On a notion of entropy in coarse geometry

Abstract The notion of entropy appears in many branches of mathematics. In each setting (e.g., probability spaces, sets, topological spaces) entropy is a non-negative real-valued function measuring the randomness and disorder that a self-morphism creates. In this paper we propose a notion of entropy, called coarse entropy, in coarse geometry, which is the study of large-scale properties of spaces. Coarse entropy is defined on every bornologous self-map of a locally finite quasi-coarse space (a recent generalisation of the notion of coarse space, introduced by Roe). In this paper we describe this new concept, providing basic properties, examples and comparisons with other entropies, in particular with the algebraic entropy of endomorphisms of monoids.


Introduction
In 1865 Clausius de ned the notion of entropy in physics, but it was only in 1948 that Shannon ([39]) introduced it in mathematics, and, more precisely, in information theory. Inspired by that concept, several other entropies have been introduced in mathematics so far. For example, let us cite Kolmogorov ([25]) and Sinai's ( [40]) measure theoretic entropy in ergodic theory, and Adler, Konheim and McAndrew's topological entropy ( [1]). Other notions of topological entropy were given by Bowen ([4]) and Hood ([22]). In algebraic dynamics, we can cite the work of Adler, Konheim and MacAndrew ( [1]), the entropy de ned by Weiss in [43], and the one introduced by Peters ( [31], and deeply studied in [12]), that was then generalised in [10] for endomorphisms of abelian groups (we refer to [8] for the de nition in the non-abelian case, while to [11] for the extension to endomorphisms of semigroups). Later, Peters in [32] gave an extension of the algebraic entropy de ned in [31] for topological automorphisms of locally compact abelian groups. This de nition was generalised by Virili ([42]) to all endomorphisms of locally compact abelian groups. This de nition can be found in [8] also for non-abelian groups. We nally mention the paper [11], where a unifying approach to several notions of entropy is provided by using normed semigroups. More recently the entropy generated by actions of amenable semigroups and groups have been studied. In particular, Ornstein and Weiss introduced topological and measure entropy of amenable group actions ( [30]), whose approach was extended to the case of actions of amenable cancellative semigroups by Ceccherini-Silberstein, Coornaert and Krieger ( [5]), Hofmann and Stoyanov studied topological entropy of locally compact semigroup actions on metric spaces ( [21]), and Dikranjan, Fornasiero and Giordano Bruno in [6] de ned and discussed the algebraic entropy of an action of an amenable cancellative semigroup on an abelian group.
The aim of this paper is to introduce a notion of entropy in coarse geometry. Coarse geometry, also known as large-scale geometry, is the study of large-scale properties of spaces. Intuitively, two spaces are considered equivalent if they look roughly alike for an observer whose point of view is getting further and further away. While some ideas already appeared for example in Mostow's rigidity theorem, and in Švarc, Milnor, and Wolf's works, a great impetus was provided by Gromov's polynomial growth theorem ( [19]). This theory found then many applications in di erent areas of mathematics, such as geometric group theory, geometric topology, coarse Baum-Connes conjecture, and Novikov conjecture. See the book of Nowak and Yu ( [29]) for a comprehensive historical introduction to the topic, and [20] for a more detailed discussion on applications to geometric group theory. Coarse geometry was initially developed for metric spaces. However, some structures, capturing and generalising the large-scale behaviour of metric spaces, appeared in literature, as a large-scale counterpart of uniform spaces ( [23]). Roe introduced coarse spaces ( [38]) following Weil's approach via entourages, Protasov and Banakh ( [35]) de ned balleans, generalising the metric ball structure, and then Protasov ([34]), and Dydak and Ho and ( [17]) independently mimed Tukey approach via covers by proposing asymptotic proximities and large-scale structures, respectively. The equivalences between these approaches are discussed in [37] and [13]. Recently, in [44], the basic notions of coarse geometry were extended to asymmetric spaces, such as quasi-metric spaces and monoids, by introducing quasi-coarse spaces. The construction there given is the large-scale counterpart of the classical notion of quasi-uniform space (see the monograph [24] and the survey [26] for a wide introduction to this topic). The morphisms between both coarse and quasi-coarse spaces that are considered are called bornologous maps.
In this paper we introduce coarse entropy hc(f ) (De nition 2.1), de ned on the class of bornologous selfmaps f (De nition 1.4) of those quasi-coarse spaces (De nition 1.1) that are locally nite, i.e., whose balls are nite. Its de nition involves a limit superior, which is not a limit even when we consider the identity map (Example 2.3), and two supremum operations. The rst one is among all entourages, while the second one is among all base points. As for the rst one, we show that it is enough to consider just a base of the quasi-coarse structure, while in the second one we just need to evaluate the points that are maximal in the reachability preorder (Proposition 2.2). We then prove basic properties of coarse entropy, such as the weak logarithmic law (Proposition 3.3), how it behaves while taking products (weak addition theorem) and coproducts (Theorem 3.5), and the monotonicity under taking invariant coarse subspaces (Corollary 3.8).
As for conjugation invariance results, that play an important role in the theory of entropy, we have to distinguish two cases. If we consider asymorphisms as isomorphisms, then we obtain a conjugation invariance result for all bornologous self-maps of locally nite quasi-coarse spaces (Corollary 3.9(a)). If, otherwise, we consider Sym-coarse equivalences, a generalisation of the notion of coarse equivalence to the realm of quasicoarse spaces, as isomorphisms then we prove the desired result for quasi-coarse spaces with bounded geometry and for some particular bornologous self-maps (Corollary 3.9(b)). Note that asymorphisms are the isomorphisms of the category QCoarse of quasi-coarse spaces and bornologous maps between them, while Symcoarse equivalences are the isomorphisms of its quotient category QCoarse/∼ Sym under the Sym-closeness relation (see [44]).
Among the maps for which the conjugation invariance result for Sym-coarse equivalences holds, of a particular interest is the identity map. We can rewrite this speci c case as follows: if X and Y are two Sym-coarsely equivalent quasi-coarse spaces with bounded geometry, then hc(id X ) = hc(id Y ) (Corollary 4.1). Moreover, for every locally nite quasi-coarse space X, hc(id X ) ∈ { , ∞} (Theorem 4.4). Those results play a key role in connecting the coarse entropy with the growth of metric spaces ( [2]), extending the known results for nitely generated groups ( [8,20]). In particular, we show that, for a monogenic metric space X, if Y is a bounded geometry skeleton of X, the coarse entropy of the identity map of Y can tell whether X has subexponential or exponential growth type (Theorem 4.9).
On monoids and groups, of a particular interest are functorial quasi-coarse structures. In [14], inspired by the notion of functorial topologies ( [7]), functorial coarse structures on groups are de ned. Informally, a functorial coarse structure F on groups associates to every group G a group coarse structure F(G) on G in such a way that, for every group homomorphism f : G → H, f : (G, F(G)) → (H, F(H)) is bornologous. In [15] functorial coarse structures on topological groups are de ned and studied. This notion can be readily extended to functorial quasi-coarse structures on monoids. Among these functorial quasi-coarse structures on monoids, the so-called monoid-quasi-coarse structure (Example 1.2(b)) plays an important role. Moreover, every monoid endowed with its monoid-quasi-coarse structure has bounded geometry. Hence it is natural to compare in this setting the coarse entropy with the algebraic entropy (in the de nition provided in [10]). It turns out that the algebraic entropy provides an upper bound to the coarse entropy, while they coincide if the endomorphism is surjective (Theorem 5.2). We also provide examples in which the two notions di er for nonsurjective endomorphisms (Examples 5.3 and 5.4). Thanks to this result and using the Pontryagin duality, we are able to connect the coarse entropy with the topological entropy and the measure entropy in particular cases (Corollaries 5.10 and 5.13).
The paper is organised as follows. In Section 1 we recall the needed background from coarse geometry, including basic de nitions (e.g., quasi-coarse and coarse spaces, morphisms, local niteness), results and examples. Section 2 is devoted to introducing the coarse entropy, discussing thoughtfully the de nition and providing the rst non-trivial examples, i.e., left shifts of monoids and groups (Example 2.6). In Section 3 we collect the basic properties of the coarse entropy, such as the logarithmic law, theorems involving products and coproducts, monotonicity under taking invariant subspaces and conjugation invariance results. The focus of Section 4 is discussing the coarse entropy of the identity, providing also connections with the growth of metric spaces. Finally, Section 5 is dedicated to describing connections with other known entropies, such as the algebraic ( §5.1), the topological and the measure entropies ( §5.2). Moreover, the coarse entropy of some group endomorphisms is computed in §5.1.

