A bridge theorem for the entropy of semigroup actions

Abstract The topological entropy of a semigroup action on a totally disconnected locally compact abelian group coincides with the algebraic entropy of the dual action. This relation holds both for the entropy relative to a net and for the receptive entropy of finitely generated monoid actions.


Introduction
The notion of topological entropy h top for continuous endomorphisms of locally compact groups is obtained by specializing the general notion of topological entropy given by Hood [29] for uniformly continuous selfmaps of uniform spaces, which was inspired by the classical topological entropy of Bowen [4] and Dinaburg [19] (see [18,27] for the details). For compact groups h top coincides with the rst notion of topological entropy by Adler, Konheim and McAndrew [1].
Since its origin, the algebraic entropy was introduced in connection to the topological entropy by means of Pontryagin duality. For a locally compact abelian group G we denote its Pontryagin dual group by G, and for a continuous endomorphism ϕ : G → G its dual endomorphism is ϕ : G → G.
In case G is a totally disconnected compact abelian group, and so G is a torsion discrete abelian group, it is known from [43] that h top (ϕ) = h alg ( ϕ). (1.1) The equality in (1.1), namely, a so-called Bridge Theorem, holds also when G is a metrizable locally compact abelian group and ϕ is an automorphism [36], and in case G is a compact abelian group [13].
In this paper we are interested in generalizing the following Bridge Theorem. Theorem 1.1 (See [14]). Let G be a totally disconnected locally compact abelian group and ϕ : G → G a continuous endomorphism. Then h top (ϕ) = h alg ( ϕ).
The Pontryagin dual groups of the totally disconnected locally compact abelian groups are precisely the compactly covered locally compact abelian groups. Recall that a topological group G is compactly covered if each Let S be a semigroup that acts on the left on a locally compact abelian group G by continuous endomorphisms and denote this left action by S γ G.
In case G is abelian, γ induces the right action de ned by γ(s) = γ(s) : G → G for every s ∈ S. We call γ the dual action of γ; see §4 for more details on the dual action and the relation of an action with its dual.
Roughly speaking, the aim of this paper is to extend Theorem 1.1 to this general setting of semigroup actions, for which there are several extensions of the previously recalled notions of topological entropy and algebraic entropy.
First of all, the topological entropy is widely studied for amenable group actions and even in more general settings (e.g., see [5,9,33,34,37,42]). On the other hand, a notion of algebraic entropy for amenable group actions on modules is developed by Virili [41], and an algebraic entropy for so c group actions by Liang [30] who proves also an instance of the Bridge Theorem in that setting.
Recently, the topological entropy of amenable semigroup actions on compact spaces was introduced by Ceccherini-Silberstein, Coornaert and Krieger [6], extending the classical notion from [1]. Analogously, in [10] the algebraic entropy of amenable semigroup actions on discrete abelian groups was de ned and investigated, generalizing classical notions and results from [15,35,43]. The de nition of these entropies of amenable semigroup actions is based on nets of non-empty nite subsets of the acting semigroup, namely, on Følner sent. The extension given by Ceccherini-Silberstein, Coornaert and Krieger [6] (see Theorem 2.1 below) of the celebrated Ornstein-Weiss Lemma [34] shows that the de nition does not depend on the choice of the Følner net.
The same approach based on nets was used by Virili [40] to introduce topological entropy and algebraic entropy for actions on locally compact abelian groups. We consider these entropies in the case of semigroup actions on locally compact (abelian) groups by continuous endomorphisms; in this case they depend on the choice of the net s of non-empty nite subsets of the acting semigroup S, so we denote them respectively by h s top and h s alg . They extend in a natural way the topological entropy h top and the algebraic entropy h alg of a single continuous endomorphism recalled above, by taking S = N and s = ({ , . . . , n}) n∈N . In §2 we give the de nitions, some useful properties and we connect these entropies with the ones for amenable semigroup actions from [6,10].
The main result in [40] is the Bridge Theorem h s top (γ) = h s alg ( γ) under the assumption that γ(s) is an automorphism for every s ∈ S. This covers the result announced in [36] in the case S = N.
We do not require the strong assumption on the action to be by automorphisms, but we assume the locally compact abelian group to be totally disconnected, and we see that the Bridge Theorem holds, obtaining the announced extension of Theorem 1.1: Theorem 1.2. Let S be an in nite monoid, s = (F i ) i∈I a net of non-empty nite subsets of S such that |F i | → ∞ and ∈ F i for every i ∈ I, G a totally disconnected locally compact abelian group, and consider the left action S γ G. Then h s top (γ) = h s alg ( γ).
See Corollary 4.3 for an application to the classical setting of amenable monoid actions.
Another kind of entropy, called receptive entropy, for semigroup actions was considered in [2,3,16,20,28]. In §3 we introduce the receptive topological entropy and the receptive algebraic entropy of actions of nitely generated monoids S on locally compact (abelian) groups. These entropies depend on the choice of a regular system Γ of S, which is a special sequence of non-empty nite subsets of S, so we denote them respectively by h Γ top and h Γ alg . Also the receptive topological entropy and the receptive algebraic entropy extend precisely the topological entropy h top and the algebraic entropy h alg of a single continuous endomorphism by taking S = N and Γ = ({ , . . . , n}) n∈N .
We nd a Bridge Theorem also for the receptive entropies generalizing Theorem 1.1: Theorem 1.3. Let S be a nitely generated monoid, Γ a regular system of S, G a totally disconnected locally compact abelian group, and consider the left action S γ G. Then h Γ top (γ) = h Γ alg ( γ).
We end the paper with some examples from [11,16].

