagin dual of K. Since the profinite abelian groups are precisely the Pontryagin duals of thetorsion abelian groups, one can announce this also in the following form (we call this kind of result a BridgeTheorem):

Theorem 1.1. [30] Let G be a torsion abelian group and φ ∈ End(G). Then ent(φ) = htop(φ).

1

Later on Peters [21] extended the above definition of algebraic entropy (we denote the Peters’ algebraicentropy by h) to automorphisms φ of arbitrary abelian groups using, instead of Tn(φ, F ) = F + φ(F ) + . . .+φn−1(F ), the “negative part of the orbit” T−

n (φ, F ) = F + φ−1(F ) + . . . + φ−n+1(F ) and instead of finitesubgroups F of G just non-empty finite subsets F of G.

Peters proved another Bridge Theorem connecting the topological entropy and the algebraic entropy h bymeans of the Pontryagin duality:

Theorem 1.2. [21] Let G be a countable abelian group and φ ∈ Aut(G). Then h(φ) = htop(φ).

Since T−n (φ, F ) = Tn(φ

−1, F ), the definition by Peters for automorphisms φ of abelian groupsG can be givenalso using Tn(φ, F ). The approach using Tn(φ, F ) has the advantage to be applicable also to endomorphisms φ(whereas T−

n (φ, F ) may give rise to an infinite subset of G when φ is not injective, so one cannot make recourseto |T−

n (φ, F )| to define the algebraic entropy). So we define the algebraic entropy h for endomorphisms φ ofabelian groups G as follows.

For a non-empty subset F of G and for any positive integer n, the n-th φ-trajectory of F is

Tn(φ, F ) = F + φ(F ) + . . .+ φn−1(F ),

and the φ-trajectory of F is T (φ, F ) =∑n∈N φ

n(F ). For F finite, let

H(φ, F ) = limn→∞

log |Tn(φ, F )|

n(1.3)

be the algebraic entropy of φ with respect to F (this limit exists as proved in Corollary 2.2). The algebraicentropy of φ is

h(φ) = supF∈[G]<ω

H(φ, F ).

In particular, for endomorphisms of torsion abelian groups, h coincides with the already defined algebraicentropy ent. More precisely, ent(φ) = h(φ ↾t(G), where t(G) denotes the torsion subgroup of G.

In Section 2 we prove the basic properties of h (see Lemma 2.7 and Proposition 2.8). These propertiesare counterparts of the properties of ent. Moreover we give many examples, starting from the fact that theidentical homomorphism has zero algebraic entropy. For ent it is obvious that the identical homomorphismhas entropy 0, but for h the proof (given in [2]) requires some more effort.

One of our main aims in this paper is to prove the Addition Theorem for the algebraic entropy h, namely,the following

Theorem 1.3 (Addition Theorem). Let G be an abelian group, φ ∈ End(G), H a φ-invariant subgroup of Gand φ : G/H → G/H the endomorphism induced by φ. Then h(φ) = h(φ ↾H) + h(φ).

Since h coincides with ent for endomorphisms of torsion abelian group, the Addition Theorem for entproved in [5] for endomorphisms of torsion abelian groups covers the torsion case of Theorem 1.3.

It is convenient to adopt the following notation: for an abelian group G, φ ∈ End(G) and H a φ-invariantsubgroup of G, we write ATh(G,φ,H) to indicate that the Addition Theorem 1.3 holds for the triple (G,φ,H).

In Sections 2, 3 and 4 we first give some technical results, which permit to reduce the proof of the AdditionTheorem 1.3 to appropriate particular cases. In particular, Lemma 3.2 is a reduction to the case of countableabelian groups. Moreover, we consider many properties of the algebraic entropy of endomorphisms of torsion-free abelian groups, still with the aim of proving the Addition Theorem 1.3. For example, for a torsion-freeabelian group G and φ ∈ End(G), we see that h(φ) = h(φ), where φ : D(G) → D(G) is the extension of φ tothe divisible hull D(G) of G (see Proposition 2.12). This allows to reduce the study of the algebraic entropyof the endomorphisms of torsion-free abelian groups to the case of divisible abelian groups. Moreover, we canreduce to the case of divisible torsion-free abelian group of finite rank. Now the endomorphism φ of G can besupposed to be injective (by Proposition 4.5) and then φ is also surjective. Finally this particular case of anautomorphism of Qn can be managed through the Algebraic Yuzvinski Formula:

Theorem 1.4 (Algebraic Yuzvinski Formula). For n ∈ N+ an automorphism φ of Qn is described by a matrixA ∈ GLn(Q). Then

h(φ) = log s+∑

|αi|>1

log |αi|, (1.4)

where αi are the eigenvalues of A and s is the least common multiple of the denominators of the coefficientsof the (monic) characteristic polynomial of A.

2

The counterpart of this formula was proved by Yuzvinsky [31] (see also [16]) for the topological entropy(see Theorem 7.4) and it implies (1.4) for h in view of Theorem 1.2. This proof is given in [2].

Let us thickly underline that the proof of the Addition Theorem 1.3 heavily uses the Algebraic YuzvinskiFormula (1.4) (already in the finite-rank torsion-free case). Needless to say, the value of this achievement willbe much higher if a purely algebraic proof of the Algebraic Yuzvinski Formula (1.4) would be available.

The proof of the Addition Theorem 1.3 is given in Section 5.

In Section 6 we prove the following Uniqueness Theorem, inspired by the Uniqueness Theorem proved in[5] for ent in the class of torsion abelian groups. It was inspired by the Uniqueness Theorem for the topologicalentropy by Stoyanov [26].

For any abelian group K the (right) Bernoulli shift βK : K(N) → K(N) is defined by

(x0, x1, x2, . . .) 7→ (0, x0, x1, . . .).

Theorem 1.5 (Uniqueness Theorem). The algebraic entropy h of the endomorphisms of the abelian groups ischaracterized as the unique collection h = {hG : G abelian group} of functions hG : End(G)→ R+ such that:

(a) hG is invariant under conjugation for every abelian group G;

(b) if φ ∈ End(G) and the group G is a direct limit of φ-invariant subgroups {Gi : i ∈ I}, then hG(φ) =supi∈I hGi(φ ↾Gi);

(c) the Addition Theorem holds for h;

(d) hK(N)(βK) = log |K| for any finite abelian group K;

(e) the Algebraic Yuzvinski Formula holds for hQ restricted to the automorphisms of Q.

Moreover, we see how this result can be deduced by a theorem of Vamos on length functions [19, 27].

In Section 7 we generalize Theorems 1.1 and 1.2 to all endomorphisms of all abelian groups:

Theorem 1.6 (Bridge Theorem). Let G be an abelian group and φ ∈ End(G). Then h(φ) = htop(φ).

To prove this theorem we use its weaker forms proved by Weiss and Peters (see Theorems 1.1 and 1.2respectively) and we apply the Addition Theorem 1.3 for the algebraic entropy and the Addition Theorem 7.3for the topological entropy.

In Section 8 we first recall the Mahler measure, which is an important invariant studied in number theoryand arithmetic geometry. Moreover we see that the problem of determining the infimum of the positive valueof the algebraic entropy is equivalent to the famous Lehmer Problem (see Corollary 8.8).

Notation and terminology

We denote by Z, N, N+, Q and R respectively the set of integers, the set of natural numbers, the set of positiveintegers, the set of rationals and the set of reals. For m ∈ N+, we use Z(m) for the finite cyclic group of orderm.

Let G be an abelian group. With a slight divergence with the standard use, we denote by [G]<ω the setof all non-empty finite subsets of G. If H is a subgroup of G, we indicate this by H ≤ G. The subgroup oftorsion elements of G is t(G), while D(G) denotes the divisible hull of G. For a cardinal α we denote by G(α)

the direct sum of α many copies of G, that is⊕

αG.Moreover, End(G) is the ring of all endomorphisms of G. We denote by 0G and idG respectively the

endomorphism of G which is identically 0 and the identity endomorphism of G. Moreover, for k ∈ Z, letµk : G→ G be the endomorphism of G defined by µk(x) = kx for every x ∈ G. If φ ∈ End(G), then we denote

by φ ∈ End(D(G)) the unique extension of φ to D(G).

2 Properties and examples

First of all we have to show that, for G an abelian group, φ ∈ End(G) and F ∈ [G]<ω , the limit definingH(φ, F ) exists. We start proving that {log |Tn(φ, F )| : n ∈ N+} is a subadditive sequence.

Lemma 2.1. Let G be an abelian group, φ ∈ End(G) and F ∈ [G]<ω. For n ∈ N+, let cn = log |Tn(φ, F )|.Then cn+m ≤ cn + cm for every n,m ∈ N+.

3

Proof. By definition

Tn+m(φ, F ) = F + φ(F ) + . . .+ φn−1(F ) + φn(F ) + . . .+ φn+m−1(F )

= Tn(φ, F ) + φn(Tm(φ, F )).

Consequently,

cn+m = log |Tn+m(φ, F )|

≤ log(|Tn(φ, F )||Tm(φ, F )|)

= log |Tn(φ, F )| + log |Tm(φ, F )|

= cn + cm.

Corollary 2.2. Let G be an abelian group, φ ∈ End(G) and F ∈ [G]<ω. Then the limit H(φ, F ) =

limn→∞log |Tn(φ,F )|

n exists and H(φ, F ) = infn∈N+

log |Tn(φ,F )|n .

Proof. By Lemma 2.1 the sequence {cn : n ∈ N+}, where cn = log |Tn(φ, F )|, is subadditive. Then thesequence { cnn : n ∈ N+} has limit and limn→∞

cnn = infn∈N+

cnn by a known fact from Calculus, due to

Fekete.

Remark 2.3. Let G be an abelian group and φ ∈ End(G). The function H(φ,−) is non-decreasing, thatis, H(φ, F ) ≤ H(φ, F ′) for every F, F ′ ∈ [G]<ω with F ⊆ F ′. Therefore, calculating h(φ), it is possible tosuppose without loss of generality that 0 ∈ F . Indeed,

h(φ) = sup{H(φ, F ) : F ∈ [G]<ω, 0 ∈ F}.

Intuitively, for any abelian group G, the zero endomorphism 0G and the identity idG have zero algebraicentropy. In fact,

Example 2.4. If G is an abelian group, then h(0G) = 0 and h(idG) = 0. While it is clear from the definitionthat h(0G) = 0, to prove that h(idG) = 0 requires some more effort. In fact, in [2] this is proved showingthat the idG-trajectories of the non-empty finite subsets F of G have polynomial growth; more precisely, forevery F ∈ [G]<ω , there exists PF (t) ∈ Z[t] such that |Tn(φ, F )| ≤ PF (n) for every n ∈ N+ (for idG one takesPF (t) = (t+ 1)|F |).

Let G be an abelian group and φ ∈ End(G). A point x ∈ G is a periodic point of φ if there existsn ∈ N+ such that φn(x) = x. Moreover, x ∈ G is a quasi-periodic point of φ if there exist n > m in N suchthat φn(x) = φm(x). We say that φ is locally (quasi-)periodic if every x ∈ G is a (quasi-)periodic point ofφ. Moreover, φ is periodic if there exists n ∈ N+ such that φn(x) = x for every x ∈ G. Analogously, φ isquasi-periodic if there exist n > m in N such that φn(x) = φm(x) for every x ∈ G.

It is possible to prove the following lemma, making use only of the definition of algebraic entropy.

Lemma 2.5. Let G be an abelian group and φ ∈ End(G). If φ is locally quasi-periodic, then h(φ) = 0.

Proof. Let F ∈ [G]<ω and assume without loss of generality that 0 ∈ F (see Remark 2.3). By hypothesis,there exists m ∈ N+ such that φm(F ) ⊆ Tm(φ, F ). Consequently, Tn(φ, F ) = Tm(φ, F ) for every n ∈ N,n ≥ m. Hence H(φ, F ) = 0. By the arbitrariness of F , this proves h(φ) = 0.

In particular, every endomorphism of a finite abelian group has zero algebraic entropy.

Let G be an abelian group and φ ∈ End(G); the hyperkernel of φ is

ker∞ φ =⋃

n∈N+

kerφn.

The subgroup ker∞ φ is φ-invariant and also invariant for inverse images. Hence the induced endomorphismφ : G/ ker∞ φ → G/ ker∞ φ is injective. Since φ ↾ker∞ φ is locally nilpotent for every, and locally nilpotentimplies locally quasi-periodic, the following is an immediate consequence of Lemma 2.5.

Corollary 2.6. Let G be an abelian group and φ ∈ End(G). Then h(φ ↾ker∞ φ) = 0.

4

In the next lemma we show that h is monotone under taking restriction to invariant subgroups and undertaking induced endomorphisms on quotients over invariant subgroups.

Lemma 2.7. Let G be an abelian group, φ ∈ End(G), H a φ-invariant subgroup of G and φ : G/H → G/Hthe endomorphism induced by φ. Then h(φ) ≥ max{h(φ ↾H), h(φ)}.

Proof. For every F ∈ [H ]<ω, obviously H(φ ↾H , F ) = H(φ, F ), so H(φ ↾H , F ) ≤ h(φ). Hence h(φ ↾H) ≤ h(φ).Now assume that F ∈ [G/H ]<ω and F = π(F0) for some F0 ∈ [G]<ω , where π : G→ G/H is the canonical

projection. Then π(Tn(φ, F0)) = Tn(φ, F ) for every n ∈ N+. Therefore, H(φ, F0) ≥ H(φ, F ) and by thearbitrariness of F this proves h(φ) ≥ h(φ).

The properties in the next proposition are the basic ones for h. They are the typical properties of theknown entropies. Indeed, similar properties holds for the algebraic entropy ent, for enti, and also for thetopological entropy (see Fact 7.1), which gave the inspiration. In the case of the algebraic entropy h they wereproved in [21] for automorphisms, and we extend them for endomorphisms.

Proposition 2.8. Let G be an abelian group and φ ∈ End(G).

