Entropy on abelian groups

We introduce the algebraic entropy for endomorphisms of arbitrary abelian groups, appropriately modifying existing notions of entropy. The basic properties of the algebraic entropy are given, as well as various examples. The main result of this paper is the Addition Theorem showing that the algebraic entropy is additive in appropriate sense with respect to invariant subgroups. We give several applications of the Addition Theorem, among them the Uniqueness Theorem for the algebraic entropy in the category of all abelian groups and their endomorphisms. Furthermore, we point out the delicate connection of the algebraic entropy with the Mahler measure and Lehmer Problem in Number Theory.


Introduction
Inspired by the notion of entropy invented by Clausius in thermodynamics in the fifties of the nineteenth century, Shannon introduced the notion of entropy in Information Theory by the end of the forties of the last century. A couple of years later, Kolmogorov and Sinai introduced the notion of (measure) entropy in ergodic theory. By appropriate modification of their definition, Adler, Konheim, and McAndrew [1] obtained the notion of topological entropy h top (ψ) of a continuous self-map ψ : X → X of a compact topological space X (see Section 7 for the definition).
The compact groups and their continuous endomorphisms provide an instance where both the measure and the 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 K is measure preserving if and only if ψ is surjective. It was established by Berg that for a surjective continuous endomorphism 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 (as ψ is not measure preserving), while h top (ψ) still makes sense. However, as observed in [26], the restriction of ψ to the subgroup Im ∞ ψ = n∈N ψ n (K) is surjective and h top (ψ) = h top (ψ ↾ Im ∞ ψ ) (see Lemma 7.6). This allows us to say that in this sense the topological entropy and the measure entropy of the continuous endomorphisms of compact groups coincide.
Using ideas briefly sketched in [1], Weiss [30] developed the definition of algebraic entropy for endomorphisms 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 T n (φ, F ) = F + φ(F ) + . . . + φ n−1 (F ) be the n-th φ-trajectory of F (while T (φ, F ) = n∈N φ n (F ) is the φ-trajectory of F ). Moreover, the limit (which is the algebraic entropy of φ with respect to F ) 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, where K is the Pontryagin dual of K. Since the profinite abelian groups are precisely the Pontryagin duals of the torsion abelian groups, one can announce this also in the following form (we call this kind of result a Bridge Theorem): Theorem 1.1. [30] Let G be a torsion abelian group and φ ∈ End(G). Then ent(φ) = h top ( φ).
Peters proved another Bridge Theorem connecting the topological entropy and the algebraic entropy h by means of the Pontryagin duality: Theorem 1.2. [21] Let G be a countable abelian group and φ ∈ Aut(G). Then h(φ) = h top ( φ).
Since T − n (φ, F ) = T n (φ −1 , F ), the definition by Peters for automorphisms φ of abelian groups G can be given also using T n (φ, F ). The approach using T n (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 recourse to |T − n (φ, F )| to define the algebraic entropy). So we define the algebraic entropy h for endomorphisms φ of abelian 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 T n (φ, F ) = F + φ(F ) + . . . + φ n−1 (F ), and the φ-trajectory of F is T (φ, F ) = n∈N φ n (F ). For F finite, let be the algebraic entropy of φ with respect to F (this limit exists as proved in Corollary 2.2). The algebraic entropy of φ is In particular, for endomorphisms of torsion abelian groups, h coincides with the already defined algebraic entropy 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 properties are counterparts of the properties of ent. Moreover we give many examples, starting from the fact that the identical homomorphism has zero algebraic entropy. For ent it is obvious that the identical homomorphism has 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 Since h coincides with ent for endomorphisms of torsion abelian group, the Addition Theorem for ent proved 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-invariant subgroup of G, we write AT h (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 Addition Theorem 1.3 to appropriate particular cases. In particular, Lemma 3.2 is a reduction to the case of countable abelian groups. Moreover, we consider many properties of the algebraic entropy of endomorphisms of torsionfree abelian groups, still with the aim of proving the Addition Theorem 1.3. For example, for a torsion-free abelian group G and φ ∈ End(G), we see that h(φ) = h( φ), where φ : D(G) → D(G) is the extension of φ to the divisible hull D(G) of G (see Proposition 2.12). This allows to reduce the study of the algebraic entropy of the endomorphisms of torsion-free abelian groups to the case of divisible abelian groups. Moreover, we can reduce to the case of divisible torsion-free abelian group of finite rank. Now the endomorphism φ of G can be supposed to be injective (by Proposition 4.5) and then φ is also surjective. Finally this particular case of an automorphism of Q n can be managed through the Algebraic Yuzvinski Formula: Theorem 1.4 (Algebraic Yuzvinski Formula). For n ∈ N + an automorphism φ of Q n is described by a matrix A ∈ GL n (Q). Then h(φ) = log s + |αi|>1 log |α i |, (1.4) where α i are the eigenvalues of A and s is the least common multiple of the denominators of the coefficients of the (monic) characteristic polynomial of A.