Acknowledgments
This work was partially supported by Programma SIR 2014 by MIUR (project GADYGR, number RBSI14V2LI, cup G22I15000160008). The author thanks also the National Group for Algebraic and Geometric Structures, and their Applications (GNSAGA -INdAM).
It is a pleasure to thank the referee for the careful reading and the useful comments and suggestions.

Notation and terminology
For a semigroup S, denote by P(S) the power set of S and by P n (S) the family of all non-empty nite subsets of S. In case S is a monoid, we denote by the neutral element of S. When we consider an action γ of a semigroup S on a group G we mean that this action is by continuous endomorphisms, that is, γ(s) : G → G is a continuous endomorphism for every s ∈ S.
For every F ∈ P n (S) and U ∈ C(G), the F-cotrajectory of U ∈ C(G) with respect to γ is If G is abelian, we de ne also the F-trajectory of U with respect to γ to be

Entropy relative to a net
In this section we consider the topological entropy and the algebraic entropy from [40]. First we recall some basic de nitions related to amenability.
Let S be a semigroup. For every s ∈ S denote by Ls : S → S the left multiplication x → sx and by Rs : S → S the right multiplication x → xs. The semigroup S is left cancellative (respectively, right cancellative) if Ls (respectively, Rs) is injective for every s ∈ S.
A semigroup S is left amenable if there exists a map µ : P(S) → [ , ] such that: The semigroup S is right amenable if S op is left amenable. The semigroup S is amenable if it is both left amenable and right amenable. Every commutative semigroup is amenable (see [7,8]).
Analogously Let S be a semigroup. A function f : P n (S) → R ≥ is: The following is the counterpart of Ornstein-Weiss Lemma for cancellative left amenable semigroups.

. Topological entropy
De nition 2.2 (See [40]). Let S be a semigroup, s = (F i ) i∈I a net in P n (S), G a locally compact group, µ a Haar measure on G and S γ G a left action. For U ∈ C(G), let The topological entropy relative to s of γ is h s Therefore, in order to compute h s top , it su ces to consider a local base at of G contained in C(G). When the locally compact group G is totally disconnected, B(G) is a local base at of G by van Dantzig Theorem [38], so we have the following useful property.

Proposition 2.3. Let S be a semigroup, G a totally disconnected locally compact group, µ a Haar measure on G and S
The following result was given in [18] for the case S = N. It shows that for U ∈ B(G) one can avoid the use of Haar measure to compute the topological entropy.
Proposition 2.4. Let S be an in nite monoid, s = (F i ) i∈I a net in P n (S) such that |F i | → ∞ and ∈ F i for every i ∈ I, G a locally compact group, µ a Haar measure on G and S γ G a left action. If U ∈ B(G), then Hence, If in addition the group G is compact, we obtain the following result.
Proof. This follows from Proposition 2.4 and the fact that, since K is compact, [K : U] is nite.
This result implies that, when a cancellative left amenable monoid acts on a compact group, this topological entropy coincides with the one introduced in [6], that in this paper we call hcov (see Proposition 2.8).
We start recalling the de nition of hcov. Let S be a cancellative left amenable semigroup, let C be a compact topological space, and consider a left action S γ C by continuous maps, that is, γ(s) : C → C is a continuous selfmap for every s ∈ S.
Let U = {U j } j∈J and V = {V k } k∈K be two open covers of C. One says that V re nes U, denoted by U ≺ V, if for every k ∈ K there exists j ∈ J such that V k ⊆ U j . Moreover, For an open cover U of C and for every F ∈ P n (S), let U γ,F = s∈F γ(s) − (U). So, xed an open cover U of C, consider the function For every U, this function is non-decreasing, subadditive, right subinvariant and uniformly bounded on singletons (see [6,Proposition 5.2]). So by applying Theorem 2.1, we have that the following de nition is well posed. Proof. We have to prove that Note that, for s ∈ F and k ∈ K, if k ∈ γ(s) − (kU) then γ(s) − (kU) = k γ(s) − (U). Consider z s∈F γ(s) − (U) ∈ ζ (C F (γ, U)). Then there exists s∈F γ(s) − (ks U) ∈ ζ (U) γ,F such that z ∈ s∈F γ(s) − (ks U). It follows that z s∈F γ(s) − (U) = s∈F γ(s) − (ks U), and so that ζ (C F (γ, U)) ⊆ ζ (U) γ,F . Assume now that s∈F γ(s) − (ks U) ∈ ζ (U) γ,F is non-empty. It is straightforward to verify that s∈F γ(s) − (ks U) = z s∈F γ(s) − (U) for every z ∈ s∈F γ(s) − (ks U). This proves that ζ (U) γ,F \ {∅} ⊆ ζ (C F (γ, U)), and so concludes the proof.
In the following results we consider an in nite cancellative left amenable monoid S. So every left Følner net s = (F i ) i∈I of S necessarily satis es |F i | → ∞, and then we can assume without loss of generality that ∈ F i for every i ∈ I.