(a) If H is another abelian group, η ∈ End(H) and φ and η are conjugated (i.e., there exists an isomorphismξ : G→ H such that η = ξφξ−1), then h(φ) = h(η).

(b) For every k ∈ N+, h(φk) = kh(φ). If φ is an automorphism, then h(φk) = |k|h(φ) for every k ∈ Z.

(c) If G is a direct limit of φ-invariant subgroups {Gi : i ∈ I}, then h(φ) = supi∈I h(φ ↾Gi).

(d) If G = G1 ×G2 and φ = φ1 × φ2 with φi ∈ End(Gi), i = 1, 2, then h(φ1 × φ2) = h(φ1) + h(φ2).

Proof. (a) For F ∈ [G]<ω and n ∈ N+, Tn(η, ξ(F )) = ξ(F ) + ξ(φ(F )) + . . . + ξ(φn−1(F )). Since ξ is anisomorphism, |Tn(φ, F )| = |Tn(η, ξ(F ))|, and so H(φ, F ) = H(η, ξ(F )). This proves that h(φ) = h(η).

(b) Fix k ∈ N+. First we prove the inequality h(φk) ≤ kh(φ). Let F ∈ [G]<ω , assuming without loss ofgenerality that 0 ∈ F (see Remark 2.3), and let n ∈ N+. Then Tn(φ

k, F ) ⊆ Tkn−k+1(φ, F ) and so

H(φk, F ) = limn→∞

log |Tn(φk, F )|

n

≤ limn→∞

log |Tkn−k+1(φ, F )|

n

= limn→∞

log |Tkn−k+1(φ, F )|

kn− k + 1· limn→∞

kn− k + 1

n

= k limn→∞

log |Tkn−n+1(φ, F )|

kn− k + 1

= kH(φ, F ).

Therefore, h(φk) ≤ kh(φ).To prove the converse inequality h(φk) ≤ kh(φ), let F ∈ [G]<ω and n ∈ N+. Let F1 = Tk(φ, F ) and note

that Tn(φk, F1) = Tkn(φ, F ). Then

1

kH(φk, F1) = lim

n→∞

log |Tn(φk, F1)|

kn= lim

n→∞

log |Tkn(φ, F )|

kn= H(φ, F ).

We can conclude that h(φk) ≥ kh(φ).Now assume that φ is an automorphism. It suffices to prove that h(φ−1) = h(φ). Let F ∈ [G]<ω

and n ∈ N+. Note that Tn(φ−1, F ) = φ−n+1Tn(φ, F ); in particular, |Tn(φ

−1, F )| = |Tn(φ, F )|, as φ is anautomorphism. This yields H(φ−1, F ) = H(φ, F ), hence h(φ−1) = h(φ).

(c) By Lemma 2.7, h(φ) ≥ h(φ ↾Gi) for every i ∈ I and so h(φ) ≥ supi∈I h(φ ↾Gi).To prove the converse inequality let F ∈ [G]<ω. Since G = lim

−→{Gi : i ∈ I} and {Gi : i ∈ I} is a directed

family, there exists j ∈ I such that F ⊆ Gj . Then H(φ, F ) = H(φ ↾Gj , F ) ≤ h(φ ↾Gj ). This proves thath(φ) ≤ supi∈I h(φ ↾Gi).

(d) Fix Fi ∈ [Gi]<ω, for i = 1, 2. Then

Tn(φ, F1 × F2) = Tn(φ1, F1)× Tn(φ2, F2),

and soH(φ, F1 × F2) = H(φ1, F1) +H(φ2, F2). (2.1)

5

Consequently, h(φ) ≥ h(φ1) + h(φ2). Since every F ∈ [G]<ω is contained in some F1 × F2, for Fi ∈ [Gi]<ω,

i = 1, 2, and so H(φ, F ) ≤ H(φ, F1 × F2), hence (2.1) proves also that h(φ) ≤ h(φ1) + h(φ2).

The next is a direct consequence of Proposition 2.8(b).

Corollary 2.9. Let G be an abelian group and φ ∈ End(G). Then:

(a) h(φ) = 0 if and only if h(φk) = 0 for some k ∈ N+, and

(b) h(φ) =∞ if and only if h(φk) =∞ for some k ∈ N+.

The following is a fundamental example in the theory of algebraic entropy. Indeed, the value of h on theright Bernoulli shift is one of the conditions that give uniqueness of h, as we will show in Section 6.

Example 2.10. It is proved in [5] that h(βZ(p)) = ent(βZ(p)) = log p, for every prime p. Equivalently,h(βK) = ent(βK) = log |K| for every finite abelian group K.

Then h(βZ) =∞. Indeed, let G = Z(N); for every prime p, the subgroup pG of G is βZ-invariant, so inducesan endomorphism βZ : G/pG → G/pG. Since G/pG ∼= Z(p)(N), and βZ is conjugated to βZ(p) through this

isomorphism, h(βZ) = h(βZ(p)) = log p by Proposition 2.8(a). Therefore h(βZ) ≥ log p for every prime p byLemma 2.7, and so h(βZ) =∞.

Assume thatK is an infinite abelian group. IfK is non-torsion, thenK contains a subgroup C ∼= Z, soK(N)

contains the βK-invariant subgroup C(N) isomorphic to Z(N). Hence, by Lemma 2.7, Proposition 2.8(a) andthe previous part of this example, h(βK) ≥ h(βC) = h(βZ) =∞. If K is torsion, then K contains arbitrarilylarge finite subgroups H . Consequently, K(N) contains the βK-invariant subgroup H(N). By Lemma 2.7 andthe first part of this example, h(βK) ≥ h(βH) = log |H | for every H . So h(βK) =∞.

Hence, for any abelian group K, h(βK) = log |K|, with the usual convention that log |K| = ∞, if |K| isinfinite.

Lemma 2.11. Let G be a torsion-free abelian group and φ ∈ End(G). If G = V (φ, g) for some g ∈ G andh(φ) <∞, then G has finite rank.

Proof. Assume that r(G) is infinite. This entails 〈φm(g)〉 ∩ Tm(φ, g) = 0 for every m ∈ N+, so Tm(φ, g) =⊕m−1k=1

⟨φk(x)

⟩and G =

⊕k∈N

⟨φk(x)

⟩∼= Z(N). Since φ is conjugated to βZ through this isomorphism, by

Proposition 2.8(a) and Example 2.10 we conclude that h(φ) = h(βZ) =∞.

The next result reduces the computation of the algebraic entropy of endomorphisms of torsion-free abeliangroups to the case of endomorphisms of divisible abelian groups.

Proposition 2.12. Let G be a torsion-free abelian group and φ ∈ End(G). Then h(φ) = h(φ).

Proof. It is obvious that h(φ) ≤ h(φ) by Lemma 2.7.Let F ∈ [D(G)]<ω . There exists m ∈ N+ such that mF ⊆ G. The automorphism µm of D(G) commutes

with φ. Hence, for n ∈ N+, Tn(φ,mF ) = Tn(φ, µm(F )) = µm(Tn(φ, F )); in particular, |Tn(φ,mF )| =

|Tn(φ, F )|. Therefore, H(φ,mF ) = H(φ, F ). By the arbitrariness of F , this gives h(φ) ≥ h(φ).

The following example shows that Proposition 2.12 may fail if G is not the torsion-free.

Example 2.13. Let G = Z(2)(N). Then h(βZ(2)) = ent(βZ(2)) = log 2. For D(G) = Z(2∞)(N) instead,

βZ(2) = βZ(2∞) has h(βZ(2∞)) = ent(βZ(2∞)) =∞ (see Example 2.10).

Recall that, if G is a torsion-free abelian group, then a subgroup H of G is essential if and only if for everyx ∈ G \ {0} there exists k ∈ Z such that kx ∈ H \ {0}.

Corollary 2.14. Let G be a torsion-free abelian group, φ ∈ End(G), H a φ-invariant subgroup of G andφ : G/H → G/H the endomorphism induced by φ. Then the purification H∗ of H in G is φ-invariant and

h(φ ↾H) = h(φ ↾H∗). Consequently, if H is an essential subgroup of G, then

(a) h(φ) = h(φ ↾H);

(b) h(φ) <∞ implies h(φ) = 0.

6

Proof. For the first assertion see [24, Lemma 3.3(a)]. Consider the divisible hull D(H) of H . We can assume

without loss of generality that H∗ is a subgroup of D(H). Let φ : D(H) → D(H) denote the common

(unique) extension of φ ↾H and φ ↾H∗. Proposition 2.12 applies to the pairs D(H), H and D(H), H∗, giving

h(φ ↾H) = h(φ ↾D(H)) = h(φ ↾H∗).

(a) follows from the first assertion, since G = H∗.

(b) Fix a prime p. Since G/H is torsion, it suffices to see that h(φ ↾(G/H)[p]) = 0 [5, Proposition 1.18]. To

this end we have to show that every x ∈ (G/H)[p] has finite trajectory under φ. By Lemma 2.11 V (φ, x) ≤ Ghas finite rank, say n ∈ N. Then there exist ki ∈ Z, i = 0, . . . , n, such that

∑ni=0 kiφ

i(x) = 0. Since G istorsion-free, we can assume without loss of generality that at least one of these coefficient is not divisible by

p. Now projecting in G/H we conclude that∑n

i=0 kiφi(x) = 0 is a non-trivial linear combination in (G/H)[p].

Hence x ∈ (G/H)[p] has finite trajectory under φ.

In the following example we calculate the algebraic entropy of the endomorphisms of Z and Q, and of themultiplications of torsion-free abelian groups. In particular, item (a) immediately shows a difference with thetorsion case. For an abelian group G and φ ∈ End(G), φ is integral if there exists f(t) ∈ Z[t] \ {0} such thatf(φ) = 0. According to [5, Lemma 2.2], if G is torsion, φ integral implies ent(φ) = 0. On the other hand, fork > 1 the endomorphism µk : Z → Z in (a) of the next example is integral over Z (as µk(x) − kx = 0 for allx ∈ Z), nevertheless, h(µk) = log k > 0.

Example 2.15. (a) For every k ∈ N+, µk : Z→ Z has h(µk) = log k.For k = 1 this follows from Example 2.4. Assume k > 1 and let F0 = {0, 1, . . . , k − 1} ∈ [Z]<ω, n ∈ N+.

Every m ∈ N with m < kn can be uniquely written in the form m = f0+f1k+ . . .+fn−1kn−1 with all fi ∈ F0.

Then Tn(µk, F0) = {m ∈ N : m < kn}, and so |Tn(µk, F0)| = kn. Consequently H(µk, F0) = log k and thisyields h(µk) ≥ log k.

To prove the converse inequality h(µk) ≤ log k, fix m ∈ N+ and let Fm = {0,±1,±2, . . . ,±m} ∈ [Z]<ω .Then |x| ≤ mkn for every x ∈ Tn(µk, Fm). So |Tn(µk, Fm)| ≤ 3mkn, hence H(µk, Fm) ≤ log k. Since eachF ∈ [Z]<ω is contained in some Fm for some m ∈ N+, we obtain h(µk) ≤ log k.

(b) Let φ ∈ End(Q). Then φ = µr, with r ∈ Q. If r = 0,±1, then h(φ) = 0 by Example 2.4. ApplyingProposition 2.8(b), we may assume that r > 1. Let r = a

b , where (a, b) = 1. Then h(φ) = log a.

To prove that h(φ) ≥ log a, take F0 = {0, 1, . . . , a− 1}. Let us check that all sums f0 + f1ab + . . .+ fn−1

an−1

bn−1 ,with fi ∈ F0, are pairwise distinct. Indeed, assume that

f0 + f1a

b+ . . .+ fn−1

an−1

bn−1= f ′

0 + f ′1

a

b+ . . .+ f ′

n−1

an−1

bn−1(2.2)

for some fi, f′j ∈ F0. Then f0b

n−1 + f1abn−2 + . . .+ fn−1a

n−1 = f ′0bn−1 + f ′

1abn−2+ . . .+ f ′

n−1an−1, so that a

divides f0bn−1 − f ′

0bn−1 = bn−1(f0 − f ′

0). As (a, b) = 1, we conclude that a divides f0 − f ′0 and this obviously

entails f0 = f ′0. Consequently, (2.2) gives f1+f2(

ab )+. . .+fn−1(

ab )n−2 = f ′

1+f′2(ab )+. . .+f

′n−1(

ab )n−2. Now an

obvious induction argument applies. Therefore, this shows that |Tn(φ, F0)| = an, and so Hn(φ, F0) = n log a.Thus h(φ) ≥ log a.

To prove the inequality h(φ) ≤ log a, note that the subgroup H of Q formed by all fractions having asdenominators powers of b (i.e., the subring of Q generated by 1

b ), is φ-invariant.Since H ⊇ Z, H is essential in Q, and so h(φ) = h(φ ↾H) by Corollary 2.14(a). Now for any m ∈ N+

consider Fm = {± rbm : 0 ≤ r ≤ mbm}. So Fm =

⟨1bm

⟩∩ [−m,m], where the interval [−m,m] is taken in H .

Let us observe that φk(Fm) ⊆⟨

1bm+k

⟩∩ [−mak

bk ,mak

bk ], consequently

Tn(φ, Fm) ⊆M + . . .+M︸ ︷︷ ︸n

where M =⟨

1bm+n−1

⟩∩ [−man−1

bn−1 ,man−1

bn−1 ]. Therefore, |Tn(φ, Fm)| ≤ 2nbm+n−1man−1

bn−1 . Hence,

log |Tn(φ, Fm)| ≤ log 2n+ (m+ n− 1) log b+ logm+ (n− 1)(log a− log b) = log 2n+m log b+ (n− 1) log a.

Thus H(φ, Fm) ≤ log a. Since every F ∈ [H ]<ω is contained in Fm for some m ∈ N+, this proves thath(φ) = h(φ ↾H) ≤ log a.

(c) Let n ∈ N+.

7

(i) For k ∈ N+ and µk : Zn → Zn, h(µk) = n log k.