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 Yuzvinski Formula (1.4) (already in the finite-rank torsion-free case). Needless to say, the value of this achievement will be 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 topological entropy by Stoyanov [26].
For any abelian group K the (right) Bernoulli shift β K : Theorem 1.5 (Uniqueness Theorem). The algebraic entropy h of the endomorphisms of the abelian groups is characterized as the unique collection h = {h G : G abelian group} of functions h G : End(G) → R + such that: (a) h G is invariant under conjugation for every abelian group G; (c) the Addition Theorem holds for h; (d) h K (N) (β K ) = log |K| for any finite abelian group K; (e) the Algebraic Yuzvinski Formula holds for h Q restricted to the automorphisms of Q.
Moreover, we see how this result can be deduced by a theorem of Vámos 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).
To prove this theorem we use its weaker forms proved by Weiss and Peters (see Theorems 1.1 and 1.2 respectively) and we apply the Addition Theorem 1.3 for the algebraic entropy and the Addition Theorem 7.3 for the topological entropy.
In Section 8 we first recall the Mahler measure, which is an important invariant studied in number theory and arithmetic geometry. Moreover we see that the problem of determining the infimum of the positive value of 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 positive integers, the set of rationals and the set of reals. For m ∈ N + , we use Z(m) for the finite cyclic group of order m.
Let G be an abelian group. With a slight divergence with the standard use, we denote by [G] <ω the set of all non-empty finite subsets of G. If H is a subgroup of G, we indicate this by H ≤ G. The subgroup of torsion 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 α G.
Moreover, End(G) is the ring of all endomorphisms of G. We denote by 0 G and id G respectively the endomorphism of G which is identically 0 and the identity endomorphism of G. Moreover, for k ∈ Z, let µ k : G → G be the endomorphism of G defined by µ k (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).

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 defining H(φ, F ) exists. We start proving that {log |T n (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 c n = log |T n (φ, F )|. Then c n+m ≤ c n + c m for every n, m ∈ N + .

Proof. By definition
Consequently, Corollary 2.2. Let G be an abelian group, φ ∈ End(G) and F ∈ [G] <ω . Then the limit H(φ, F ) = lim n→∞ Proof. By Lemma 2.1 the sequence {c n : n ∈ N + }, where c n = log |T n (φ, F )|, is subadditive. Then the sequence { cn n : n ∈ N + } has limit and lim n→∞ cn n = inf n∈N+ cn n by a known fact from Calculus, due to Fekete.
, it is possible to suppose without loss of generality that 0 ∈ F . Indeed, Intuitively, for any abelian group G, the zero endomorphism 0 G and the identity id G have zero algebraic entropy. In fact, Example 2.4. If G is an abelian group, then h(0 G ) = 0 and h(id G ) = 0. While it is clear from the definition that h(0 G ) = 0, to prove that h(id G ) = 0 requires some more effort. In fact, in [2] this is proved showing that the id G -trajectories of the non-empty finite subsets F of G have polynomial growth; more precisely, for every F ∈ [G] <ω , there exists P F (t) ∈ Z[t] such that |T n (φ, F )| ≤ P F (n) for every n ∈ N + (for id G one takes P F (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 exists n ∈ N + such that φ n (x) = x. Moreover, x ∈ G is a quasi-periodic point of φ if there exist n > m in N such that φ n (x) = φ m (x). We say that φ is locally (quasi-)periodic if every x ∈ G is a (quasi-)periodic point of φ. Moreover, φ is periodic if there exists n ∈ N + such that φ n (x) = x for every x ∈ G. Analogously, φ is quasi-periodic if there exist n > m in N such that φ n (x) = φ m (x) for every x ∈ G.