Background on large-scale geometry
In this section we recall some notions of the classical large-scale geometry. We will give them in a more general setting, namely, the one of quasi-coarse spaces. Quasi-coarse spaces were recently introduced in [44] as a large-scale counterpart of quasi-uniform spaces in order to encode the large-scale properties of asymmetric spaces, e.g., quasi-metric spaces and monoids.
De nition 1.1 ([38, 44]). Let X be a set. A family E ⊆ P(X × X) is a quasi-coarse structure on X if it satis es the following properties: , closed under taking subsets and nite unions); (c) for every E, F ∈ E, the composite of E with F belongs to E (i.e., In that case, the pair (X, E) is a coarse space.
The entourages of a quasi-coarse space (X, E) can be seen as generalised radii as follows: for every E ∈ E, and x ∈ X, we call the subset E[x] = {y ∈ X | (x, y) ∈ E} as the ball centred in x with radius E. Moreover, if A ⊆ X, Let us present two main examples of quasi-coarse spaces. • d(x, y) ≤ d(x, z) + d(z, y), for every x, y, z ∈ X (triangle inequality).
As usual, we adopt the convention that ∞ + a = a + ∞ = ∞ and a ≤ ∞ for all a ∈ R ≥ ∪ {∞}. For the sake of simplicity, we refer to (X, d) as a quasi-metric space.
For every R > , we de ne a particular subset of the square X × X, called the strip of width R, as follows: where B(x, R) denotes the closed ball centred in x with radius R. Then the family {S R | R > } is a base of the so-called metric quasi-coarse structure E d . A metric quasi-coarse structure E d is a coarse structure if the extended-pseudo-quasi-metric d satis es also the symmetry property: d(x, y) = d(y, x), for every x, y ∈ X. However, there are non-symmetric quasi-metric spaces inducing coarse structures, as shown in [44]. (b) Let M be a monoid. An ideal I of subsets of M is a monoid ideal if, for every K, L ∈ I, KL = {kl | k ∈ K, l ∈ L} ∈ I. Let I be a monoid ideal on M. For every K ∈ I, de ne the strip of width K as the subset If I is a group ideal, then E I is a coarse structure. This construction can be also found in [27]. For example, for every group G, [G] <ω is a group ideal, inducing the group-coarse structure. Let us present an important example of group ideal on topological groups. If G is a topological group, then the family rC(G) of all relatively compact subsets of G is a group ideal and E rC(G) is called compact-group-coarse structure. The functorial properties of the compact-group-coarse structure have been widely studied in [14] and [15].
(where inf ∅ = ∞). In [44] was noted that, if Σ and ∆ are two nite generating sets of the nitely generated monoid M, Let now M be a nitely generated group, and Σ be a nite symmetric (i.e., Σ − = Σ) generating set. Then the map d Σ : M×M → R ≥ de ned as in (2) is a left-invariant metric on G (i.e., for every g, h, k ∈ G, d Σ (kg, kh) = d Σ (g, h)), called word metric associated to Σ. Note that d Σ induces the discrete topology, and then . For more details about the word metric of nitely generated groups, see, for example, [20].
If (X, E) is a quasi-coarse space and Y is a subset of X, then Y can be endowed with the subspace quasi-coarse An important large-scale notion is the one of connectedness. Let (X, E) be a quasi-coarse space and x, y ∈ X. We de ne a preorder ↓, called reachability preorder on X induced by E as follows: x ↓ y if {(x, y)} ∈ E (equivalently, if there exists E ∈ E such that y is contained in the ball centred in x with radius E). We call an element x ∈ X a maximum if it is a maximum in the preorder ↓ (i.e., for every y ∈ X, x ↓ y). A quasi-coarse space is connected if every point is a maximum or, equivalently, if E = X × X. For example, for a monoid endowed with its nitary-quasi-coarse structure, the identity is a maximum. Note that, for a coarse space X, there exists a maximum if and only if X is connected.
Let (X, E) be a quasi-coarse space. If x ∈ X, de ne Q ↓ . Then a point x ∈ X is a maximum if and only if Q ↓ X (x) = X. If X is a coarse space, then, for every A ⊆ X, Q ↓ X (A) = Q ↑ X (A) and we will denote it by Q X (A).
A quasi-coarse space (X, E) is locally nite if, for every E ∈ E and x ∈ X, E[x] is nite. Moreover, it has bounded geometry if there exists a map δ : E → N such that, for every E ∈ E, |E[x]| ≤ δ(E). For example, every monoid endowed with the nitary-quasi-coarse structure has bounded geometry. In order to de ne the morphisms, we need the notion of Sym-closeness. Let f , g : S → (X, E) be two maps from a set to a quasi-coarse space. Then f and g are Sym-close (and we denote it by • an asymorphism if f is bijective and both f and f − are bornologous; • a Sym-coarse equivalence if one of the following equivalent conditions holds: • f is a large-scale surjective coarse embedding, • f is bornologous and there exists another bornologous map g : Let us add some remarks concerning the morphisms introduced in De nition 1.4.