(ii) For r = ab ∈ Q with a > b > 0, and µr : Qn → Qn, h(µr) = n log a.

To verify (i) and (ii), it suffices to apply Proposition 2.8(d) and (a) and (b) respectively.

(d) Let G be a torsion-free abelian group and consider µk : G→ G for some k ∈ N+. Then

h(µk) =

{r(G) log k if r(G) is finite,

∞ if r(G) is infinite.

Let α = r(G). Then D(G) ∼= Q(α), with µk conjugated to the multiplication by k µQk : Q(α) → Q(α), and G

has a subgroup H isomorphic to Z(α), which is µk-invariant, with µk ↾H conjugated to the multiplication byk µZ

k : Z(α) → Z(α). Assume that α ∈ N. By (ii) of (c) and Proposition 2.8(a), h(µk) = h(µQk ) = α log k. By

(i) of (c) and Proposition 2.8(a), h(µk ↾H) = h(µZk) = α log k. Then h(µk) = α log k by Lemma 2.7. If α is

infinite, by Lemma 2.7 and in view of the finite case, h(µk ↾H) = h(µZk) > n log k for every n ∈ N. Hence

h(µk ↾H) =∞ and so h(µk) =∞ by Lemma 2.7.

In item (b) of the above example we have given the explicit computation of the entropy of µr : Q → Q,with r = a

b > 1 and (a, b) = 1. One can also apply the Algebraic Yuzvinski Formula (1.4); indeed, the uniqueeigenvalue of µr is a

b > 1, and so (1.4) gives h(µr) = log a. This formula was given by Abramov for the

topological entropy of the automorphisms of Q.

Example 2.16. Fix k ∈ Z and consider the automorphism φ : Z2 → Z2 defined by φ(x, y) = (x + ky, y) forall (x, y) ∈ Z2. Then h(φ) = 0.

Letm ∈ N+ and Fm = {0,±1,±2, . . . ,±m}×{0,±1,±2, . . . ,±m} ∈ [Z2]<ω. Every F ∈ [Z2]<ω is containedin Fm for some m ∈ N+. Therefore it suffices to show that H(φ, Fm) = 0. One can prove by inductionthat, for every n ∈ N+, Tn(φ, Fm) is contained in a parallelogram with sides 2nm and nm(2 + nk − k), so|Tn(φ, Fm)| ≤ 2n2m2(2 + nk − k). Thus H(φ, Fm) = 0.

Let G be an abelian group and φ ∈ End(G). For F ⊆ G, the trajectory T (φ, F ) needs not be a subgroupof G. So let

V (φ, F ) = 〈φn(F ) : n ∈ N〉 = 〈T (φ, F )〉 .

This is the smallest φ-invariant subgroup containing T (φ, F ) (and so also F ). If F = {g} we denote V (φ, {g})simply by V (φ, g). For F ∈ [G]<ω, V (φ, F ) =

∑g∈F V (φ, g). Note that V (φ, F ) is the Z[t]-module generated

by F . Indeed, G has structure of Z[t]-module given by φ: the multiplication by t is defined by tx = φ(x) forevery x ∈ G. This will be discussed with more details in Section 6.

Lemma 2.17. Let G be an abelian group and φ ∈ End(G). Assume that G = V (φ, F ) for some F ∈ [G]<ω,i.e., G is finitely generated (by F ) as a Z[t]-module.

(a) Every subgroup and every quotient of G is finitely generated as a Z[t]-module.

(b) If φ is locally periodic, then φ is periodic.

(c) There exists m ∈ N+ such that mt(G) = 0.

Proof. (a) It follows from the fact that Z[t] is Noetherian.

(b) There exists m ∈ N+ such that φm ↾F= idF . Since F generates G, φm = idG.

(c) By (a) t(G) is finitely generated as a Z[t]-module, that is t(G) = V (φ, F ′) for some F ′ ∈ [G]<ω. SinceF ′ is finite, there exists m ∈ N+ such that mF = 0. Then mt(G) = 0.

Lemma 2.18. Let G be an abelian group and φ ∈ End(G). Then G = lim−→{V (φ, F ) : F ∈ [G]<ω}, and so

h(φ) = sup{h(φ ↾V (φ,F )) : F ∈ [G]<ω}.

Proof. The family {V (φ, F ) : F ∈ [G]<ω} is a direct system; indeed, for every F1, F2 ∈ [G]<ω , V (φ, F1) ∪V (φ, F2) ⊆ V (φ, F1 ∪ F2). That h(φ) = sup{h(φ ↾V (φ,F )) : F ∈ [G]<ω} follows from Proposition 2.8(c).

Since h is defined “locally”, in some sense this lemma permits to reduce to the case G = V (φ, F ) for someF ∈ [G]<ω , that is, G is finitely generated (by F ) as a Z[t]-module. By Lemma 2.17(a) every subgroup andevery quotient of G is finitely generated as a Z[t]-module.

8

Proposition 2.19. Let G be a countable abelian group, φ ∈ End(G) and H a φ-invariant subgroup of G.Then there exists a family {Ln : n ∈ N} ⊆ [G]<ω, such that:

h(φ) = limn→∞

h(φ ↾V (φ,Ln)),

h(φ ↾H) = limn→∞

h(φ ↾H∩V (φ,Ln)), and

h(φ) = limn→∞

h(φ ↾π(Ln)).

Proof. We prove that, whenever G is a countable and φ ∈ End(G),

(i) there exists {Fn : n ∈ N} ⊆ [G]<ω such that G is increasing union of the V (φ, Fn);

(ii) consequently, h(φ) = limn→∞ h(φ ↾V (φ,Fn)).

Let G = {gn : n ∈ N}, and for every n ∈ N let Fn = {g0, . . . , gn}. Then G is increasing union of the Fn andconsequently of the V (φ, Fn). By Proposition 2.8(d) h(φ) = supn∈N h(φ ↾V (φ,Fn)) = limn→∞ h(φ ↾V (φ,Fn)), as{h(φ ↾V (φ,Fn)) : n ∈ N} is a non-decreasing sequence.

By (i) and (ii) applied to H and φ ↾H there exists {F ′n : n ∈ N} ⊆ [H ]<ω, such that H =

⋃n∈N V (φ ↾H , F

′n),

where this is an increasing union, and h(φ) = limn→∞ h(φ ↾V (φ↾H ,F ′

n)). Let Ln = Fn ∪ F ′

n. Then {Ln : n ∈N} ⊆ [G]<ω is such that G =

⋃n∈N V (φ, Ln), where this is an increasing union. By (ii)

h(φ) = limn→∞

h(φ ↾V (φ,Ln)).

Since V (φ ↾H , F′n) ⊆ H ∩ V (φ, Ln) for every n ∈ N, H =

⋃n∈N(H ∩ V (φ, Ln)), where this is an increasing

union, and so by (ii)h(φ ↾H) = lim

n→∞h(φ ↾H∩V (φ,Ln)).

For π : G → G/H the canonical projection, G/H =⋃n∈N π(V (φ, Ln)) =

⋃n∈N V (φ, π(Ln)), where {π(Ln) :

n ∈ N} ⊆ [G/H ]<ω and this is an increasing union. By (ii) applied to G/H and φ,

h(φ) = limn→∞

h(φ ↾π(Ln)),

and this concludes the proof.

3 The club of supports, the skew products and various relations

for the Addition Theorem

Definition 3.1. Let G be an abelian group and φ ∈ End(G). An entropy support of (G,φ) is a countableφ-invariant subgroup S of G such that h(φ ↾S) = h(φ).

Clearly, every countable φ-invariant subgroup of G containing an entropy support, will have the sameproperty, that is, such a subgroup is not uniquely determined. In the sequel we denote by S(G,φ) the familyof all entropy supports of the pair (G,φ).

The next lemma shows that S(G,φ) is not empty, i.e., there exists at least one (and then infinitely many)of such subgroups.

Lemma 3.2. Let G be an abelian group and φ ∈ End(G). Then there exists a entropy support of (G,φ).

Proof. Since h(φ) = supF∈[G]<ω H(φ, F ) is in R+∪{∞}, there exists a family {Fn : n ∈ N} ⊆ [G]<ω such thath(φ) = supn∈NH(φ, Fn). Let S = V (φ,

⋃n∈N Fn), which is a countable φ-invariant subgroup of G, such that

h(φ) = supn∈NH(φ, Fn) = supn∈NH(φ ↾S , Fn) = h(φ ↾S).

The family S(G,φ) is a club. Let us recall that a family C of countable subsets of an infinite set X is aclub (for closed unbounded) if it is closed for countable increasing unions and if every countable subset of Xis contained in an element of C.

The following is a first reduction for the proof of the Addition Theorem 1.3 to countable abelian groups.

Proposition 3.3. Assume that ATh(G,φ,H) holds for every countable abelian group G, φ ∈ End(G) andH a φ-invariant subgroup of G. Then ATh(G,φ,H) holds for every abelian group G, φ ∈ End(G) and H aφ-invariant subgroup of G.

9

Proof. Let G be an abelian group, φ ∈ End(G), H a φ-invariant subgroup of G, φ : G/H → G/H theendomorphism induced by φ and π : G → G/H be the canonical projection. Let S ∈ S(G,φ), SH ∈S(H,φ ↾H) and S ∈ S(G/H, φ). We can assume without loss of generality that S ⊇ SH and π(S) ⊇ S. ThenS′H = kerπ ↾S= S ∩ H ⊇ SH and S′

H ∈ S(H,φ ↾H). Consequently, S′H is a φ-invariant subgroup of S such

that S/S′H∼= π(S). Let φ ↾S : S/S′

H → S/S′H be the endomorphism induced by φ ↾S , which is conjugated to

φ ↾π(S). By hypothesis, and by Proposition 2.8(a),

h(φ ↾S) = h(φ), h(φ ↾S) = h(φ) and h(φ ↾S′

H) = h(φ ↾H). (3.1)

Since S is countable, by hypothesis ATh(S, φ ↾S, S′H) holds, and hence (3.1) implies that ATh(G,φ,H) holds

as well.

Let K, H be abelian groups and φ1 ∈ End(K), φ2 ∈ End(H). The direct product π = φ1 × φ2 : K ×H →K×H is defined by π(x, y) = (φ1(x), φ2(y)) for every pair (x, y) ∈ K×H . For an homomorphism s : K → H ,the skew product of φ1 and φ2 via s is φ : K ×H → K ×H defined by

φ(x, y) = (φ1(x), φ2(y) + s(x)) for every (x, y) ∈ K ×H. (3.2)

We say that the homomorphism s is associated to the skew product φ.Clearly, H = 0×H is a φ-invariant subgroup of K×H and the induced endomorphism of K ∼= (K×H)/H

is precisely φ1. When s = 0 one obtains the usual direct product endomorphism φ = φ1 × φ2. Let us see thatthe skew products arise precisely in such a circumstance:

Remark 3.4. If G is an abelian group and φ ∈ End(G), suppose to have a φ-invariant subgroup H of G thatsplits as a direct summand, that is G = K ×H . Let us see that φ is a skew product. Indeed, let ι : G/H → Kbe the natural isomorphism and let φ : G/H → G/H be the induced endomorphism. Denote by φ1 : K → Kthe endomorphism φ1 = ιφι−1 of K, and let φ2 = φ ↾H . Then there exists a homomorphism sφ : K → H suchthat φ(x) = (φ1(x), sφ(x)) for every x ∈ K. Now φ is the skew product of φ1 and φ2 via this sφ.

A natural instance to this effect are fully invariant subgroups. For example, when D is a divisible groupand φ ∈ End(D), then H = t(D) is fully invariant, so necessarily φ-invariant. Thus, φ is a skew product.

In the sequel, for a skew product φ : G = K ×H → K ×H , we denote by φ1 : K → K the endomorphismof K conjugated to the induced endomorphism φ : G/H → G/H and we let φ2 = φ ↾H . The direct productπφ = φ1 × φ2 is the direct product associated to the skew product φ. We can extend to G = K × H thehomomorphism sφ : K → H associated to the skew product, defining it to be 0 on H . This allows us toconsider sφ ∈ End(G ×H) and speak of the composition s2φ = 0, as well as φ = πφ + sφ in the ring End(G).In other words, the difference sφ = φ − πφ measures how much the skew product φ fails to coincide with itsassociated direct product πφ.

Example 3.5. Let K be a torsion-free abelian group and let T be a torsion abelian group. Then everyφ ∈ End(K × T ) is a skew product.

Proposition 3.6. Let G be an abelian group and φ ∈ End(G). Assume that G = K × T , with T torsion andφ-invariant, and suppose that φ is a skew product such that sφ(K) is finite. Then ATh(G,φ, T ) holds.

Proof. Let φ : G/T → G/T be the endomorphism induced by φ. By Proposition 2.8(d), h(πφ) = h(φ1)+h(φ2).Moreover, by definition, φ2 = φ ↾T and φ1 is conjugated to φ, so that h(φ) = h(φ1) by Proposition 2.8(a).Then it suffices to prove that

h(φ) = h(πφ). (3.3)

Let F ∈ [G]<ω. Then F ⊆ F1 × F2, for some F1 ∈ [K]<ω and F2 ∈ [T ]<ω. We can assume without loss ofgenerality that (0, 0) ∈ F1 × F2 and that F2 is a subgroup of T with F2 ⊇ sφ(K). To conclude the proof, itsuffices to show that, for n ∈ N+,

Tn(πφ, F1 × F2) = Tn(φ, F1 × F2). (3.4)

We have πnφ(F1 × F2) = φn1 (F1)× φn2 (F2) and so Tn(πφ, F1 × F2) = Tn(φ1, F1)× Tn(φ2, F2).One can prove by induction that, for every x ∈ K and every n ∈ N+,

φn(x, 0) = (φn1 (x), φn−12 (sφ(x)) + φn−2

2 (sφ(φ1(x))) + . . .+ φ2(sφ(φn−21 (x))) + sφ(φ

n−11 (x))).