It is possible to prove the following lemma, making use only of the definition of algebraic entropy.
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 The subgroup ker ∞ φ is φ-invariant and also invariant for inverse images. Hence the induced endomorphism φ : G/ ker ∞ φ → G/ ker ∞ φ is injective. Since φ ↾ ker ∞ φ is locally nilpotent for every, and locally nilpotent implies locally quasi-periodic, the following is an immediate consequence of Lemma 2.5.
In the next lemma we show that h is monotone under taking restriction to invariant subgroups and under taking 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 : Now assume that F ∈ [G/H] <ω and F = π(F 0 ) for some F 0 ∈ [G] <ω , where π : G → G/H is the canonical projection. Then π(T n (φ, F 0 )) = T n (φ, F ) for every n ∈ N + . Therefore, H(φ, F 0 ) ≥ H(φ, F ) and by the arbitrariness 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 the known entropies. Indeed, similar properties holds for the algebraic entropy ent, for ent i , and also for the topological entropy (see Fact 7.1), which gave the inspiration. In the case of the algebraic entropy h they were proved in [21] for automorphisms, and we extend them for endomorphisms. Proposition 2.8. Let G be an abelian group and φ ∈ End(G).
<ω , assuming without loss of generality that 0 ∈ F (see Remark 2.3), and let n ∈ N + . Then T n (φ k , F ) ⊆ T kn−k+1 (φ, F ) and so To prove the converse inequality h(φ k ) ≤ kh(φ), let F ∈ [G] <ω and n ∈ N + . Let F 1 = T k (φ, F ) and note that T n (φ k , F 1 ) = T kn (φ, F ). Then We can conclude that h(φ k ) ≥ kh(φ). Now assume that φ is an automorphism. It suffices to prove that h( and so 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(φ k ) = 0 for some k ∈ N + , and The following is a fundamental example in the theory of algebraic entropy. Indeed, the value of h on the right Bernoulli shift is one of the conditions that give uniqueness of h, as we will show in Section 6.
Assume that K is an infinite abelian group. If K is non-torsion, then K contains a subgroup C ∼ = Z, so K (N) contains the β K -invariant subgroup C (N) isomorphic to Z (N) . Hence, by Lemma 2.7, Proposition 2.8(a) and the previous part of this example, h(β K ) ≥ h(β C ) = h(β Z ) = ∞. If K is torsion, then K contains arbitrarily large finite subgroups H. Consequently, K (N) contains the β K -invariant subgroup H (N) . By Lemma 2.7 and the first part of this example, Hence, for any abelian group K, h(β K ) = log |K|, with the usual convention that log |K| = ∞, if |K| is infinite.
Lemma 2.11. Let G be a torsion-free abelian group and φ ∈ End(G). If G = V (φ, g) for some g ∈ G and h(φ) < ∞, then G has finite rank.
The next result reduces the computation of the algebraic entropy of endomorphisms of torsion-free abelian groups 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).
The following example shows that Proposition 2.12 may fail if G is not the torsion-free.
Recall that, if G is a torsion-free abelian group, then a subgroup H of G is essential if and only if for every Corollary 2.14. Let G be a torsion-free abelian group, φ ∈ End(G), H a φ-invariant subgroup of G and φ : G/H → G/H the endomorphism induced by φ. Then the purification Proof. For the first assertion see [24,  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) ≤ G has finite rank, say n ∈ N. Then there exist k i ∈ Z, i = 0, . . . , n, such that n i=0 k i φ i (x) = 0. Since G is torsion-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 .
has finite trajectory under φ.
In the following example we calculate the algebraic entropy of the endomorphisms of Z and Q, and of the multiplications of torsion-free abelian groups. In particular, item (a) immediately shows a difference with the torsion case. For an abelian group G and φ ∈ End For k = 1 this follows from Example 2.4. Assume k > 1 and let To prove that h(φ) ≥ log a, take F 0 = {0, 1, . . . , a − 1}. Let us check that all sums Now an obvious induction argument applies. Therefore, this shows that |T n (φ, F 0 )| = a n , and so H n (φ, F 0 ) = n log a.