Remark 1.5. (a)
We say that two quasi-coarse spaces are Sym-coarsely equivalent if there exists a Sym-coarse equivalence between them. In [44] there is another characterisation of Sym-coarsely equivalent quasicoarse spaces. Namely, two quasi-coarse spaces X and Y are Sym-coarsely equivalent if and only if there exist two Sym-large subspaces X ⊆ X and Y ⊆ Y such that X and Y are asymorphic. (b) Let f : M → N be a homomorphism between two monoids (i.e., for every x, y ∈ M, f (xy) = f (x)f (y) and f (e M ) = e N ). Then f is bornologous provided that both monoids are endowed with their monoid-quasicoarse structures. A sharper computation is useful: for every K ∈ [M] <ω , In the case of groups, this "functorial property" is de ned and widely studied in [14].
Let us now introduce an important class of quasi-coarse spaces.
where E n is the composite of n copies of E. In particular, every monogenic quasi-coarse space is metrisable since it has a countable base (see [44]). For example, every nitely generated monoid is monogenic if it is endowed with its word quasi-metric (see Remark 1.3). In fact, let M be a monoid generated by the nite subset e ∈ Σ. Then E M is generated by the entourage E Σ . The notion of monogenicity is invariant under Sym-coarse equivalence, as Lemma 1.6 states.
Proof. Since monogenicity is trivially preserved under taking asymorphisms, according to Remark 1.5(a), it is enough to consider the case when Y is a Sym-large subspace of X (and Suppose that X is monogenic and E ∈ E X is an entourage such that {E n | n ∈ N} is a base of E X . Let F ∈ E X | Y . Then there exists n F ∈ N such that F ⊆ E n F . Let (x, y) ∈ F. Then there exists z = x, z , . . . , zn = y ∈ X such that (z i , z i+ ) ∈ E, for every i = , . . . , n − . Moreover, for every i = , . . . , n − , there exists z i ∈ Y such that (z i , z i ) ∈ M. Then, if we de ne z = x and z n = y, for every i = , . . . , n − , Conversely, suppose that Y is monogenic and {E n | n ∈ N} is a base of E Y , for some E ∈ E Y . By using a similar argument, it is easy to show that {(M • E • M) n | n ∈ N} is a base of E X .
In particular, Lemma 1.6 implies that a Sym-large subspace of a quasi-coarse space X is monogenic if and only if X itself is monogenic. Note that monogenicity is not inherited by arbitrary subspaces. For example, while N, endowed with the usual euclidean metric coarse structure, is monogenic, the subspace A = {n | n ∈ N} ⊆ N is not monogenic.
Before concluding this section, we introduce two categorical constructions. See [44] for a more comprehensive treatment of the category of quasi-coarse spaces.
Let (X, E X ) and (Y , E Y ) be two quasi-coarse spaces. Let p X : ). Moreover, f × g is bornologous if and only if both f and g are bornologous.
Let {(X k , E k )} k∈I be a family of quasi-coarse spaces and X = k X k be the disjoint union of their supports. For every k ∈ I, denote by i k : X k → X the canonical inclusion. For every J ∈ [I] <ω and φ : J → k∈I E k such that, for every k ∈ J, φ(k) ∈ E k , we de ne Then the coproduct quasi-coarse structure k E k on X is de ned by the base consisting of the elements E J,φ , where J and φ are as above. If, for every k ∈ I, f k :