10

Since sφ(K) ⊆ F2, we conclude that

φn(x, 0) ∈ (φn1 (x), 0) + [0× (φn−12 (F2) + φn−2

2 (F2) + . . .+ φ2(F2) + F2)] = (φn1 (x), 0) + [0× Tn(φ2, F2)];

as 0× Tn(φ2, F2) = Tn(φ, 0 × F2) is a subgroup of G, we deduce

φn(x, 0) ∈ (φn1 (x), 0) + [0× Tn(φ2, F2)] and (φn1 (x), 0) ∈ φn(x, 0) + Tn(φ, 0 × F2). (3.5)

Fix m ∈ N and an m-tuple a0, a1, . . . , am−1 ∈ F1. Applying (3.5) to an for n = 0, 1, . . . ,m− 1 we get

m−1∑

n=0

φn(an, 0) ∈m−1∑

n=0

(φn1 (an), 0) + [0× Tm(φ2, F2)] ⊆ Tm(φ1, F1)× Tm(φ2, F2)

andm−1∑

n=0

(φn1 (an), 0) ∈m−1∑

n=0

φn(an, 0) + Tm(φ, 0 × F2) ⊆ Tm(φ, F1 × 0) + Tm(φ, 0 × F2).

In other words,

Tm(φ, F1 × 0) ⊆ Tm(φ1, F1)× Tm(φ2, F2) and Tm(φ1, F1)× 0 ⊆ Tm(φ, F1 × 0) + Tm(φ, 0 × F2).

As 0× Tm(φ2, F2) = Tm(φ, 0× F2) is a subgroup of G, we can write it also as

Tm(φ, F1 × F2) = Tm(φ, F1 × 0) + Tm(φ, 0× F2) ⊆ Tm(φ1, F1)× Tm(φ2, F2) = Tm(πφ, F1 × F2)

and

Tm(πφ, F1 × F2) = Tm(φ1, F1)× Tm(φ2, F2) ⊆ Tm(φ, F1 × 0) + Tm(φ, 0 × F2) = Tm(φ, F1 × F2).

This proves (3.4), which gives the thesis.

Let G be an abelian group, φ ∈ End(G) and H,K φ-invariant subgroups of G. Let π : G → G/H bethe canonical projection and φ : G/H → G/H the endomorphism induced on the quotient by φ. Then thesubgroup π(K) of G/H is φ-invariant. Since π(K) is isomorphic to K/H ∩K, and the induced endomorphismφ ↾K : K/(H ∩K)→ K/(H ∩K) is conjugated to φ ↾π(K) through this isomorphism,

h(φ ↾π(K)) = h(φ ↾K) (3.6)

holds by Proposition 2.8(a).

In the sequel ∧ stays for conjunction.

Proposition 3.7. Let G be an abelian group, φ ∈ End(G) and H,K φ-invariant subgroups of G. Then:

ATh(G,φ,K) ∧ ATh(H,φ ↾H , H ∩K) ∧ ATh(K,φ ↾K , H ∩K)∧

∧ATh(G/H, φH , (H +K)/H) ∧ATh(G/K, φK , (H +K)/K) =⇒ ATh(G,φ,H).

Proof. The situation is described by the following diagram involving the pairs (G,H) and (G/K, (H+K)/K):

Hφ↾H //

��

��

H��

��

(H +K)/Kφ↾(H+K)/K //

��

��

(H +K)/K��

��G

φ //

����

G

����

G/KφK //

����

G/K

����G/H

φH // G/H (G/K)/((H +K)/K)φK // (G/K)/((H +K)/K)

Our hypotheses imply that:

(i) h(φ) = h(φ ↾K) + h(φK);

(ii) h(φ ↾H) = h(φ ↾H∩K) + h(φ ↾H);

11

(iii) h(φ ↾K) = h(φ ↾H∩K) + h(φ ↾K);

(iv) h(φH) = h(φH ↾(H+K)/H ) + h(φH);

(v) h(φK) = h(φK ↾(H+K)/K) + h(φK).

The isomorphism (G/K)/((H +K)/K) ∼= (G/H)/((H +K)/H) commutes with

φK : (G/K)/((H +K)/K)→ (G/K)/((H +K)/K) and φH : (G/H)/((H +K)/H)→ (G/H)/((H +K)/H),

so we get

h(φH) = h(φK), (3.7)

applying Proposition 2.8(c). Moreover

h(φ ↾H) = h(φ ↾H+K/K) and h(φ ↾K) = h(φ ↾H+K/H) (3.8)

by (3.6). Applying (i), (iii), (v), (3.7), (3.8), (ii) and (iv),

h(φ) = h(φ ↾K) + h(φK)

= (h(φ ↾H∩K) + h(φ ↾K)) + (h(φK ↾(H+K)/K) + h(φK))

= h(φ ↾H∩K) + h(φ ↾H+K/H) + h(φ ↾H) + h(φH)

= (h(φ ↾H∩K) + h(φ ↾H)) + (h(φ ↾H+K/H) + h(φH))

= h(φ ↾H) + h(φH).

Then h(φ) = h(φ ↾H) + h(φH).

Corollary 3.8. Let G be an abelian group and φ ∈ End(G) and H,K φ-invariant subgroups of G with H ⊆ K.Then:

(a) ATh(G,φ,H) ∧ ATh(K,φ ↾K , H) ∧ATh(G/H, φ,K/H) =⇒ ATh(G,φ,K); and

(b) ATh(G,φ,K) ∧ATh(K,φ ↾K , H) ∧ ATh(G/H, φ,K/H) =⇒ ATh(G,φ,H).

Corollary 3.9. Let G be an abelian group, φ ∈ End(G) and H,K φ-invariant subgroups of G such thatG = H +K. Then

(a) ATh(G,φ,H ∩K) ∧ATh(H,φ ↾H , H ∩K) =⇒ ATh(G,φ,H);

(b) ATh(G,φ,K) ∧ATh(H,φ ↾H , H ∩K) ∧ ATh(K,φ ↾K , H ∩K) =⇒ ATh(G,φ,H).

Proof. (a) We are going to apply Corollary 3.8(a) to the triple H ∩K ⊆ H ⊆ G. By hypothesis ATh(G,φ,H ∩K) and ATh(H,φ ↾H , H ∩ K) hold. For the last triple (G/H ∩ K,φ,H/H ∩ K) note that H/H ∩ K is aφ-invariant subroup of G/H ∩ K and G/H ∩K ∼= t(G)/H ∩ K × H/H ∩ K is a splitting of G/H ∩ K intoa direct sum of two φ-invariant subgroups. Therefore, ATh(G/H ∩K,φ,H/H ∩K) holds, as the φ-invariantsubgroup H/H ∩K has a direct summand that is a φ-invariant subgroup.

(b) is obvious from Proposition 3.7.

4 The Addition Theorem in the torsion-free case

The next properties, frequently used in the sequel, are easy to prove.

Lemma 4.1. Let G be an abelian group and H a subgroup of G.

(a) If H is divisible, then H is pure.

(b) If G is divisible, then H is divisible if and only if H is pure.

(c) If G is torsion-free, then the subgroup H is pure in G if and only if G/H is torsion-free.

We start by showing that the Addition Theorem 1.3 holds for automorphisms of Qn and its invariantdivisible subgroups, applying the Algebraic Yuzvinski Formula (1.4).

Proposition 4.2. Let m ∈ N+, φ ∈ Aut(Qm) and H be a pure (i.e., divisible) φ-invariant subgroup of Qm.Then ATh(Qm, φ,H) holds.

12

Proof. LetD = Qm and r(H) = k ∈ N, that is,H ∼= Qk. Then one can choose a basis B = {v1, . . . , vk, vk+1, . . . , vm}of D such that BH = {v1, . . . , vk} is a basis of H and the matrix of φ with respect to B has the followingblock-wise form:

A =

(A1 B0 A2

),

where A1 is the matrix of φ ↾H with respect to BH . Let π : D → D/H be the canonical projection andφ : D/H → D/H the endomorphism induced by φ. Let B = {π(vk+1), . . . , π(vm)}, which is a basis ofD/H ∼= Qm−k. Then A2 is the matrix of φ with respect to B. Let α1, . . . , αk be the eigenvalues of A1 and letαk+1, . . . , αm be the eigenvalues of A2. Then α1, . . . , αn are the eigenvalues of A.

Let χ and χ1, χ2 ∈ Q[x] be the characteristic polynomials of A and A1, A2 respectively. Then χ = χ1χ2.Let s1 and s2 be the least common multiples of the denominators of the coefficients of χ1 and χ2 respectively.This means that p1 = s1χ1 and p2 = s2χ2 ∈ Z[x] are primitive. By Gauss Lemma p = p1p2 is primitive andso for s = s1s2 the polynomial p = sχ ∈ Z[x] is primitive. Now the Algebraic Yuzvinski Formula (1.4) appliedto φ, φ ↾H , φ gives

h(φ) = log s+∑

1≤i≤m,|αi|>1

log |αi|

= log(s1s2) +∑

1≤i≤k,|αi|>1

log |αi|+∑

k+1≤i≤m,|αi|>1

log |αi|

=

log s1 +

∑

1≤i≤k,|αi|>1

log |αi|

+

log s2 +

∑

k+1≤i≤m,|αi|>1

log |αi|

= h(φ ↾H) + h(φ).

The next is a consequence of Corollary 2.14.

Corollary 4.3. Let G be a torsion-free abelian group, φ ∈ End(G) and H a φ-invariant essential subgroup ofG. Then ATh(G,φ,H) holds.

Proof. By Corollary 2.14(a), h(φ) = h(φ ↾H). If h(φ) =∞, this implies that ATh(G,φ,H) holds. If h(φ) <∞,then h(φ) = 0 by Corollary 2.14(b) and so ATh(G,φ,H) holds as well.

An immediate consequence of this corollary is that for G a torsion-free abelian group, φ ∈ End(G), H aφ-invariant subgroup of G, and φ : G/H → G/H the endomorphism induced by φ, G/H is torsion impliesthat ATh(G,φ,H) holds. Indeed, G/H torsion yields H essential in G.

The next is another consequence of Proposition 2.12 and Corollary 2.14. It shows that the verification ofthe Addition Theorem 1.3 for torsion-free abelian groups and their pure subgroups can be reduced to the caseof divisible ones.

Corollary 4.4. Let G be a torsion-free abelian group, φ ∈ End(G) and H a pure φ-invariant subgroup

of G. Then h(φ) = h(φ), h(φ ↾H) = h(φ ↾D(H)) and h(φ) = h(φ), where φ : G/H → G/H and φ :D(G)/D(H)→ D(G)/D(H) are the induced endomorphism. In particular, ATh(G,φ,H) holds if and only if

ATh(D(G), φ, D(H)) holds.

Proof. Since the purification H∗ of H in D(G) is divisible in view of Lemma 4.1, H∗ = D(H) and H =

D(H) ∩G. By Proposition 2.12, h(φ) = h(φ) and h(φ ↾H) = h(φ ↾D(H)). Let π : D(G)→ D(G)/D(H) be the

canonical projection. Then π(G) is essential in D(G)/D(H) and h(φ) = h(φ ↾π(G)) by Corollary 2.14(a). Since

G/H ∼= π(G), and φ is conjugated to φ ↾π(G) through this isomorphism, h(φ) = h(φ ↾π(G)) by Proposition

2.8(a). Hence h(φ) = h(φ).

If the abelian group G is torsion-free, then for φ ∈ End(G) the subgroup ker∞ φ is also pure. The nextresult will reduce the computation of the entropy of endomorphisms of a finite-rank torsion-free divisibleabelian groups to the case of injective ones.

Proposition 4.5. Let G be a torsion-free abelian group of finite rank and φ ∈ End(G). Then h(φ) = h(φ),where φ : G/ ker∞ φ→ G/ ker∞ φ. Moreover, ATh(G,φ, ker∞ φ) holds.

13

Proof. Suppose first that G = D is divisible. Since D has finite rank, ker∞ φ has finite rank as well. Thenthere exists n ∈ N, such that ker∞ φ = kerφn. Let γ = φn and γ : D/ kerγ → D/ kerγ the endomorphisminduced by γ. Then h(γ) = nh(φ) by Proposition 2.8(b). Since γ = φ

n, it follows that h(γ) = nh(φ) again by

Proposition 2.8(b). So, if we prove that h(γ) = h(γ), it will follow that h(φ) = h(φ).This shows that we can suppose without loss of generality that ker∞ φ = kerφ; let φ : D/ kerφ→ D/ kerφ

be the endomorphism induced by φ. From [10, Section 58, Theorem 1] it follows that

D ∼= kerφ× φ(D). (4.1)

Corollary 2.6 gives h(φ ↾kerφ) = 0 and so h(φ) = h(φ ↾φ(D)) by Proposition 2.8(d). Since φ(D) ∼= D/ kerφ,

and φ ↾φ(D) and φ are conjugated by this isomorphism, h(φ ↾φ(D)) = h(φ) by Proposition 2.8(a). Hence

h(φ) = h(φ).

We consider now the general case. Since ker∞ φ is pure in D(G), it is divisible by Lemma 4.1. Moreover

ker∞ φ is essential in ker∞ φ. Indeed, let x ∈ ker∞ φ, i.e., there exists n ∈ N+ such that φn(x) = 0. Since G is

essential in D(G) there exists k ∈ Z such that kx ∈ G \ {0}. Moreover, φn(kx) = φn(kx) = kφn(x) = 0 and so

kx ∈ ker∞ φ \ {0}). It follows that ker∞ φ = D(ker∞ φ). By the first part of the proof, that is, by the divisible

case, h(φ) = h(φ), where φ : D(G)/ ker∞ φ → D(G)/ ker∞ φ is the induced endomorphism. By Proposition

2.12 h(φ) = h(φ) and by Corollary 4.4 h(φ) = h(φ). Hence h(φ) = h(φ).