Proposition 2.8. Let S be an in nite cancellative left amenable monoid, K a totally disconnected compact group and S
γ K a left action. If s = (F i ) i∈I is a left Følner net of S such that ∈ F i for every i ∈ I, then for U ∈ B(K), Hcov(γ, ζ (U)) = H s top (γ, U), and so hcov(γ) = h s top (γ).
Proof. By de nition, by Lemma 2.7, by (2.2) and by Corollary 2.5, we have To prove the second assertion, let U be an open cover of K. Since B(K) is a local base at of K by van Dantzig Theorem [38], there exists U ∈ B(K) such that U ≺ ζ (U). Therefore, in view of (2.1), we have the required equality.

. Algebraic entropy
De nition 2.10 (See [40]). Let S be a semigroup, s = (F i ) i∈I a net in P n (S), G a locally compact abelian group, µ a Haar measure on G and G α S a right action. For U ∈ C(G), let The algebraic entropy relative to s of α is h s alg (α) = sup{H s alg (α, U) | U ∈ C(G)}.
The map H s alg (γ, −) is monotone, that is, if U, V ∈ C(G) and U ⊆ V, then H s alg (α, U) ≤ H s alg (α, V). Therefore, in order to compute h s alg , it su ces to consider a co nal subfamily of C(G). When the locally compact abelian group G is compactly covered, B(G) is a co nal in C(G) by [14, Proposition 2.2], so we have the following result.
Proposition 2.11. Let S be a semigroup, s = (F i ) i∈I a net in P n (S), G a compactly covered locally compact abelian group, µ a Haar measure on G and G α S a right action. Then The following result was proved for the case S = N in [22]. It shows that for U ∈ B(G) one can avoid the use of Haar Haar measure to compute the algebraic entropy. As above, we note that, since S is an in nite cancellative left amenable monoid, every left Følner net s = (F i ) i∈I of S necessarily satis es |F i | → ∞, and so we can assume without loss of generality that ∈ F i for every i ∈ I. Hence, In the discrete case we compare this algebraic entropy with the algebraic entropy ent introduced in [10] for a right action α of a cancelletive left amenable semigroup on a discrete abelian group A. For U ∈ B(A) (i.e., U is a nite subgroup of A), the function for any left Følner net s = (F i ) i∈I in P n (S). The algebraic entropy of α is ent(α) = sup{ent(α, U) | U ∈ B(A)}.
The next result follows immediately from the de nitions and Proposition 2.11.
De nition 3.1. For a nitely generated monoid S, a regular system of S is a sequence Γ = (Nn) n∈N of elements of P n (S) such that N = { }, and N i · N j ⊆ N i+j for every i, j ∈ N. (3.1) In particular, for a regular system Γ = (Nn) n∈N of a nitely generated monoid S, we have that Nn ⊆ N n+ for every n ∈ N.
Clearly, if S is a nitely generated monoid and N is a nite set of generators of S with ∈ N , then (N n ) n∈N of S, with N = { }, is a regular system such that S = n∈N Nn. This is the inspiring fundamental example of a regular system of a nitely generated monoid.

. Topological receptive entropy
We start introducing the following notion of receptive topological entropy, imitating the topological entropy in De nition 2.2.