De nition of coarse entropy
Let (X, E) be a quasi-coarse space and f : X → X be a bornologous self map. Then, for every x ∈ X and E ∈ E, de ne the following families of subsets recursively as follows: , More explicitly, for every n ∈ N, x ∈ X, E ∈ E, which is called the n+ -coarse trajectory Tn(f , x, E) with respect to x and E. When there is no risk of ambiguity, we will simply call it n + trajectory. Note that, if X is locally nite, for every bornologous self-map, every trajectory is a nite subset according to (6). Before de ning the coarse entropy, in the notation above, let us focus a bit more on the entourages of the form (f n × f n )(E). For every self-map g : X → X and every subset A ⊆ X, we have that In (7) implies that the trajectories can decrease.
Note that, for every n ∈ N \ { }, x ∈ x, and E ∈ E, In fact, if y / ∈ f n− (X), then, for every other z ∈ X, (z, y) / ∈ (f n− × f n− )(E). Let us de ne the coarse entropy.
De nition 2.1. Let (X, E) be a locally nite quasi-coarse space and f : X → X be a bornologous self map. If x ∈ X, E ∈ E, and n ∈ N \ { }, we de ne The value H loc c (f , x) and hc(f ) are called the local entropy of f in x and the coarse entropy of f , respectively.
Proposition 2.2 discusses more in detail the two supremum operations in (10).
, which shows the desired inequality. The second part of the assertion trivially follows.
The reader may wonder if the limit superior in (9) is a limit or not. In Example 2.3 we provide a locally nite metric space (Example 2.3(a)) and a metric space with bounded geometry (Example 2.3(c)) such that, for suitable inputs, the sequence dn de ned in (9) has no limit.
i= of edges such that x = x and xn = y. The vertex set V can be endowed with the path metric d, i.e., for every pair of distinct points x, y ∈ V, d(x, y) is the length of the shortest path connecting x and y provided that it exists, otherwise it is ∞. If X is connected as a graph, then the metric d does not take the value ∞. In this example, for the sake of simplicity, we write X for the associated metric space, even though writing V is more correct.
(a) We want to de ne a non-directed graph where (m, n), (m , n ) ∈ V, if and only if |m − m | = . In order to de ne the set V, we need to inductively construct a sequence {Kn}n of natural numbers. Let K = . Suppose that we have de ned K , . . . , K m− . Then Endow X with its path metric and then with the induced metric coarse structure. In particular, X is locally nite, even though it has not bounded geometry. Let us consider the map id X . Since X is connected, in order to compute its coarse entropy, we can just consider the trajectories centred in ( , ). Thanks to the de nition of X, there exists a strictly increasing sequence (an)n of natural numbers such that  (11) to The induced coarse space X is still locally nite. However, it can be easily proved that Hc(id X , ( , ), ) = ∞. (c) We want to provide now an example of a metric space with bounded geometry for which the sequence dn does not have a limit even when we consider the identity map.
We will de ne a non-directed graph X and endow it with its path metric. For every n ∈ N, denote by and k n+ = min{k ∈ N | |X n+ | + k ≤ (diam(X n+ ) + k)}, and X n+ = (X n+ L k n+ )/ ≈ n , where ≈ n is the nest equivalence relation satisfying j kn+ (a kn+ ) ≈ n y k n+ .
For the sake of simplicity, for every n ∈ N, in (13) and (14) we have identi ed the points of D kn+ and L k n+ with their images in X n D kn+ and in X n+ L k n+ , respectively. Then the graph X is the direct limit of the family {Xn} n∈N of nite graphs (with the family of obvious inclusion maps). In Figure 1 a representation of this space is provided. The metric space associated with the graph X has bounded geometry since every vertex has degree at most . Similarly to what we have done for item (a), it is not hard to prove that, in the notation of (9), if E = E and x = y , then lim inf n→∞ dn = < log ≤ lim sup n→∞ dn .
Hence the sequence {dn}n has no limit.
In Example 2.3(b) we provided an example of a locally nite coarse space (X, E) for which there exist x ∈ X and E ∈ E with Hc(id X , x, E) = ∞. Note that, if (Y , E ) is a quasi-coarse space with bounded geometry, then, for every point y ∈ Y and every E ∈ E , Hc(id Y , y, E ) < ∞. More precisely, if δ(E ) is a uniform bound to the cardinality of the balls with radius E , then However, the answer to the following question is not known.
Cellular coarse spaces are precisely those with asymptotic dimension ( [37]). If X is cellular, then hc(id) = since the trajectories stabilise. More precisely, for every choice of E ∈ E, and x ∈ X, the n-th trajectory is contained in a subset, namely E [x], which is bounded from x and thus nite. Note that a monoid, endowed with the monoid-quasi-coarse structure is cellular if and only if it is locally nite (i.e., for every K ∈ [M] <ω , the submonoid generated by K is still nite).
Example 2.6. Let M be a monoid endowed with the monoid-quasi-coarse structure and x ∈ M. We want to discuss the entropy of the left shift f = s λ x : y → xy. First of all, let us note that, for every K ∈ [M] <ω , which shows, in particular, that f is bornologous. Since the neutral element e is a maximum in M, then we just need to consider the trajectories with respect to e (Proposition 2.2).
(a) Suppose that x is an invertible element of M. Then f is an asymorphism with inverse s λ x − . Moreover, it is easy to see that (15) becomes an equality and thus Remark 2.5(a) implies that hc(f ) = hc(id M ). In particular this equality holds if M is a group. (b) Let M be left-cancellative (i.e., for every y ∈ M, the left shift s λ y is injective) and commutative. Split , according to (7). Hence, without loss of generality, we can assume that K = xF, for some non-empty F ∈ [M] <ω . By using induction, (7) T n+ (f , e, E K ) = a n+ B(e, (n + )m − (n + )) + a n bB(e, (n + )m − (n + )) = a n B(e, nm + m − n). (16) If n = , then (16) is trivial. Suppose now that (16) holds for some n ∈ N. Then Proof. Items (a) and (b) trivially follow from (8)