We can prove now that the Addition Theorem 1.3 holds in the case of torsion-free countable finite-rankabelian groups and pure invariant subgroups. Indeed, the next proposition generalizes Proposition 4.2 toendomorphisms. As far as we work in the torsion-free context, it seems natural to consider mainly puresubgroups that allows us to remain in the class even under passage to quotients.

Proposition 4.6. Let n ∈ N+, φ ∈ End(Qn) and H a pure (i.e., divisible) φ-invariant subgroup of Qn. Thenh(φ) <∞ and ATh(Qn, φ,H) holds.

Proof. Let D = Qn. We prove first that we can assume without loss of generality that φ is injective. Indeed,consider φ : D/ ker∞ φ → D/ ker∞ φ. Then φ is injective and D/ ker∞ φ is divisible and torsion-free asker∞ φ is pure in D (see Lemma 4.1(c)). Therefore, D/ ker∞ φ ∼= Qm for some m ∈ N, m ≤ n, and φ is anautomorphism of D/ ker∞ φ. By Proposition 4.5 h(φ) = h(φ). Hence h(φ) <∞ by Theorem 1.4.

Let π : D → D/ ker∞ φ be the canonical projection. Then π(H) = (H + ker∞ φ)/ ker∞ φ is a φ-invariantpure (i.e., divisible) subgroup of D/ ker∞ φ and we have the following two diagrams, where φH : D/H → D/H

is the endomorphism induced by φ and φπ(H) : (D/ ker∞ φ)/π(H)→ (D/ ker∞ φ)/π(H) is the endomorphism

induced by φ.

Hφ↾H //

��

��

H��

��

π(H)φ↾π(H) //

��

��

π(H)��

��D

φ //

����

D

����

D/ ker∞ φ // φ //

����

D/ ker∞ φ

����D/H

φH // D/H (D/ ker∞ φ)/π(H)φπ(H) // (D/ ker∞ φ)/π(H)

By Proposition 4.5

(i) ATh(D,φ, ker∞ φ) holds.

Since ker∞ φ ↾H= H ∩ ker∞ φ is φ ↾H -invariant and is pure in H , by Proposition 4.5

(ii) ATh(H,φ ↾H , H ∩ ker∞ φ) holds.

Since h(φ ↾ker∞ φ) = 0 by Corollary 2.6,

(iii) ATh(ker∞ φ, φ ↾ker∞ φ, H ∩ ker∞ φ) holds.

Since ker∞ φ = (H + ker∞ φ)/H , by Proposition 4.5,

(iv) ATh(D/H, φH , (H + ker∞ φ)/H) holds.

Assume that

14

(v) ATh(D/ ker∞ φ, φ, (H + ker∞ φ)/ ker∞ φ) holds.

So the hypotheses of Proposition 3.7 are satisfied and it yields that ATh(D,φ,H) holds. This shows that wecan assume without loss of generality that φ is injective. Then φ is surjective as well, hence the conclusionfollows from Proposition 4.2.

The following is a clear consequence of Proposition 4.6 and Corollary 4.4.

Corollary 4.7. Let G be a torsion-free abelian group of finite rank, φ ∈ End(G) and H a pure φ-invariantsubgroup of G. Then ATh(G,φ,H) holds.

In the remaining part of this section we discuss other consequences of Propositions 2.12 and 4.6.

Corollary 4.8. Let G be a torsion-free abelian group and φ ∈ End(G).

(a) If G has finite rank, then h(φ) <∞.

(b) If G = V (φ, g) for some g ∈ G, then G has finite rank if and only if h(φ) <∞.

Proof. (a) Since D(G) ∼= Qn, for n ∈ N+, Proposition 4.6 gives h(φ) < ∞, an so h(φ) = h(φ) < ∞ byProposition 2.12.

(b) When G = V (φ, g) has finite rank, then h(φ) <∞ by (a). On the other hand, h(φ) <∞ implies r(G)finite by Lemma 2.11.

Corollary 4.8(b) allows for a clear picture about the entropies of endomorphisms of torsion-free abeliangroups. Indeed, for every element g of a torsion-free abelian group G,

h(φ ↾V (φ,g)) =∞ if and only if r(V (φ, g)) is infinite.

Corollary 4.9. Let G be a torsion-free abelian group and φ ∈ End(G). If h(φ ↾V (φ,g)) =∞, then h(φ ↾V (φ,z)

) =∞ for every z ∈ V (φ, g) \ {0}.

Proof. By Corollary 4.8(b), h(φ ↾V (φ,g)) = ∞ implies r(V (φ, g)) infinite. For z ∈ V (φ, g) \ {0}, it is easy tosee that r(V (φ, z)) is infinite as well, and so h(φ ↾V (φ,z)) =∞ again by Corollary 4.8(b).

In other words, for every φ-invariant subgroup H of V (φ, g) one has the following surprising dychotomy:

either H = 0 or h(φ ↾H) =∞.

5 Proof of the Addition Theorem

Lemma 5.1. Let G be a torsion-free abelian group of finite rank. Then mG has finite index in G for everym ∈ N+.

Proof. Let n = r(G). Let m ∈ N+ and write it as product of primes, that is, m = p1 · . . . · pr. We proceed byinduction on r ∈ N+. Let r = 1, that is, m = p is a prime. Since r(pG) = n, we can think that Zn ⊆ pG ⊆ Qn.Consequently, G/pG ∼= (G/Zn)/(pG/Zn). Since G/Zn ⊆ Qn/Zn ∼=

⊕q(Z(q

∞))n, G/Zn ∼=⊕

q Fq, whereFq ≤ Z(q∞))n for every prime q. Then Fq = Z(q∞)nq×Lq, for some nq ∈ N with nq ≤ n and Lq ≤ Fq finite (see[7, Section 25.1]). For every prime q 6= p, pFq = Fq, soG/pG ∼= (

⊕q Fq)/(pFp×

⊕q 6=p Fq)

∼= Fp/pFp ∼= Lp/pLp.Now Lp is finite and so G/pG ∼= Lp/pLp is finite as well, and this shows that pG has finite index in G.

Assume now that the assertion holds for r and that m = p1 · . . . · pr+1. By inductive hypothesis, G′ =p2 · . . . · pr+1G has finite index in G, and by the case r = 1, p1G

′ has finite index in G′. Then mG has finiteindex in G.

It is now possible to prove in the following proposition that the Addition Theorem 1.3 holds with respectto the torsion subgroup. We assume that the group is countable, but this hypothesis is removable in view ofProposition 3.3.

Proposition 5.2. Let G be a countable abelian group and φ ∈ End(G). Then ATh(G,φ, t(G)) holds.

15

Proof. Suppose that there exists g ∈ G such that V (φ, g) has infinite rank. Let π : G → G/t(G) be thecanonical projection on the torsion-free abelian group G/t(G), and let φ : G/t(G) → G/t(G) be the inducedendomorphism. Then π(V (φ, g)) = V (φ, π(g)) has infinite rank and so h(φ ↾V (φ,π(g))) = ∞ by Lemma 2.11.

Therefore, h(φ) =∞ and h(φ) =∞ by Lemma 2.7.Suppose now that V (φ, g) has finite rank for every g ∈ G. Let F ∈ [G]<ω. Then

GF = V (φ, F )

has finite rank and we show thatATh(GF , φ ↾GF , t(GF )) holds. (5.1)

By Lemma 2.17(c) there exists m ∈ N+ such that mt(GF ) = 0. Then there exists a finite-rank torsion-freesubgroup K of GF such that GF ∼= K × t(GF ) (in view of a theorem by Kulikov, see [7, Section 27.5]). Sincet(GF ) is a φ-invariant subgroup of GF that splits, by Remark 3.4 this gives rise to a skew product, that is,there exists a homomorphism sφ : K → t(GF ) such that φ(x, y) = (φ1(x), φ2(y)+sφ(x)) for every (x, y) ∈ GF ,where φ1 : K → K is conjugated to φ : GF /t(GF ) → GF /t(GF ) by the isomorphism K ∼= GF /t(GF ), andφ2 = φ ↾t(GF ). We show that sφ(K) is finite. In fact, let π : K → K/mK be the canonical projectionand let ψ : K/mK → t(GF ) be defined by ψ(π(x)) = sφ(x) for every x ∈ K. Since sφ(mK) = msφ(K) ⊆mt(GF ) = 0, so sφ(mK) = 0. Thus ψ is well-defined and sφ = ψ ◦ π. Now K/mK is finite by Lemma 5.1, sosφ(GF ) = ψ(K/mK) is finite as well. Then Proposition 3.6 gives (5.1), which is equivalent to

h(φ ↾GF ) = h(φ ↾t(GF )) + h(φ ↾GF ), (5.2)

where φ ↾GF : GF /t(GF )→ GF /t(GF ) is the induced endomorphism. By (3.6)

h(φ ↾GF ) = h(φ ↾GF /t(GF )).

By Proposition 2.19 there exists {Ln : n ∈ N} ⊆ [G]<ω , such that:

h(φ) = limn→∞

h(φ ↾V (φ,Ln)),

h(φ ↾t(G)) = limn→∞

h(φ ↾t(G)∩V (φ,Ln)) = limn→∞

h(φ ↾t(V (φ,Ln)), and

h(φ) = limn→∞

h(φ ↾π(Ln)) = limn→∞

h(φ ↾V (φ,Ln)).

Consequently, by (5.2) and these equalities,

h(φ) = limn→∞

h(φ ↾V (φ,Ln))

= limn→∞

h(φ ↾t(V (φ,Ln))) + limn→∞

h(φ ↾V (φ,Ln))

= h(φ ↾t(G)) + h(φ),

that is, ATh(G,φ, t(G)) holds.

Lemma 5.3. Let G be a countable abelian group, φ ∈ End(G) and H a φ-invariant subgroup of G such thatG/H is torsion. Then ATh(G,φ,H) holds.

Proof. We will apply Proposition 3.7 with K = t(G). Let φH : G/H → G/H and φt(G) : G/t(G) → G/t(G)be the endomorphisms induced by φ. We have that

(i) ATh(G,φ, t(G)) and

(ii) ATh(H,φ ↾H , t(H)) hold by Proposition 5.2;

(iii) ATh(t(G), φ ↾t(G), t(H)) and

(iv) ATh(G/H, φH , (H + t(G))/H) hold because t(G) and G/H are torsion [5];

(v) ATh(G/t(G), φt(G), (H+t(G))/t(G)) holds by Corollary 4.3, asG/t(G) is torsion-free and (H+t(G))/t(G)is essential in G/t(G), being G/(H + t(G)) torsion as a quotient of the torsion group G/H .

Now Proposition 3.7 with K = t(G) applies to conclude the proof.

Proposition 5.4. Let G be a countable torsion-free abelian group, φ ∈ End(G) and let H be a φ-invariantsubgroup of G. Then ATh(G,φ,H) holds true.

16

Proof. Assume there exists g ∈ G such that V (φ, g) has infinite rank. Then h(φ ↾V (φ,g)) = ∞ by Corollary4.8(b) and so h(φ) = ∞ by Lemma 2.7. If V (φ, g) ∩ H is non-zero, then h(φ ↾H) = ∞ by Corollary 4.9and Lemma 2.7. Then h(φ) = h(φ ↾H) = ∞ and in particular ATh(G,φ,H) holds. Assume now thatV (φ, g) ∩ H = 0. Let π : G → G/H be the canonical projection. Therefore, V (φ, g) projects injectively inthe quotient G/H and so π(V (φ, g)) = V (φ, π(G)) has infinite rank. So h(φ) = ∞ by Corollary 4.8(b) andLemma 2.7. Hence h(φ) = h(φ) =∞ by Lemma 2.7 and in particular ATh(G,φ,H) holds.

We show now thatif G has finite rank, then ATh(G,φ,H) holds true. (5.3)

According to Proposition 2.12 h(φ) = h(φ). Since G is essential in D(G), G/H is essential in D(G)/H . Let

φ : G/H → G/H and φ : D(G)/H → D(G)/H be the induced endomorphisms, and note that φ ↾G/H= φ.

By Corollary 2.14(a), h(φ) = h(φ). Then it suffices to prove that (D(G), φ,H) holds. Since D(H) is φ-

invariant and D(H)/H = t(D(G)/H), hence ATh(D(G)/H, φ,D(G)/H) holds by Proposition 5.2. Moreover,

ATh(D(H), φ ↾D(H), H) holds by Lemma 5.3, and ATh(D(G), φ, D(H)) holds by Proposition 4.6. Therefore,

Corollary 3.8(b) applies to conclude that ATh(D(G), φ, H) holds.

So, going back to the general case, we can suppose now that V (φ, g) has finite rank for every g ∈ G. Inparticular V (φ, F ) has finite rank for every F ∈ [G]<ω . By (5.3) ATh(V (φ, F ), φ ↾V (φ,F ), H ∩ V (φ, F )) holdsfor every F ∈ [G]<ω, that is

h(φ ↾V (φ,F )) = h(φ ↾H∩V (φ,F )) + h(φ ↾V (φ,F )),

where φ ↾V (φ,F ) : V (φ, F )/(H ∩ V (φ, F ))→ V (φ, F )/(H ∩ V (φ, F )) is the induced homomorphism. By (3.6)

h(φ ↾V (φ,F )) = h(φ ↾V (φ,F )/(H∩V (φ,F ))).

By Proposition 2.19 there exists a family {Ln : n ∈ N} of finite subsets of G, such that:

h(φ) = limn→∞

h(φ ↾V (φ,Ln)),

h(φ ↾H) = limn→∞

h(φ ↾H∩V (φ,Ln)), and

h(φ) = limn→∞

h(φ ↾π(Ln)) = limn→∞

h(φ ↾V (φ,Ln)).

Consequently

h(φ) = limn→∞

h(φ ↾V (φ,Ln))

= limn→∞

h(φ ↾H∩V (φ,Ln)) + limn→∞

h(φ ↾V (φ,Ln))

= h(φ ↾H) + h(φ),

and this gives the thesis.