To prove the inequality h(φ) ≤ log a, note that the subgroup H of Q formed by all fractions having as denominators powers of b (i.e., the subring of Q generated by Thus (c) Let n ∈ N + .
(i) For k ∈ N + and µ k : Z n → Z n , h(µ k ) = n log k.
(ii) For r = a b ∈ Q with a > b > 0, and µ r : Q n → Q n , h(µ r ) = 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 µ k : G → G for some k ∈ N + . Then In item (b) of the above example we have given the explicit computation of the entropy of µ r : Q → Q, with r = a b > 1 and (a, b) = 1. One can also apply the Algebraic Yuzvinski Formula (1.4); indeed, the unique eigenvalue of µ r is a b > 1, and so (1.4) gives h(µ r ) = 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 : Therefore it suffices to show that H(φ, F m ) = 0. One can prove by induction that, for every n ∈ N + , T n (φ, F m ) is contained in a parallelogram with sides 2nm and nm(2 This is the smallest φ-invariant subgroup containing T (φ, F ) (and so also F ).
-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) for every 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).
(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.
Proof. (a) It follows from the fact that Lemma 2.18. Let G be an abelian group and φ ∈ End(G).
Since h is defined "locally", in some sense this lemma permits to reduce to the case G = V (φ, F ) for some F ∈ [G] <ω , that is, G is finitely generated (by F ) as a Z[t]-module. By Lemma 2.17(a) every subgroup and every quotient of G is finitely generated as a Z[t]-module. 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 {L n : n ∈ N} ⊆ [G] <ω , such that: Proof. We prove that, whenever G is a countable and φ ∈ End(G), Fn) ).
Let G = {g n : n ∈ N}, and for every n ∈ N let F n = {g 0 , . . . , g n }. Then G is increasing union of the F n and consequently of the V (φ, F n ). By Proposition 2.
, where this is an increasing union, and so by (ii) n ∈ N} ⊆ [G/H] <ω and this is an increasing union. By (ii) applied to G/H and φ, 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 countable Clearly, every countable φ-invariant subgroup of G containing an entropy support, will have the same property, that is, such a subgroup is not uniquely determined. In the sequel we denote by S(G, φ) the family of 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).
The family S(G, φ) is a club. Let us recall that a family C of countable subsets of an infinite set X is a club (for closed unbounded) if it is closed for countable increasing unions and if every countable subset of X is 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. Proof. Let G be an abelian group, φ ∈ End(G), H a φ-invariant subgroup of G, φ : G/H → G/H the endomorphism induced by φ and π : G → G/H be the canonical projection. Let S ∈ S(G, φ), S H ∈ S(H, φ ↾ H ) and S ∈ S(G/H, φ). We can assume without loss of generality that S ⊇ S H and π(S) ⊇ S. Then which is conjugated to φ ↾ π(S) . By hypothesis, and by Proposition 2.8(a), Since S is countable, by hypothesis AT h (S, φ ↾ S , S ′ H ) holds, and hence (3.1) implies that AT h (G, φ, H) holds as well.
Let K, H be abelian groups and We say that the homomorphism s is associated to the skew product φ.
Let us see that the 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 that splits as a direct summand, that is G = K × H. Let us see that φ is a skew product. Indeed, let ι : G/H → K be the natural isomorphism and let φ : A natural instance to this effect are fully invariant subgroups. For example, when D is a divisible group and 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 φ 1 : K → K the endomorphism of K conjugated to the induced endomorphism φ : G/H → G/H and we let φ 2 = φ ↾ H . The direct product π φ = φ 1 × φ 2 is the direct product associated to the skew product φ. We can extend to G = K × H the homomorphism s φ : K → H associated to the skew product, defining it to be 0 on H. This allows us to consider s φ ∈ End(G × H) and speak of the composition s 2 φ = 0, as well as φ = π φ + s φ in the ring End(G). In other words, the difference s φ = φ − π φ measures how much the skew product φ fails to coincide with its associated direct product π φ .