and from the fact that X is locally nite and thus F[x] and E[x]
are nite. Finally, it is enough to show item (c) in order to prove item (d). If x / ∈ Q ↑ X (f n (X)), then Q ↓ X (x)∩ f n (X) = ∅. Hence, (8) implies that, T n+ (f , x, E) ⊆ Q ↓ X (x) ∩ f n− (X) = ∅, for every E ∈ E. Morevoer, for every k > n, f k (X) ⊆ f n (X), and thus T k (f , x, E) = ∅. Hence, Hc(f , x, E) = and H loc c (f , x) = .
Let X be a locally nite quasi-coarse space, and f :

Corollary 3.2. Let (X, E) be a locally nite quasi-coarse space and f : X → X be a bornologous self-map. The subspace Y = n Q ↑ X (f n (X)) is f -invariant and hc(f ) = hc(f | Y ).
Proof. Let y ∈ Y. Then, for every n ∈ N, there exists xn ∈ X such that {(y, f n (xn))} ∈ E. In particular, since f is bornologous, {(f (y), f n+ (xn))} ∈ E, which implies that y ∈ Q ↑ X (f n+ (X)). Since the chain {Q ↑ X (f n (X)) | n ∈ N} is decreasing, f (y) ∈ Q ↑ X (f (X)) and thus the rst statement is proved. The equality follows from Proposition 3.1(d).