Corollary 5.5. Let G be a countable abelian group, φ ∈ End(G) and let H be a φ-invariant subgroup of G.Then ATh(G,φ,H) holds when:

(a) H is a torsion subgroup of G;

(b) H contains the torsion subgroup t(G);

(c) there exists a φ-invariant subgroup K of G such that H ∩K ⊆ t(G) and G = H +K;

(d) G = H + t(G).

Proof. (a) The subgroup t(G)/H ofG/H is precisely t(G/H), so both ATh(G/H, φ, t(G)/H) and ATh(G,φ, t(G))hold by our hypothesis. On the other hand, ATh(t(G), φ ↾t(G), H) holds as t(G) is torsion [5]. Now Corollary3.8(b) applies to the triple H ⊆ t(G) ⊆ G.

(b) Note that ATh(G,φ, t(G)) and ATh(H,φ ↾H , t(G)) hold by Proposition 5.2, and ATh(G/t(G), φ,H/t(H))holds by Proposition 5.4. By Corollary 3.8(a) applied to the triple t(G) ⊆ H ⊆ G, ATh(G,φ,H) holds.

(c) Follows directly from (a) and Corollary 3.9(a).

(d) Follows immediately from (c) with K = t(G).

17

We can now prove the Addition Theorem 1.3:

Proof of Theorem 1.3. By Proposition 3.3 we can suppose that G is countable. According to Corollary5.5(b) we have that ATh(G,φ, t(G) +H) holds, while ATh(t(G) +H,φ ↾t(G)+H , H) holds by Corollary 5.5(d).

Finally, ATh(G/H, φ, (t(G) + H)/H) holds by Corollary 5.5(a) as the subgroup (t(G) + H)/H of G/H istorsion. Now Corollary 3.8 applies to the triple H ⊆ t(G) +H ⊆ G to conclude the proof.

6 The Uniqueness Theorem

We start this section proving the Uniqueness Theorem 1.5 for the algebraic entropy h in the category of allabelian groups.

Proof of Theorem 1.5. Let h∗ = {h∗G : G abelian group} be a collection of functions h∗G : End(G) → R+

satisfying (a) – (e) from Theorem 1.5. We have to show that h∗(φ) = h(φ) for every abelian group G andφ ∈ End(G).

(i) If G is torsion, then h∗(φ) = ent(φ) = h(φ) by the Uniqueness Theorem for ent proved in [5, Theorem6.1].

(ii) It suffices to consider the case when G is torsion-free. Indeed, let φ : G/t(G) → G/t(G) be theendomorphism induced by φ, where G/t(G) is torsion-free. By (c) h∗(φ) = h∗(φ ↾t(G)) + h∗(φ) and by

the Addition Theorem 1.3 h(φ) = h(φ ↾t(G)) + h(φ). Since h∗(φ ↾t(G)) = h(φ ↾t(G)) by (i), it follows that

h∗(φ) = h(φ) if h∗(φ) = h(φ).

(iii) If G is torsion-free and r(G) = n is finite, then h∗(φ) = h(φ). Indeed, D(G) ∼= Qn and D(G)/G

is torsion. Let φ : D(G)/G → D(G)/G be the endomorphism induced by φ : D(G) → D(G). By the

Algebraic Yuzvinski Formula (1.4) and by (e) h∗(φ) = h(φ) and the value is finite. Since D(G)/G is torsion,

h∗(φ) = h(φ) by (i). By (c) h∗(φ) = h∗(φ) + h∗(φ) and by the Addition Theorem 1.3 h(φ) = h(φ) + h(φ).

Then h∗(φ) = h∗(φ)− h∗(φ) = h(φ)− h(φ) = h(φ).

(iv) If G is torsion-free, G = V (φ, g) for some g ∈ G and r(G) is infinite, then h∗(φ) =∞ = h(φ). In fact,G ∼=

⊕n∈N 〈φ

n(g)〉 ∼=⊕

n∈N Z. Moreover, φ is conjugated to βZ through this isomorphism and so h(φ) = ∞by Proposition 2.8(a) and Example 2.10. Also h∗(φ) = ∞. Indeed, h∗(φ) = h∗(βZ) by (a). Moreover,h∗(βZ(p)) = log p, for every prime p, by (d). For G = Z(N) and for every prime p, the subgroup pG of G is

βZ-invariant, so induces an endomorphism βZ : G/pG→ G/pG. Since G/pG ∼= Z(p)(N), and βZ is conjugatedto βZ(p) through this isomorphism, h∗(βZ) = h∗(βZ(p)) = log p by (a). Therefore, h∗(βZ) ≥ log p for everyprime p by (c), and so h∗(βZ) =∞.

(v) If G is torsion-free and G = V (φ, F ) for some F ∈ [G]<ω, then h∗(φ) = h(φ). To prove this, letF = {f1, . . . , fk}. Then G = V (φ, f1) + . . . + V (φ, fk). If r(V (φ, fi)) is finite for every i ∈ {1, . . . , k},then r(G) is finite as well, and h∗(φ) = h(φ) by (iii). If r(V (φ, fi)) is infinite for some i ∈ {1, . . . , k}, thenh∗(φ ↾V (φ,fi)) =∞ = h(φ ↾V (φ,fi)) by (iv). By (c) and by Lemma 2.7, we have h∗(φ) =∞ = h(φ).

(vi) Consider now the general case. By (ii) we can suppose without loss of generality that G is torsion-free.By Lemma 2.18 G = lim

−→{V (φ, F ) : F ∈ [G]<ω}. By (v) h∗(φ ↾V (φ,F )) = h(φ ↾V (φ,F )) for every F ∈ [G]<ω .

Therefore, (b) and Proposition 2.8(c) give h∗(φ) = supF∈[G]<ω h∗(φ ↾V (φ,F )) = supF∈[G]<ω h(φ ↾V (φ,F )) =

h(φ).

Note that the logarithmic law is not among the properties necessary to give uniqueness of h in the categoryof all abelian groups. This is different from the behavior of ent. Moreover, this means that the logarithmiclaw follows automatically from the other properties.

It is possible to prove the Uniqueness Theorem 1.5 also in a less direct way, that is, using a known theoremby Vamos on length functions [27]. We explain this alternative proof in the remaining part of this section.

Let R be a unitary commutative ring. We denote by ModR the category of all R-modules and theirhomomorphisms. An invariant i : ModR → R+ ∪ {∞} is a function such that i(0) = 0 and i(M) = i(M ′) ifM ∼=M ′ in ModR. For M ∈ModR, denote by F(M) the family of all finitely generated submodules of M .

Definition 6.1. [19, 27] Let R be a unitary commutative ring. A length function L of ModR is an invariantL : ModR → R+ ∪ {∞} such that:

(a) L(M) = L(M ′) + L(M ′′) for every exact sequence 0→M ′ →M →M ′′ → 0 in ModR;

18

(b) L(M) = sup{L(F ) : F ∈ F(M)}.

An invariant satisfying (b) is said additive and an invariant with (c) is called upper continuous. So a lengthfunction is an additive upper continuous invariant of ModR.

Consider the category AbGrp of all abelian groups and their homomorphisms. As done more generallyin [3], we introduce the category FlowAbGrp of flows of AbGrp. The objects of FlowAbGrp (namely, thealgebraic flows) are the pairs (G,φ) with G ∈ AbGrp and φ ∈ End(G). A morphism u : (G,φ) → (H,ψ)in FlowAbGrp between two algebraic flows (G,φ) and (H,ψ) is an homomorphism u : G→ H such that thediagram

Gφ

−−−−→ G

u

yyu

H −−−−→ψ

H

(6.1)

in AbGrp commutes. Two algebraic flows (G,φ) and (H,ψ) are isomorphic in FlowAbGrp if the morphismu : G→ H in (6.1) is an isomorphism in AbGrp.

By [3, Theorem 3.2]FlowAbGrp

∼= ModZ[t]. (6.2)

This isomorphism of categories is given by the functor F : FlowAbGrp →ModZ[t], associating to an algebraicflow (G,φ) the Z[t]-module Gφ, where Gφ is G with the structure of Z[t]-module given by the multiplicationtx = φ(x) for every x ∈ G. Moreover, for a morphism u : (G,φ) → (H,ψ) in FlowAbGrp, F (u) = u : Gφ →Hψ is an homomorphism of Z[t]-modules. In the opposite direction consider the functor F ′ : ModZ[t] →FlowAbGrp, which associates to M ∈ModZ[t] the algebraic flow (M,µt), where µt(x) = tx for every x ∈M .If u : M → N is an homomorphism in ModZ[t], then F ′(u) = u : (M,µt) → (N,µt) is a morphism inFlowAbGrp.

Remark 6.2. By the isomorphism (6.2), every function f defined on endomorphisms of abelian groups canbe viewed as a function f : FlowAbGrp → R+ ∪ {∞} or equivalently as f : ModZ[t] → R+ ∪ {∞} by lettingf(M) = f(M,µt) = f(µt). In particular, this holds for the algebraic entropy h : ModZ[t] → R+ ∪ {∞}.

So we have the following

Proposition 6.3. The algebraic entropy h is a length function of ModZ[t].

Proof. With respect to Definition 6.1, h satisfies (a) by Example 2.4, (b) by the Addition Theorem 1.3 and(c) by Proposition 2.8(c).

It is proved in [27] that the values of a length function L of ModR are determined by its values on thefinitely generated R-modules. In case R is a Noetherian commutative ring, L is determined by its valuesL(R/p) for prime ideals p of R [27, Corollary of Lemma 2].

Second proof of Theorem 1.5. Let h∗ = {h∗G : G abelian group} be a collection of functions h∗G : End(G)→R+ satisfying (a) – (e) from Theorem 1.5. Let R = Z[t]. By Remark 6.2 h∗ can be viewed as a functionh∗ : ModR → R+ ∪ {∞}. Then h∗ is a lenght function of ModR by (b) and (c). So in view of Proposition6.3 and of the above mentioned results from [27] it suffices only to check that

h∗(R/p) = h(R/p) for all prime ideals p of R.

Let us recall that R has Krull dimension 2. More precisely, the non-zero prime ideals p of R are eitherminimal of maximal. In particular, if p is a minimal prime ideal of R, then p = (f(t)), where f(t) ∈ R isirreducible (either f(t) = p is a prime in Z, or f(t) is irreducible with deg f(t) > 0). On the other hand, amaximal ideal m of R is of the form m = (p, f(t)), where p is a prime in Z and f(t) ∈ R has deg f(t) > 0 andis irreducible modulo p.

(i) For p = 0, we prove that h∗(R) =∞ = h(R).To this end we have to show that µt : R → R has h∗(µt) = ∞ = h(µt). Indeed, R is isomorphic to Z(N)

and µt is conjugated to βZ through this isomorphism. By Example 2.10 h(βZ) = ∞ and so h(µt) = ∞ byProposition 2.8(a). As shown in (iv) of the first proof of Theorem 1.5, also h∗(βZ) = ∞ and so h∗(µt) = ∞by (a). In particular, h∗(µt) =∞ = h(µt).

(ii) For p = m a maximal ideal of R, we see now that h∗(R/m) = 0 = h(R/m).

19

Indeed, m = (p, f(t)), where p ∈ Z is a prime and f(t) ∈ R is irreducible modulo p. Moreover, R/m ∼=Z(p)[t]/(fp(t)), where fp(t) is the reduction of f(t) modulo p. This shows that R/m is finite. Hence h∗(R/m) =h(R/m) by the Uniqueness Theorem for the algebraic entropy of endomorphisms of torsion abelian groupsproved in [5, Theorem 6.1], and h(R/m) = 0 by Lemma 2.5.

(iii) So it remains to see that h∗(R/p) = h(R/p) when p is a minimal prime ideal of R.

Assume first that p = (p) for some prime p ∈ Z. We show that h∗(R/p) = log p = h(R/p).Indeed, R/(p) ∼= Z(p)[t], Z(p)[t] ∼= Z(p)(N) and µt : Z(p)[t] → Z(p)[t] is conjugated to βZ(p) through this

isomorphism. By (a) and (d) h∗(µt) = h∗(βZ(p)) = log p, while Proposition 2.8(a) and Example 2.10 yieldh(µt) = h(βZ(p)) = log p.

Suppose now that p = (f(t)), where f(t) = a0 + a1t + . . . + an−1tn−1 + ant

n ∈ Z[t] is irreducible withdeg f(t) = n > 0. We verify that h∗(R/p) = h(R/p) =

∑|αi|>1 log |αi|, where αi are the roots of f(t).

Let M = R/(f(t)). Moreover let J = Q[t]f(t) be the principal ideal generated by f(t) in Q[t] andD = Q[t]/J . Let π : Q[t] → D be the canonical projection. Since J ∩ R = (f(t)), π induces an injectivehomomorphism M → D and we can think without loss of generality that M is a subgroup of D (identifyingM with π(R)). Now D ∼= Qn as abelian groups. Since r(M) ≥ n, M is essential in D. Consider µt : D → Dand µt ↾M= µt : M → M . By Corollary 4.8 h(µt) < ∞ and so Corollary 2.14(b) implies h(µt) = 0, whereµt = µt : D/M → D/M is induced by µt. Since D/M is torsion, h∗(µt) = h(µt) = 0 by the UniquenessTheorem for the algebraic entropy of endomorphisms of torsion abelian groups proved in [5, Theorem 6.1].By (c), h∗(µt) = 0 implies h∗(µt ↾M ) = h∗(µt). Moreover, h(µt ↾M ) = h(µt) by Corollary 2.14(a). So wehave to check that h∗(µt) = h(µt), that is, h∗(D) = h(D). We have D ∼= Qn and through this isomorphismµt : D → D is conjugated to the automorphism φ of Qn given by the companion matrix

C(f) =

0 0 . . . 0 − a0an

1 0 . . . 0 − a1an

0 1. . . 0 − a2

an...

. . .. . .

......

0 0 . . . 1 −an−1

an

.