Example 3.5. Let K be a torsion-free abelian group and let T be a torsion abelian group. Then every φ ∈ 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 and φ-invariant, and suppose that φ is a skew product such that Then it suffices to prove that We can assume without loss of generality that (0, 0) ∈ F 1 × F 2 and that F 2 is a subgroup of T with F 2 ⊇ s φ (K). To conclude the proof, it suffices to show that, for n ∈ N + , We have π n φ (F 1 × F 2 ) = φ n 1 (F 1 ) × φ n 2 (F 2 ) and so T n (π φ , F 1 × F 2 ) = T n (φ 1 , F 1 ) × T n (φ 2 , F 2 ). One can prove by induction that, for every x ∈ K and every n ∈ N + , Fix m ∈ N and an m-tuple a 0 , a 1 , . . . , a m−1 ∈ F 1 . Applying (3.5) to a n for n = 0, 1, . . . , m − 1 we get In other words, and 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 be the canonical projection and φ : G/H → G/H the endomorphism induced on the quotient by φ. Then the subgroup π(K) of G/H is φ-invariant. Since π(K) is isomorphic to K/H ∩ K, and the induced endomorphism 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: Proof. The situation is described by the following diagram involving the pairs (G, H) and (G/K, (H + K)/K): Our hypotheses imply that: The applying Proposition 2.8(c). Moreover by (3.6). Applying (i), (iii), (v), (3.7), (3.8), (ii) and (iv), 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: Corollary 3.9. Let G be an abelian group, φ ∈ End(G) and H, K φ-invariant subgroups of G such that

The Addition Theorem in the torsion-free case
The next properties, frequently used in the sequel, are easy to prove.   . . , v k } is a basis of H and the matrix of φ with respect to B has the following block-wise form: Let χ and χ 1 , χ 2 ∈ Q[x] be the characteristic polynomials of A and A 1 , A 2 respectively. Then χ = χ 1 χ 2 . Let s 1 and s 2 be the least common multiples of the denominators of the coefficients of χ 1 and χ 2 respectively. This means that p 1 = s 1 χ 1 and p 2 = s 2 χ 2 ∈ Z[x] are primitive. By Gauss Lemma p = p 1 p 2 is primitive and so for s = s 1 s 2 the polynomial p = sχ ∈ Z[x] is primitive. Now the Algebraic Yuzvinski Formula (1.4) applied to φ, φ ↾ H , φ gives The next is a consequence of Corollary 2.14. An immediate consequence of this corollary is that for G a torsion-free abelian group, φ ∈ End(G), H a φ-invariant subgroup of G, and φ : G/H → G/H the endomorphism induced by φ, G/H is torsion implies that AT h (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 of the Addition Theorem 1.3 for torsion-free abelian groups and their pure subgroups can be reduced to the case of divisible ones. 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 next result will reduce the computation of the entropy of endomorphisms of a finite-rank torsion-free divisible abelian groups to the case of injective ones.

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 = ker φ; let φ : D/ ker φ → D/ ker φ be the endomorphism induced by φ. From [10, Section 58, Theorem 1] it follows that D ∼ = ker φ × φ(D). 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 φ n (x) = 0. Since G is essential in D(G) there exists k ∈ Z such that kx ∈ G \ {0}. Moreover, φ n (kx) = φ n (kx) = k φ n (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 We can prove now that the Addition Theorem 1.3 holds in the case of torsion-free countable finite-rank abelian groups and pure invariant subgroups. Indeed, the next proposition generalizes Proposition 4.2 to endomorphisms. As far as we work in the torsion-free context, it seems natural to consider mainly pure subgroups that allows us to remain in the class even under passage to quotients. Proposition 4.6. Let n ∈ N + , φ ∈ End(Q n ) and H a pure (i.e., divisible) φ-invariant subgroup of Q n . Then h(φ) < ∞ and AT h (Q n , φ, H) holds.
Assume that So the hypotheses of Proposition 3.7 are satisfied and it yields that AT h (D, φ, H) holds. This shows that we can assume without loss of generality that φ is injective. Then φ is surjective as well, hence the conclusion follows 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-invariant subgroup of G. Then AT h (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).
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 ) = ∞.

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 every m ∈ N + .