Proposition 3.3 (Weak logarithmic law).
Let (X, E) be a locally nite quasi-coarse space and f : X → X be a bornologous self-map. If f is surjective, for every k > , hc(f k ) ≤ k · hc(f ).
Proof. Fix a positive integer k > . Then, for every n ∈ N, x ∈ X, and ∆ X ⊆ E ∈ E, Tn(f k , x, E) ⊆ T kn−k+ (f , x, E) since the surjectivity of f implies that ∆ X ⊆ (f s × f s )(E), for every s ∈ N, and so from which the stated inequality follows.

Question 3.4. Does the opposite inequality in Proposition 3.3 hold?
The next result, Theorem 3.5, states that the coarse entropy behaves as expected in relation with ( nite) products and coproducts of quasi-coarse spaces. We consider only nite products since arbitrary products of locally nite quasi-coarse spaces are not necessarily locally nite. Then and thus from which the equality Hc(f × g, (x, y), E × F) = Hc(f , x, E) + Hc(g, y, F) and the desired claim follow.
Item (b) trivially follows from the observation that, for every n ∈ N\{ }, i j (x) ∈ k X k , and E J,φ ∈ k E k , de ned as in (4), where J ∈ [I] <ω and φ : J → k E k with the desired properties, Conjugation results are particularly important in developing entropies (see also Remark 3.10). The nal part of this section is devoted to prove conjugation results for the coarse entropy.
Proof. Let x ∈ X, E ⊆ E X , and x ∈ X. Then, by applying Lemma 3.6 and the commutativity of (17), for every The computation shows that because of the assumption on the bers of h. Then, since both E ∈ E and x ∈ X are arbitrary, hc(f ) ≤ hc(g). Then hc(f ) ≤ hc(g).
Proof. We want to apply Theorem 3.7. If h is injective, then we can set K = .
If h is large-scale injective, then for every x ∈ X.
As an immediate consequence of Corollary 3.8 we have the monotonicity of the coarse entropy under taking invariant subspaces. Let (X, E) be a locally nite quasi-coarse space, f : X → X be a bornologous self-map, and Y be a f -invariant subset of X, then hc(f | Y ) ≤ hc(f ). Moreover, the same result implies, in the case of coarse spaces with bounded geometry, the monotonicity of the coarse entropy under taking coarse embeddings. In fact, in the notation of Corollary 3.8, if h is a coarse embedding, then item (b) is ful lled. From Corollary 3.8 the following important invariance result trivially follows.

Corollary 3.9 (Invariance under conjugation).
Let (X, E) and (Y , E Y ) be two locally nite quasi-coarse spaces, f : X → X and g : Y → Y be two bornologous self-maps, and h : X → Y be a map such that the diagram (17) commutes. Suppose, moreover, that one of the following properties holds: (a) h is an asymorphism; (b) X and Y have bounded geometry, h is a Sym-coarse equivalence with a Sym-coarse inverse k : Y → X such that f • k = k • g. Then hc(f ) = hc(g). Remark 3.10. Let X be a category. We de ne the category FlowX of ows in X. As objects, it has pairs (X, f ), where X ∈ X and f : X → X is a morphism of X. Moreover, a morphism between two such pairs (X, f ) and Let QCoarse be the category of quasi-coarse spaces and bornologous maps between them ( [44]). We refer to [13] and [45] for a deep investigation of the full subcategory Coarse of QCoarse of all coarse spaces and its quotient category Coarse/∼, where ∼ is the closeness relation, i.e., the Sym-closeness relation restricted to Coarse. Denote by LF-QCoarse the full subcategory of QCoarse of locally nite quasi-coarse spaces. Consider then the category FlowLF-QCoarse. Thanks to Corollary 3.9, if (X, f ) and (Y , g) are two isomorphic ows, then hc(f ) = hc(g). Hence, hc associates a value in R ≥ ∪ {∞} to every isomorphism class of ows in FlowLF-QCoarse.