In particular, g(t) = a0an

+ a1ant + . . . + an−1

antn−1 + tn ∈ Q[t] is the characteristic polynomial of C(f). Hence

(a) and Proposition 2.8(a) give respectively h∗(µt) = h∗(φ) and h(µt) = h(φ), while (e) and the AlgebraicYuzvinski Formula (1.4) yield h∗(φ) = h(φ) =

∑|αi|>1 log |αi|, where αi are the roots of g(t), which are the

same roots of f(t).

7 The Bridge Theorem

For an abelian group G the Pontryagin dual G is Hom(G,T) endowed with the compact-open topology [20].The Pontryagin dual of an abelian group is compact. Moreover, for an endomorphism φ : G→ G, its adjointendomorphism φ : G → G is continuous. For basic properties concerning the Pontryagin duality see [6] and

[11]. For a subset A of G, the annihilator of A in G is A⊥ = {χ ∈ G : χ(A) = 0}, while for a subset B of G,the annihilator of B in G is B⊥ = {x ∈ G : χ(x) = 0 for every χ ∈ B}.

We recall the definition of the topological entropy following [1]. For a compact topological space X andfor an open cover U of X , let N(U) be the minimal cardinality of a subcover of U . Since X is compact,N(U) is always finite. Let H(U) = logN(U) be the entropy of U . For any two open covers U and V of X ,let U ∨ V = {U ∩ V : U ∈ U , V ∈ V}. Define analogously U1 ∨ . . . ∨ Un, for open covers U1, . . . ,Un of X .Let ψ : X → X be a continuous map and U an open cover of X . Then ψ−1(U) = {ψ−1(U) : U ∈ U}. The

topological entropy of ψ with respect to U is Htop(ψ,U) = limn→∞H(U∨ψ−1(U)∨...∨ψ−n+1(U))

n , and the topologicalentropy of ψ is htop(ψ) = sup{Htop(ψ,U) : U open cover of X}.

In the following fact we collect the basic properties of the topological entropy.

Fact 7.1. Let K be a compact abelian group and ψ : K → K a continuous endomorphism.

(a) If H is another compact group, η : H → H a continuous endomorphism and ψ and η are conjugated(i.e., there exists a topological isomorphism ξ : K → H such that η = ξφξ−1), then htop(ψ) = htop(η).

(b) For every k ∈ N+, htop(ψk) = k ·htop(ψ). If ψ is an automorphism, then htop(ψ

k) = |k|htop(ψ) for everyk ∈ Z.

20

(c) If K is a inverse limit of closed ψ-invariant subgroups {Ki : i ∈ I}, then htop(φ) = supi∈I htop(φ ↾Ki).

(d) If K = K1 × K2 and ψ = ψ1 × ψ2 with ψi : Ki → Ki continuous, i = 1, 2, then htop(ψ1 × ψ2) =h(ψ1) + h(ψ2).

As the right Bernoulli shift is a fundamental example for the algebraic entropy, the left Bernoulli shiftplays the same role for the topological entropy:

Example 7.2. For any compact abelian group K the (left) Bernoulli shift is Kβ : KN → KN defined by

(x0, x1, x2, . . .) 7→ (x1, x2, x3 . . .).

(a) It is a well-known fact (see [26]) that htop(Kβ) = log |K|, with the usual convention that log |K| =∞, if|K| is infinite. In particular, htop(Z(p)β) = log p, for every prime p.

(b) Moreover, Kβ = βK (see [4, Proposition 6.1]).

It was proved by Bowen and Peters that an Addition Theorem holds also for the topological entropy ofcontinuous endomorphisms of compact groups:

Theorem 7.3 (Addition Theorem). Let K be a compact abelian group, ψ : K → K a continuous endomor-phism, N a closed ψ-invariant subgroup of K and ψ : K/N → K/N the endomorphism induced by ψ. Thenhtop(ψ) = htop(ψ ↾N ) + htop(ψ).

The following is the Yuzvinski Formula for the topological entropy.

Theorem 7.4 (Yuzvinski Formula). [31] For n ∈ N+ an automorphism ψ of Qn is described by a matrixA ∈ GLn(Q). Then

htop(ψ) = log s+∑

|αi|>1

log |αi|, (7.1)

where αi are the eigenvalues of A and s is the least common multiple of the denominators of the coefficientsof the (monic) characteristic polynomial of A.

Remark 7.5. Let G be an abelian group and φ ∈ End(G). Let K = G and ψ = φ. Let also H be a φ-invariantsubgroup of G. By the Pontryagin duality, N = H⊥ is a closed ψ-invariant subgroup of K, and N⊥ = H .Moreover, we have the following commutative diagrams:

H

φ↾H

��

� � // G

φ

��

// // G/H

φ

��

K/N Koooo N?_oo

H� � // G // // G/H K/N

ψ

OO

K

ψ

OO

oooo N

ψ↾N

OO

? _oo

The second diagram is obtained by the first one applying the Pontryagin duality functor. In particular,

K/N ∼= H and N ∼= G/H . Moreover, ψ is conjugated to φ ↾H and ψ ↾N is conjugated to φ. By Proposition2.8(a),

(i) h(φ ↾H) = h(ψ),

(ii) h(φ) = h(ψ ↾N ).

The role of the hyperkernel in the case of endomorphisms of abelian groups, for continuous endomorphismsψ of compact groups K, is played by the hyperimage of ψ defined by Im∞ψ =

⋂n∈N ψ

n(K), which is a closedψ-invariant subgroup of K.

The next result shows that, as far as the computation of the value of the topological entropy of continuousendomorphisms of compact groups is concerned, one can restrict it to surjective endomorphisms.

Lemma 7.6. Let K be a compact group and ψ : K → K a continuous endomorphism. Then ψ ↾Im∞ψ issurjective and Im∞ψ is the largest closed ψ-invariant subgroup of K with this property. Moreover, h(ψ) =h(ψ ↾Im∞ψ).

Proof. The inclusion ψ(Im∞ψ) ⊆ Im∞ψ is obvious. If x ∈ Im∞ψ, then for every n ∈ N there exists xn ∈ Ksuch that x = ψn+1(xn). Let yn = ψn(xn). Then

yn ∈ ψn(K) and x = ψ(yn) for every n ∈ N. (7.2)

21

For n ∈ N, let now Yn = {yn, yn+1, . . .} ⊆ ψn(K). Since ψn(K) is compact, so closed, Yn ⊆ ψn(K) as well.By the compactness of K we have

⋂n∈N Yn 6= ∅. Let y ∈

⋂n∈N Yn ⊆ Im∞ψ. By (7.2) x = ψ(y). Hence

x ∈ ψ(Im∞ψ). This proves that ψ(Im∞ψ) ⊇ Im∞ψ, and so ψ(Im∞ψ) = Im∞ψ. Equivalently, ψ ↾Im∞ψ issurjective.

Assume that N is a closed ψ-invariant subgroup of K such that ψ(N) = N . Hence obvioulsy N ⊆ Im∞ψ.

To prove the equality h(ψ) = h(ψ ↾Im∞ψ), by the Addition Theorem 7.3 it suffices to show that h(ψ) =0, where ψ : K/Im∞ψ → K/Im∞ψ is the endomorphism induced by ψ. This follows from the fact that⋂n∈N ψ

n(K/Im∞ψ) = 0, due to the definition Im∞ψ =

⋂n∈N ψ

n(K).

We can now proof the Bridge Theorem 1.6.

Proof of Theorem 1.6. Let K = G and ψ = φ.

(i) It is possible to assume that G is torsion-free (i.e., K is connected). Indeed, consider t(G) and theendomorphism φ : G/t(G)→ G/t(G) induced by φ. Then c(K) = t(G)⊥ and so Remark 7.5 gives

t(G)

φ↾t(G)

��

� � // G

φ

��

// // G/t(G)

φ

��

K/c(K) Koooo c(K)? _oo

t(G)� � // G // // G/t(G) K/c(K)

ψ

OO

K

ψ

OO

oooo c(K)

ψ↾c(K)

OO

? _oo

where ψ : K/c(K)→ K/c(K) is the induced endomorphism. By the Addition Theorem 1.3 for the algebraicentropy h(φ) = h(φ ↾t(G)) + h(φ). By the Addition Theorem 7.3 for the topological entropy h(ψ) = h(ψ ↾c(K)

) + h(ψ). By Remark 7.5(i) and by Theorem 1.1, we have h(φ ↾t(G)) = htop(ψ). By Remark 7.5(ii) h(φ) =

h(ψ ↾c(K)). So if we show that h(ψ ↾c(K)) = htop(ψ ↾c(K)), this will imply h(φ) = htop(ψ).

(ii) We can assume that G is torsion-free of finite rank. Indeed, by (i) we can suppose that G is torsion-free.If there exists g ∈ G such that r(V (φ, g)) is infinite (in particular, V (φ, g) ∼= Z(N) and φ ↾V (φ,g) is conjugated

to βZ), then h(φ) ≥ h(φ ↾V (φ,g)) = ∞ by Lemma 2.7 and Corollary 4.8(b). Let N = V (φ, g)⊥. Then N is

a closed ψ-invariant subgroup of K such that K/N ∼= V (φ, g) ∼= TN. Moreover, the induced endomorphismψ : K/N → K/N is the adjoint of φ ↾V (φ,g). So, according to Example 7.2(b), ψ : K/N → K/N is conjugatedto Tβ, which has htop(Tβ) =∞ by Example 7.2(a). Hence htop(ψ) =∞ = h(φ).

Assume now that r(V (φ, g)) is finite for every g ∈ G. Then r(V (φ, F )) is finite for every F ∈ [G]<ω. ByLemma 2.18 h(φ) = supF∈[G]<ω h(φ ↾V (φ,F )). For every F ∈ [G]<ω let NF = V (φ, F )⊥. As noted in Remark

7.5, for every F ∈ [G]<ω, V (φ, F ) ∼= K/NF and h(φ ↾V (φ,F )) = h(ψF ), where ψF : K/NF → K/NF is the

endomorphism induced by ψ. So K = lim←−{K/NF : F ∈ [G]<ω}. If we verify that h(ψF ) = htop(ψF ) for every

F ∈ [G]<ω, then Proposition 2.8(c) and Fact 7.1(c) give

h(φ) = supF∈[G]<ω

h(φ ↾V (φ,F )) = supF∈[G]<ω

h(ψF ) = supF∈[G]<ω

htop(ψF ) = htop(ψ).

This shows that we can consider only torsion-free abelian groups of finite rank.

(iii) It suffices to prove the thesis for G a divisibile torsion-free abelian group of finite rank. Indeed, by (ii)we can assume that G is a torsion-free abelian group of finite rank n ∈ N+, i.e., K is a connected compactabelian group of dimension n. Assume without loss of generality (by Proposition 2.8(a) and Fact 7.1(a)) that

D(G) = Qn, and so that D(G) = Qn. Let ϕ = φ : Qn/G → Qn/G be the induced endomorphism, and

η =φ : Qn → Qn, N = G⊥. So Remark 7.5 gives the following corresponding diagrams:

G

φ

��

� � // Qn

φ

��

// // Qn/G

ϕ

��

K Qnoooo N?_oo

G� � // Qn // // Qn/G K

ψ

OO

Qn

η

OO

oooo N

η↾N

OO

? _oo

Then h(φ) = h(φ) by Proposition 2.12. The next step is to show that htop(ψ) = htop(η). To this end, sincehtop(η) = htop(η ↾N ) + htop(ψ) by the Addition Theorem 7.3, it suffices to prove that htop(η ↾N ) = 0. So

Qn/G is torsion, since G is essential in Qn. Therefore, h(η ↾N) = htop(η ↾N) by Theorem 1.1. By Remark

22

7.5 h(ϕ) = h(η ↾N ). Then it remains to verify that h(ϕ) = 0. Let H = Qn/G ∼=⊕

p Z(p∞)kp with each

kp ∈ N, kp ≤ n. For every m ∈ N+, the fully invariant subgroup H [m] = {x ∈ H : mx = 0} of H is finite, soh(ϕ ↾H[m]) = 0. Since H = lim

−→H [m], Proposition 2.8(c) yields h(ϕ) = supm∈N+

h(ϕ ↾H[m]) = 0.

We have seen that h(φ) = h(φ) and that htop(ψ) = htop(η). Then h(φ) = htop(η) would imply h(φ) =htop(ψ). In other words, it suffices to prove the thesis for φ ∈ End(Qn).

(iv) We can suppose that φ is injective (i.e., ψ surjective). Indeed, consider the corresponding diagramsgiven by Remark 7.5:

ker∞ φ

φ↾ker∞ φ

��

� � // G

φ

��

// // G/ ker∞ φ

φ

��

K/Im∞ψ Koooo Im∞ψ? _oo

ker∞ φ � � // G // // G/ ker∞ φ K/Im∞ψ

ψ

OO

K

ψ

OO

oooo Im∞ψ

ψ↾Im∞ψ

OO

? _oo

Indeed, Im∞ψ = (ker∞ φ)⊥. By Corollary 2.6 and the Addition Theorem 1.3 h(φ) = h(φ), where the inducedendomorphism φ : G/ ker∞ φ → G/ ker∞ φ is injective. By Lemma 7.6 htop(ψ) = htop(ψ ↾Im∞ψ), where

ψ ↾Im∞ψ is surjective. Remark 7.5 yields h(φ) = h( ψ ↾Im∞ψ). So if we prove that h( ψ ↾Im∞ψ) = htop(ψ ↾Im∞ψ),this will imply h(φ) = htop(ψ).

(v) By (iii) we can assume that G is a divisible torsion-free abelian group of finite rank, that is, G = Qn forn = r(G). By (iv) we can suppose that φ ∈ End(Qn) is injective; then φ is also surjective and so φ ∈ Aut(Qn).Therefore, h(φ) = htop(ψ) by Theorem 1.2.

Note that step (v) of the proof of the Bridge Theorem 1.6 can be proved also applying the Algebraic

Yuzvinski Formula (1.4) to φ and the Yuzvinski Formula (7.1) to φ. So if one has an independent proof ofthe Algebraic Yuzvinski Formula (1.4), this would be a proof of the Bridge Theorem 1.6 independent from theparticular case proved by Peters.