Proof. Let n = r(G). Let m ∈ N + and write it as product of primes, that is, m = p 1 · . . . · p r . We proceed by induction on r ∈ N + . Let r = 1, that is, m = p is a prime. Since r(pG) = n, we can think that Z n ⊆ pG ⊆ Q n . Consequently, G/pG ∼ = (G/Z n )/(pG/Z n ). Since G/Z n ⊆ Q n /Z n ∼ = q (Z(q ∞ )) n , G/Z n ∼ = q F q , where F q ≤ Z(q ∞ )) n for every prime q. Then F q = Z(q ∞ ) nq ×L q , for some n q ∈ N with n q ≤ n and L q ≤ F q finite (see [7,Section 25.1]). For every prime q = p, pF q = F q , so G/pG ∼ = ( q F q )/(pF p × q =p F q ) ∼ = F p /pF p ∼ = L p /pL p . Now L p is finite and so G/pG ∼ = L p /pL p 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 = p 1 · . . . · p r+1 . By inductive hypothesis, G ′ = p 2 · . . . · p r+1 G has finite index in G, and by the case r = 1, p 1 G ′ has finite index in G ′ . Then mG has finite index in G.
It is now possible to prove in the following proposition that the Addition Theorem 1.3 holds with respect to the torsion subgroup. We assume that the group is countable, but this hypothesis is removable in view of Proposition 3.3.
We show now that if G has finite rank, then AT h (G, φ, H) holds true. So, going back to the general case, we can suppose now that V (φ, g) has finite rank for every g ∈ G. In particular V (φ, F ) has finite rank for every By Proposition 2.19 there exists a family {L n : n ∈ N} of finite subsets of G, such that: Ln) ), and and this gives the thesis. (c) Follows directly from (a) and Corollary 3.9(a).
(d) Follows immediately from (c) with K = t(G).
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 Corollary 5.5(b) we have that AT h (G, φ, t(G) + H) holds, while AT h (t(G) + H, φ ↾ t(G)+H , H) holds by Corollary 5.5(d).
Finally, AT h (G/H, φ, (t(G) + H)/H) holds by Corollary 5.5(a) as the subgroup (t(G) + H)/H of G/H is torsion. Now Corollary 3.8 applies to the triple H ⊆ t(G) + H ⊆ G to conclude the proof.

The Uniqueness Theorem
We start this section proving the Uniqueness Theorem 1.5 for the algebraic entropy h in the category of all abelian 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 and φ ∈ End(G).
(vi) Consider now the general case. By (ii) we can suppose without loss of generality that G is torsion-free.
Note that the logarithmic law is not among the properties necessary to give uniqueness of h in the category of all abelian groups. This is different from the behavior of ent. Moreover, this means that the logarithmic law 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 theorem by Vámos 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 Mod R the category of all R-modules and their homomorphisms. An invariant i : Consider the category AbGrp of all abelian groups and their homomorphisms. As done more generally in [3], we introduce the category Flow AbGrp of flows of AbGrp. The objects of Flow AbGrp (namely, the algebraic flows) are the pairs (G, φ) with G ∈ AbGrp and φ ∈ End(G). A morphism u : (G, φ) → (H, ψ) in Flow AbGrp between two algebraic flows (G, φ) and (H, ψ) is an homomorphism u : It is proved in [27] that the values of a length function L of Mod R are determined by its values on the finitely generated R-modules. In case R is a Noetherian commutative ring, L is determined by its values L(R/p) for prime ideals p of R [27, Corollary of Lemma 2]. Let us recall that R has Krull dimension 2. More precisely, the non-zero prime ideals p of R are either minimal of maximal. In particular, if p is a minimal prime ideal of R, then p = (f (t)), where f (t) ∈ R is irreducible (either f (t) = p is a prime in Z, or f (t) is irreducible with deg f (t) > 0). On the other hand, a maximal 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 and is irreducible modulo p.
To this end we have to show that µ t : R → R has h * (µ t ) = ∞ = h(µ t ). Indeed, R is isomorphic to Z (N) and µ t is conjugated to β Z through this isomorphism. By Example 2.10 h(β Z ) = ∞ and so h(µ t ) = ∞ by Proposition 2.8(a). As shown in (iv) of the first proof of Theorem 1.5, also h * (β Z ) = ∞ and so h * (µ t ) = ∞ by (a). In particular, h * (µ t ) = ∞ = h(µ t ).
(ii) For p = m a maximal ideal of R, we see now that h * (R/m) = 0 = h(R/m).
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]/(f p (t)), where f p (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 groups proved 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).