Coarse entropy of the identity and growth of quasi-coarse spaces
Let us focus on the identity map of a quasi-coarse space. Corollary 3.8 implies that, the coarse entropy of the identity map is an invariant under Sym-coarse equivalence in the realm of quasi-coarse spaces with bounded geometry.
Corollary 4.1. Let X and Y be two Sym-coarsely equivalent quasi-coarse spaces with bounded geometry. Then hc(id X ) = hc(id Y ).
Proof. Let f : X → Y be a Sym-coarse equivalence. Then we can easily apply Corollary 3.8, which implies that hc(id X ) ≤ hc(id Y ). The opposite inequality can be similarly proved.
We want to show that the identity function cannot have arbitrary values. More precisely, if X is a locally nite quasi-coarse space, then hc(id X ) ∈ { , ∞}. In order to prove this result, let us recall the following folklore fact.
Proof. Since, the two sequences have non-negative values, for every n ∈ N, sup k≥n (a k b k ) ≤ sup k≥n a k · sup k≥n b k and thus the inequality (≤) in (18) follows. As for the opposite inequality, x a value < ε < l = limn→∞ an. Then there exists N ∈ N such that, for every n ≥ N, an > l − ε. Thus, for every k ≥ N, Since ε can be arbitrarily taken, (19) implies the desired inequality. Proof. If t = , then there is nothing to prove. For the sake of simplicity, we prove the result for t = , but the argument can be easily generalised. Fix n ≥ and de ne two sequences {u n k } k≥n and {v n s } s≥ n− as follows: , and v n s = log(as) s/ , for every k ≥ n and s ≥ n − .
Since, for every k ≥ n, u n k = v n k , sup k≥n u n k ≤ sup s≥ n− v n s . We want to show the opposite inequality. Let then s ≥ n − . If s = k for some k ≥ n, then v n s = u n k . Otherwise, if s is odd, v n s ≤ u n (s+ )/ and thus sup k≥n u n k = sup Then, since (20)   Let us now connect the coarse entropy with the growth of a metric space. Here we present a generalisation of the approach for metric spaces that can be found in [2]. A quasi-coarse space (X, E) has a bounded geometry skeleton if there exists a subset Y (called bounded geometry skeleton) of X such that • (Y , E| Y ) has bounded geometry; • Y is Sym-large in X. In [2], this notion is given only for metric spaces under the name quasi-lattice, and a metric space is said to have coarse bounded geometry if it has a quasi-lattice.
Not every quasi-coarse space has a bounded geometry skeleton, as the following examples show.  Let X be as in quasi-coarse space having a bounded geometry skeleton. We de ne h bg c (X) = hc(id Y ), where Y is a bounded geometry skeleton of X. Note that h bg c (X) is well-de ned. In fact, if Y and Z are two bounded geometry skeletons of X, then Y is Sym-coarsely equivalent to Z, and thus Corollary 4.1 implies that hc(id Y ) = hc(id Z ).
Because of the monotonicity of the coarse entropy under taking invariant subspaces, h bg c (X) ≤ hc(id X ). Moreover, there exists a locally nite quasi-coarse space X such that h bg c (X) < hc(id X ). In order to nd an example showing the strict inequality, we have to choose X without bounded geometry. Let X as in Example 2.3(a). Then Theorem 4.4 implies that hc(id X ) = ∞. However, Y = N × { } is Sym-large in X and hc(id Y ) = .
Let us recall the classical notion of growth type of non-decreasing functions from N to R ≥ . Let u, v : N → R ≥ be two non-decreasing functions. We say that v dominates u, and we write u ≤ v, if there are a, b ≥ and c > such that, for every n ≥ c, u(n) ≤ av(bn). Then u and v have the same growth type Let u : N → R ≥ be a non-decreasing function. Then the growth type of u is: • polynomial if u is dominated by a polynomial function of some exact degree; • sub-exponential if u does not dominate any exponential function n → a n ; • exponential if u dominates an exponential function n → a n .
We can characterise the previous classes of growth types as follows. If u is a non-decreasing function, then It is proved in [2] that the growth rate of a metric space does not depend on the bounded geometry skeleton and on the point chosen. We want to estimate the growth rate of a coarse space X having a bounded geometry skeleton Y which is monogenic. However, thanks to Lemma 1.6, the existence of such a Y ensures that X itself is monogenic. Hence we can consider monogenic quasi-coarse spaces having bounded geometry skeletons.

Relationships with other entropies
In this section we discuss the relationships of the coarse entropy with other well-known entropy notions in other branches of mathematics. In particular, we consider the coarse entropy of endomorphisms of groups and we connect it with the algebraic entropy (in §5.1) and, through the Pontryagin functor, with the topological and the measure entropy (in §5.2).