8 Computation of Peters entropy via Mahler measure

Let f(t) = a0 + a1t+ . . .+ aktk ∈ Z[t] be a primitive polynomial. Let {αi : i = 1, . . . , k} ⊆ C be the set of all

roots of f(t). The Mahler measure of f(t) is

m(f(t)) = log |ak|+∑

|αi|>1

log |αi|.

Sometimes the exponential form of Mahler measure M(f(t)) =∑ki=1 max{1, |αi|} is also considered [12];

clearly, m(f(t)) = logM(f(t)). The Mahler measure plays an important role in number theory and arithmeticgeometry (see [8, Chapter 1]).

If g(t) ∈ Q[t] is monic, then there exists a smallest s ∈ N+ such that sg(t) ∈ Z[t]; in particular, sg(t) isprimitive. So we can define the Mahler measure of g(t) as m(g(t)) = m(sg(t)).

For an algebraic number α ∈ C, the Mahler measure m(α) of α is the Mahler measure of the minimalpolynomial of α.

Lehmer [15], with the aim of generating large primes, associated to any monic polynomial f(t) ∈ Z[t] withroots α1, . . . , αk the sequence of integers

∆n(f(t)) =

k∏

i=1

|1− αni |.

The idea comes from Mersenne primes generated by the polynomial f(t) = t − 2. Lehmer was using thepolynomial f(t) = t3 − t − 1. This is the the non-reciprocal polynomial with the smallest positive Mahlermeasure [25]. The polynomial

g(t) = x10 + x9 − x7 − x6 − x5 − x4 − x3 + x+ 1

is the reciprocal polynomial with the smallest known positive Mahler measure, that is, m(g(t)) = logλ, whereλ = 1.17628 . . . is the Lehmer number [12]. Still in [12] it is noted that λ is the largest real root of g(t) and itis the only one of its algebraic conjugates outside the unit circle (i.e., λ is a Salem number). If there exists apolynomial h(t) with positive Mahler measure smaller than this, then deg h(t) ≥ 55 [17].

23

Remark 8.1. Let f(t) ∈ Z[t] \ {0} be monic with roots α1, . . . , αk. Then

m(f(t)) = limn→∞

log |∆n(f(t))|

n.

Indeed,

limn→∞

log |∆n(f(t))|

n= lim

n→∞

∑ki=1 log |1− α

ni |

n=

k∑

i=1

limn→∞

log |1− αni |

n=

∑

|αi|>1

log |αi|.

Theorem 8.2 (Kronecker Theorem). [14] Let f(t) ∈ Z[t] be a monic polynomial with roots α1, . . . , αk. If α1

is not a root of unity, then |αi| > 1 for at least one i ∈ {1, . . . , k}.

Corollary 8.3. Let f(t) ∈ Z[t] \ {0} be primitive. Then m(f(t)) = 0 if and only if f(t) is cyclotomic (i.e.,all the roots of f(t) are roots of unity). Consequently, if α is an algebraic integer, then m(α) = 0 if and onlyif α is a root of unity.

Proof. Let f(t) = a0 + a1t+ . . .+ aktk and deg f(t) = k. Assume that all roots α1, . . . , αk of f(t) are roots of

unity. Then there exists m ∈ N+ such that every αi is a root of tm − 1. By Gauss Lemma f(t) divides tm − 1in Z[t]. Therefore, ak = 1, hence m(f(t)) = 0. Suppose now that m(f(t)) = 0. Then we can suppose withoutloss of generality that f(t) is monic. Moreover, all |αi| ≤ 1 for every i ∈ {1, . . . , k}. By Theorem 8.2 each αiis a root of unity.

This determines completely the case of zero Mahler measure.

Problem 8.4 (Lehmer Problem). [15] Given any δ > 0, is there an algebraic integer whose Mahler measureis strictly between 0 and δ?

This problem is equivalent replacing algebraic integer with algebraic number. Moreover, this is equivalentto ask whether inf{m(f(t)) : f(t) ∈ Z[t] primitive} = 0. It suffices to consider monic polynomials. Indeed, ifdeg(f(t)) = k and f(t) = a0 + a1t+ . . .+ akt

k with |ak| > 1, then m(f(t)) ≥ log |ak| ≥ log 2 > 0.

It is known from [5] that the algebraic entropy ent takes values in logN+ ∪ {∞}. Then

inf{ent(φ) : G abelian group, φ ∈ End(G), ent(φ) > 0} = log 2. (8.1)

We consider now the positive values of the algebraic entropy h, that is, the real number

ε = inf{h(φ) : G abelian group, φ ∈ End(G), h(φ) > 0}.

Problem 8.5. Is ε = 0?

By the Bridge Theorem 1.6, this problem is equivalent to the major open problem about the infimum ofthe positive values of the topological entropy (see [28]).

The following is an immediate consequence of the Algebraic Yuzvinski Formula (1.4).

Corollary 8.6. Let n ∈ N+ and φ ∈ Aut(Qn). Then h(φ) = m(f(t)), where f(t) ∈ Q[t] is the (monic)characteristic polynomial of the matrix associated to φ.

Theorem 8.7. We have ε = inf{h(φ) : n ∈ N+, φ ∈ Aut(Qn), h(φ) > 0}.

Proof. (i) Let G be an abelian group, φ ∈ End(G) and φ : G/t(G) → G/t(G) the endomorphism induced byφ. Since h(φ ↾t(G)) = ent(φ ↾t(G)), we can suppose that h(φ ↾t(G)) = 0, otherwise h(φ) ≥ h(φ ↾t(G)) ≥ log 2

by Lemma 2.7 and (8.1). By the Addition Theorem 1.3 h(φ) = h(φ). In other words, we can consider onlytorsion-free abelian groups G.

(ii) By (i) assume that G is a torsion-free abelian group and φ ∈ End(G). If there exists g ∈ G suchthat r(V (φ, g)) is infinite, then h(φ) ≥ h(φ ↾V (φ,g)) = ∞ by Lemma 2.7 and Corollary 4.8(b). Then we canconsider only G such that r(V (φ, g)) is finite for every g ∈ G. Then r(V (φ, F )) is finite for every F ∈ [G]<ω .By Lemma 2.18 h(φ) = supF∈[G]<ω h(φ ↾V (φ,F )). This shows that we can consider only torsion-free abeliangroups of finite rank.

(iii) By (i) assume that G is a torsion-free abelian group and φ ∈ End(G). By Theorem 2.12 h(φ) = h(φ),and so we can reduce to divisible torsion-free abelian groups.

24

(iv) By (ii) and (iii) we can consider divisible torsion-free abelian groups G of finite rank, namely, G ∼= Qn

for some n ∈ N+. Let φ ∈ End(Qn). By Proposition 4.5 h(φ) = h(φ), where the induced endomorphismφ : Qn/ ker∞ φ→ Qn/ ker∞ φ is injective (hence surjective) and Qn/ ker∞ φ ∼= Qm for some m ∈ N, m ≤ n, asker∞ φ is pure in Qn (so Qn/ ker∞ φ is divisible and torsion-free by Lemma 4.1(c)). Therefore we can considerautmorphisms of Qn, and this gives the thesis.

Theorem 8.7 and Corollary 8.6 have the following immediate consequence.

Corollary 8.8. Problem 8.5 is equivalent to Lehmer Problem 8.4.

By Corollary 8.6 the algebraic entropy h(φ) of a φ ∈ Qn is equal to the Mahler measure of a polynomialf(t) ∈ Z[t] with non-zero constant term. Now we see the viceversa, that is, the Mahler measure of a polynomialf(t) ∈ Z[t] with non-zero constant term is equal to the algebraic entropy of an automorphism φ of Qn for somen ∈ N+. Indeed, let f(t) = a0 + a1t+ . . .+ akt

k ∈ Z[t] with deg f(t) = k and a0 6= 0. Let

C(f) =

0 0 . . . 0 − a0ak

1 0 . . . 0 − a1ak

0 1. . . 0 − a2

ak...

. . .. . .

......

0 0 . . . 1 −ak−1

ak

(8.2)

be the companion matrix associated to f(t). The characteristic polynomial of C(f) is f(t). Since detC(f) =(−1)k+2 a0

ak6= 0 by hypothesis, C(f) is the matrix associated to an automorphism φ ofQn. Thenm(f(t)) = h(φ)

by Corollary 8.6.

Remark 8.9. Let α ∈ C be an algebraic number of degree n ∈ N+ over Q and let f(t) ∈ Q[t] be its minimalpolynomial, with deg f(t) = n. Let K = Q(α); then K ∼= Qn as abelian groups. Following [32], call algebraicentropy h(α) of α the algebraic entropy h(µα), where µα is the multiplication by α inK. Let ak be the smallestpositive integer such that akf(t) = a0 + a1t+ . . .+ akt

k ∈ Z[t]. With respect to the basis {1, α, . . . , αk−1} ofK, the matrix associated to µα is the companion matrix C(f) in (8.2). The characteristic polynomial of C(f)is f(t). By Corollary 8.6, h(µα) = m(f(t)), i.e., h(α) = m(α). This means that the algebraic entropy of analgebraic number is precisely its Mahler measure.

References

[1] R. L. Adler, A. G. Konheim, M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965),309–319.

[2] D. Dikranjan and A. Giordano Bruno, The Pinsker subgroup of an algebraic flow, submitted.http://arxiv.org/abs/1006.5120

[3] D. Dikranjan and A. Giordano Bruno, Entropy in a module category, submitted.http://arxiv.org/abs/1006.5122

[4] D. Dikranjan, A. Giordano Bruno and L. Salce, Adjoint algebraic entropy, Journal of Algebra 324 (2010),442–463.

[5] D. Dikranjan, B. Goldsmith, L. Salce and P. Zanardo, Algebraic entropy of endomorphisms of abeliangroups, Trans. Amer. Math. Soc. 361 (2009), 3401–3434.

[6] D. Dikranjan, I. Prodanov and L. Stoyanov, Topological Groups: Characters, Dualities and MinimalGroup Topologies, Pure and Applied Mathematics, Vol. 130, Marcel Dekker Inc., New York-Basel, 1989.

[7] L. Fuchs, Infinite Abelian Groups, Vol. I and II, Academic Press, 1970 and 1973.

[8] G. Everest and T. Ward, Heights of polynomials and entropy in algebraic dynamics, Universitext. Springer-Verlag London, Ltd., London, 1999.

[9] P. Halmos, On automorphisms of compact groups, Bull. Amer. Math. Soc. 49 (1943), 619–624.

[10] P. Halmos, Finite-Dimensional Vector Spaces, Undergraduate Texts in Mathematics, Springer-Verlag,New York - Heidelberg - Berlin, 1987.

[11] E. Hewitt and K. A. Ross, Abstract harmonic analysis I, Springer-Verlag, Berlin-Heidelberg-New York,1963.

25

[12] E. Hironaka, What is. . . Lehmers number?, Not. Amer. Math. Soc. 56 (2009), no. 3, 374–375.

[13] D. Kerr and H. Li, Dynamical entropy in Banach spaces, Invent. Math. 162 (2005), no. 3, 649–686.

[14] L. Kronecker, Zwei Satze uber Gleichungen mit ganzzahligen Coefficienten, Jour. Reine Angew. Math. 53(1857), 173–175.

[15] D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. 34 (1933), 461–469.

[16] D. Lind and T. Ward, Automorphisms of solenoids and p-adic entropy, Ergodic Theory Dynam. Systems8 (1988) no. 3, 411-419.

[17] M. J. Mossinghoff, G. Rhin and Q. Wu, Minimal Mahler measures, Experiment. Math. 17 (2008) no.4,451–458.

[18] D. G. Northcott, Lessons on rings, modules and multiplicities, Cambridge University Press, London 1968.

[19] D. G. Northcott and M. Reufel, A generalization of the concept of lenght, Quart. J. of Math. (Oxford)(2), 16 (1965), 297–321.

[20] L. S. Pontryagin, Topological Groups, Gordon and Breach, New York, 1966.

[21] J. Peters, Entropy on discrete Abelian groups, Adv. Math. 33 (1979), 1–13.

[22] J. Peters, Entropy of automorphisms on L.C.A. groups, Pacific J. Math. 96 (1981), no. 2, 475–488.

[23] L. Salce and P. Zanardo Commutativity modulo small endomorphisms and endomorphisms of zero alge-braic entropy, Models, Modules and Abelian groups (2008), 487–497.

[24] L. Salce and P. Zanardo A general notion of algebraic entropy and the rank entropy, Forum Math. 21(2009) no. 4, 579–599.

[25] C. Smyth, On the product of conjugates outside the unit circle of an algebraic integer, Bull. London Math.Soc. 3 (1971), 169–175.

[26] L. N. Stoyanov, Uniqueness of topological entropy for endomorphisms on compact groups, Boll. Un. Mat.Ital. B (7) 1 (1987), no. 3, 829–847.

[27] P. Vamos, Additive Functions and Duality over Noetherian Rings, Quart. J. of Math. (Oxford) (2), 19(1968), 43–55.

[28] T. Ward, Group automorphisms with few and with many periodic points, Proc. Amer. Math. Soc. 133(2004), no.1, 91–96.

[29] T. Ward, Entropy of Compact Group Automorphisms, on-line Lecture Notes(http://www.mth.uea.ac.uk/∼h720/lecturenotes/).

[30] M. D. Weiss, Algebraic and other entropies of group endomorphisms, Math. Systems Theory 8 (1974/75),no. 3, 243–248.

[31] S. Yuzvinski, Metric properties of endomorphisms of compact groups, Izv. Acad. Nauk SSSR, Ser. Mat.29 (1965), 1295–1328 (in Russian). English Translation: Amer. Math. Soc. Transl. (2) 66 (1968), 63–98.

[32] P. Zanardo, Algebraic entropy of endomorphisms over local one-dimensional domains. J. Algebra Appl.8 (2009) no. 6, 759–777.

26