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 adjoint endomorphism 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 We recall the definition of the topological entropy following [1]. For a compact topological space X and for an open cover U of X, let N (U) be the minimal cardinality of a subcover of U. Since X is compact, In the following fact we collect the basic properties of the topological entropy. (b) For every k ∈ N + , h top (ψ k ) = k · h top (ψ). If ψ is an automorphism, then h top (ψ k ) = |k|h top (ψ) for every k ∈ Z.
(c) If K is a inverse limit of closed ψ-invariant subgroups {K i : i ∈ I}, then h top (φ) = sup i∈I h top (φ ↾ Ki ).
As the right Bernoulli shift is a fundamental example for the algebraic entropy, the left Bernoulli shift plays the same role for the topological entropy: (a) It is a well-known fact (see [26]) that h top ( K β) = log |K|, with the usual convention that log |K| = ∞, if |K| is infinite. In particular, h top ( Z(p) β) = log p, for every prime p.
It was proved by Bowen and Peters that an Addition Theorem holds also for the topological entropy of continuous endomorphisms of compact groups: Theorem 7.3 (Addition Theorem). Let K be a compact abelian group, ψ : K → K a continuous endomorphism, N a closed ψ-invariant subgroup of K and ψ : K/N → K/N the endomorphism induced by ψ. Then The following is the Yuzvinski Formula for the topological entropy.
Theorem 7.4 (Yuzvinski Formula). [31] For n ∈ N + an automorphism ψ of Q n is described by a matrix A ∈ GL n (Q). Then h top (ψ) = log s + |αi|>1 log |α i |, where α i are the eigenvalues of A and s is the least common multiple of the denominators of the coefficients of 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-invariant subgroup 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: 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 Proposition 2.8(a), The role of the hyperkernel in the case of endomorphisms of abelian groups, for continuous endomorphisms ψ of compact groups K, is played by the hyperimage of ψ defined by Im ∞ ψ = n∈N ψ n (K), which is a closed ψ-invariant subgroup of K.
The next result shows that, as far as the computation of the value of the topological entropy of continuous endomorphisms 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 is surjective and Im ∞ ψ is the largest closed ψ-invariant subgroup of K with this property. Moreover, h(ψ) = h(ψ ↾ Im ∞ ψ ).
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 the endomorphism φ : G/t(G) → G/t(G) induced by φ. Then c(K) = t(G) ⊥ and so Remark 7.5 gives t(G) where ψ : K/c(K) → K/c(K) is the induced endomorphism. By the Addition Theorem 1.3 for the algebraic entropy 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) ) = h top (ψ). By Remark 7.5 (ii) We can assume that G is torsion-free of finite rank. Indeed, by (i) we can suppose that G is torsion-free.
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 . By If we verify that h( ψ F ) = h top (ψ F ) for every F ∈ [G] <ω , then Proposition 2.8(c) and Fact 7.1(c) give 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 compact abelian group of dimension n. Assume without loss of generality (by Proposition 2.8(a) and Fact 7.1(a)) that D(G) = Q n , and so that D(G) = Q n . Let ϕ = φ : Q n /G → Q n /G be the induced endomorphism, and η = φ : Q n → Q n , N = G ⊥ . So Remark 7.5 gives the following corresponding diagrams: Then h(φ) = h( φ) by Proposition 2.12. The next step is to show that h top (ψ) = h top (η). To this end, since h top (η) = h top (η ↾ N ) + h top (ψ) by the Addition Theorem 7.3, it suffices to prove that h top (η ↾ N ) = 0. So Q n /G is torsion, since G is essential in Q n . Therefore, h( η ↾ N ) = h top (η ↾ N ) by Theorem 1. We have seen that h(φ) = h( φ) and that h top (ψ) = h top (η). Then h( φ) = h top (η) would imply h(φ) = h top (ψ). In other words, it suffices to prove the thesis for φ ∈ End(Q n ).