. Relationship with the algebraic entropy
Let A standard approach to prove that the limit in (22) exists is by using Fekete's Lemma (see, for example, [8]). We refer to [8] for a comprehensive survey on the algebraic entropy.
In this subsection, every monoid is endowed with its monoid-quasi-coarse structure. Since every monoid endomorphism is automatically bornologous, it is natural to compare its algebraic entropy with its coarse entropy.
Hence Hc(f , e, E K ) ≤ H alg (f , K) and thus , for every K ∈ [M] <ω , and thus the inclusion in (23) becomes an equality, which proves the desired formula.
Thanks to Theorem 5.2, we can reinterpret what we have obtained so far as generalisations, in the case of a surjective homomorphism of monoids, of classical results in the realm of the algebraic entropy. Results that can be seen from this point of view are Proposition 2.2(a), Remark 2.5(c), Proposition 3.3, Theorem 3.5(a), Corollary 3.9, Theorem 4.4, and Theorem 4.9 in the case of nitely generated groups (see [8] for a comprehensive survey on algebraic entropy containing the mentioned results).
Let M be a monoid and g : M → M be an endomorphism. We want to specialise formula (7) in this setting. Let K be a nite subset of M. Then   By induction, it is easy to see that the sequence {gn}n is bounded by m.
If n = , then (27) is satis ed. Suppose that for some n ∈ N, (27) holds. Then, applying (24), Note that, in the notation of Example 2.6(c), even though n f n (M) = ∅, hc(f ) = ∞, but f is not an endomorphism.

. Relationship with the topological and the measure entropy
In this subsection we want to relate the coarse entropy with the topological entropy and the measure entropy, using the Pontryagin duality.
Let ϑτ( T ) be the family of all open neighbourhoods of the identity of the torus T. Consider a topological abelian group G, and denote by G the family of all continuous characters χ : G → T. With the pointwise operation, G is actually an abelian group. We can endow G with the compact-open topology τ, which is de ned by the following base of neighbourhoods of the null-character ∈ G: for every compact subset K ⊆ G and every U ∈ ϑτ( T ), W G (K, U) = {χ ∈ G | χ(K) ⊆ U}.
With this topology, G is a topological abelian group, called the dual group of G. For every continuous homomorphism f : G → H between topological abelian groups, there exists a continuous homomorphism f : H → G de ned by the law f (χ) = χ • f , for every χ ∈ H. The Pontryagin-van Kampen duality theorem states that the functor · : L → L, where L is the category of locally compact groups, induces a duality (see [33] for details). In particular, recall that, a locally compact abelian group G is discrete if and only if G is compact.
The Pontryagin duality is a powerful tool to translate concepts and results from the small-scale geometry to the large-scale geometry of locally compact abelian groups. In particular, we cite a result due to Nicas and Rosenthal ([28]): for every locally compact abelian group G, the covering dimension of G ( [18]) is equal to the asymptotic dimension (the large-scale counterpart of the covering dimension, see, for example, [3,29]) of G, where G is endowed with its compact-group-coarse structure. Other examples of those connections can be found in [14]. Let us just recall the following result.

Proposition 5.7 ([15]). Let G be a locally compact abelian group. Then (G, E rC(G) ) is metrisable (i.e., there exists a metric d on G such that E rC(G) = E d ) if and only if G is metrisable as a topological group.
In the realm of entropies, the Pontryagin duality plays an important role since it connects the algebraic entropy with the topological entropy (see Theorem 5.9).
Let f : X → X be a self-map of a set X and U a cover of X. Then we denote by f − (U) = {f − (U) | U ∈ U}. Moreover, if V is another cover of X, then we de ne U ∨ V = {U ∩ V | U ∈ U, V ∈ V}.
Let now X be a compact space. Denote by cov(X) the family of all open covers of X. For every cover U ∈ cov(X), denote by N(U) the minimum of the cardinalities of nite subcovers of U. For every U ∈ cov(X), de ne the entropy of U as H(U) = log N(U).
De nition 5.8 ([1]). Let f : X → X be a continuous self-map of a compact space. For U ∈ cov(X), de ne the topological entropy of f with respect to U as n .
Then the topological entropy of f is h top (f ) = sup{H top (f , U) | U ∈ cov(X)}.
Theorem 5.9 (Bridge theorem, [9]). Let f : G → G be an endomorphism of a group G. Then h alg (f ) = h top ( f ).
By combining Theorems 5.2 and 5.9 we obtain the following result. Moreover, if f is surjective, then hc(f ) = h alg (f ) = h top ( f ).
Before ending the section, let us connect the coarse entropy also with the measure entropy. Let (X, B, µ) be a measure space and ξ = {A i | i = , . . . , n} be a measurable partition of X. De ne the entropy of ξ by H(ξ ) = − n i= µ(A i ) log(µ(A i )).
De nition 5.11 ([25, 40]). Let X be a measure space and f : X → X be a measure preserving map. If ξ is a measurable partition of X, the measure entropy of f with respect to ξ is The measure entropy of f is hmes(f ) = sup{Hmes(f , ξ ) | ξ measurable partition of X}.
In the realm of compact groups, the following result was proved by Stoyanov. Proof. Corollary 5.10 states the rst two equalities. Since G is countable and discrete, (G, E rC(G) ) = (G, E G ) is metrisable ( [16]). Hence Proposition 5.7 implies that G is metrisable as topological group. Moreover, since f is an isomorphism, also f is an isomorphism and so we can apply Theorem 5.12 to get the claim.