De nition 3.2.
Let S be a nitely generated monoid, Γ = (Nn) n∈N a regular system of S, G a locally compact group, µ a Haar measure on G and S γ G a left action. For U ∈ C(G), let The receptive topological entropy with respect to Γ of γ is h Γ top (γ) = sup{ H Γ top (γ, U) | U ∈ C(G)}.
Proceeding as above we obtain the following results, we omit their proofs since they are the same of the proofs of Proposition 2.3, Proposition 2.4, Corollary 2.5 and Proposition 2.8, respectively.

Proposition 3.3. Let S be a nitely generated monoid, Γ = (Nn) n∈N a regular system of S, G a totally disconnected locally compact group, µ a Haar measure on G and S γ G a left action. Then
In the following result we see that, for U ∈ B(G), we can compute H Γ top (γ, U) avoiding the use of Haar measure. If in addition the group is compact, we obtain the following result. We recall now the de nition of receptive topological entropy from [3], which naturally extends the classical topological entropy hcov from [1].
De nition 3.6. Let S be a nitely generated monoid, Γ = (Nn) n∈N a regular system of S, C a compact space and S γ C a left action. For an open cover U of C, let The receptive topological entropy with respect to Γ of γ is h Γ cov (γ) = sup{ H Γ cov (γ, U) | U open cover of C}.
As a consequence of Corollary 3.5, the two notions of receptive topological entropy recalled in this section coincide when they are both available: Proposition 3.7. Let S be a nitely generated monoid, Γ = (Nn) n∈N a regular system of S, K a totally disconnected compact group and S γ K a left action. If U ∈ B(K), then

. Algebraic receptive entropy
We give a notion of receptive algebraic entropy, imitating the algebraic entropy in De nition 2.10.
De nition 3.8. Let S be a nitely generated monoid, Γ = (Nn) n∈N a regular system of S, G a locally compact abelian group, µ a Haar measure on G and G α S a right action. For U ∈ C(G), let The receptive algebraic entropy with respect to Γ of α is h Γ alg (α) = sup{ H Γ alg (α, U) | U ∈ C(G)}.
Proceeding as above we obtain the following results. We omit their proofs since they are the same of the proofs of Proposition 2.11 and Proposition 2.12.
Proposition 3.9. Let S be a nitely generated monoid, Γ = (Nn) n∈N a regular system of S, G a compactly covered locally compact abelian group, µ a Haar measure on G and G α S a right action. Then In the following result we see that, for U ∈ B(G), we can compute H Γ alg (α, U) avoiding the use of Haar measure.

The entropy of the dual action
Let G be a locally compact abelian group and denote by G its Pontryagin dual group. For a continuous homomorphism ϕ : G → H, where H is another locally compact abelian group, let ϕ : H → G be the dual of ϕ, de ned by ϕ(χ) = χ • ϕ for every χ ∈ H.
If G is a locally compact abelian group and U is a closed subgroup of G, the annihilator of U in G is the closed subgroup Moreover, G is topologically isomorphic to G, so in the sequel we shall simply identify G with G; under this identi cation we have that (U ⊥ ) ⊥ = U (4.1) for every closed subgroup U of G.
We give a proof of the following basic fact for reader's convenience. Proof.
Since U is open in G, the quotient G/U is discrete, therefore Let S be a semigroup and G a locally compact abelian group and consider the left action S γ G. Then γ induces the right action G γ S, de ned by In fact, xed s, t ∈ S, since γ(st) = γ(s)γ(t), we have that γ(st) = γ(st) = γ(s)γ(t) = γ(t) γ(s) = γ(t) γ(s). Analogously, let S be a semigroup and G a locally compact abelian group and consider the right action G α S.
Then α induces the left action S α G, de ned by In fact, xed By Pontryagin duality, γ = γ and α = α.
The following technical lemma is a key step in the proof of the Bridge Theorem, see [14] for more details in the case S = N. Proof. Let U ∈ B(G). By Pontryagin duality theory, Since This gives the required equality.
We are in position to prove the two versions of the Bridge Theorem stated in the introduction.
By As an application of Theorem 1.2, together with Proposition 2.8 and Proposition 2.14, we obtain the following counterpart of Weiss' Bridge Theorem for amenable semigroup actions.

Corollary 4.3. Let S be an in nite cancellative left amenable monoid, K a totally disconnected compact abelian group and S γ K a left action. Then hcov(γ) = ent( γ).
The proof of Theorem 1.3 is analogous to that of Theorem 1.2, one applies Proposition 3.4, Proposition 3.10, Proposition 3.3 and Proposition 3.9.
We end with two important examples for the computation of the values of the entropies considered in this paper. Then h alg (σ) = log p (see [11]).