(iv) We can suppose that φ is injective (i.e., ψ surjective). Indeed, consider the corresponding diagrams given by Remark 7.5: Indeed, Im ∞ ψ = (ker ∞ φ) ⊥ . By Corollary 2.6 and the Addition Theorem (v) By (iii) we can assume that G is a divisible torsion-free abelian group of finite rank, that is, G = Q n for n = r(G). By (iv) we can suppose that φ ∈ End(Q n ) is injective; then φ is also surjective and so φ ∈ Aut(Q n ). Therefore, h(φ) = h top (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 of the Algebraic Yuzvinski Formula (1.4), this would be a proof of the Bridge Theorem 1.6 independent from the particular case proved by Peters. Sometimes the exponential form of Mahler measure M (f (t)) = k i=1 max{1, |α i |} is also considered [12]; clearly, m(f (t)) = log M (f (t)). The Mahler measure plays an important role in number theory and arithmetic geometry (see [8,Chapter 1]).

Computation of Peters entropy via Mahler measure
If g(t) ∈ Q[t] is monic, then there exists a smallest s ∈ N + such that sg(t) ∈ Z[t]; in particular, sg(t) is primitive. 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 minimal polynomial of α.
Lehmer [15], with the aim of generating large primes, associated to any monic polynomial f (t) ∈ Z[t] with roots α 1 , . . . , α k the sequence of integers The idea comes from Mersenne primes generated by the polynomial f (t) = t − 2. Lehmer was using the polynomial f (t) = t 3 − t − 1. This is the the non-reciprocal polynomial with the smallest positive Mahler measure [25]. The polynomial is the reciprocal polynomial with the smallest known positive Mahler measure, that is, m(g(t)) = log λ, where λ = 1.17628 . . . is the Lehmer number [12]. Still in [12] it is noted that λ is the largest real root of g(t) and it is the only one of its algebraic conjugates outside the unit circle (i.e., λ is a Salem number). If there exists a polynomial h(t) with positive Mahler measure smaller than this, then deg h(t) ≥ 55 [17]. Indeed, lim n→∞ log |∆ n (f (t))| n = lim log |α i |.
This determines completely the case of zero Mahler measure.
By the Bridge Theorem 1.6, this problem is equivalent to the major open problem about the infimum of the positive values of the topological entropy (see [28]).
The following is an immediate consequence of the Algebraic Yuzvinski Formula (1.4).
(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.
(iv) By (ii) and (iii) we can consider divisible torsion-free abelian groups G of finite rank, namely, G ∼ = Q n for some n ∈ N + . Let φ ∈ End(Q n ). By Proposition 4.5 h(φ) = h(φ), where the induced endomorphism φ : Q n / ker ∞ φ → Q n / ker ∞ φ is injective (hence surjective) and Q n / ker ∞ φ ∼ = Q m for some m ∈ N, m ≤ n, as ker ∞ φ is pure in Q n (so Q n / ker ∞ φ is divisible and torsion-free by Lemma 4.1(c)). Therefore we can consider autmorphisms of Q n , and this gives the thesis. Theorem 8.7 and Corollary 8.6 have the following immediate consequence. By Corollary 8.6 the algebraic entropy h(φ) of a φ ∈ Q n is equal to the Mahler measure of a polynomial f (t) ∈ Z[t] with non-zero constant term. Now we see the viceversa, that is, the Mahler measure of a polynomial f (t) ∈ Z[t] with non-zero constant term is equal to the algebraic entropy of an automorphism φ of Q n for some n ∈ N + . Indeed, let f (t) = a 0 + a 1 t + . . . + a k t k ∈ Z[t] with deg f (t) = k and a 0 = 0. Let be the companion matrix associated to f (t). The characteristic polynomial of C(f ) is f (t). Since det C(f ) = (−1) k+2 a0 a k = 0 by hypothesis, C(f ) is the matrix associated to an automorphism φ of Q n . Then m(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 minimal polynomial, with deg f (t) = n. Let K = Q(α); then K ∼ = Q n as abelian groups. Following [32], call algebraic entropy h(α) of α the algebraic entropy h(µ α ), where µ α is the multiplication by α in K. Let a k be the smallest positive integer such that a k f (t) = a 0 + a 1 t + . . . + a k t k ∈ Z[t]. With respect to the basis {1, α, . . . , α k−1 } of K, the matrix associated to µ α is the companion matrix C(f ) in (8.2). The characteristic polynomial of C(f ) is f (t). By Corollary 8.6, h(µ α ) = m(f (t)), i.e., h(α) = m(α). This means that the algebraic entropy of an algebraic number is precisely its Mahler measure.