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An analogue of Fekete's lemma for subadditive functions on cancellative amenable semigroups

We prove an analogue of Fekete's lemma for subadditive right-subinvariant functions defined on the finite subsets of a cancellative left-amenable semigroup. This extends results previously obtained in the case of amenable groups by E. Lindenstrauss and B. Weiss and by M. Gromov.


Introduction
Fekete's lemma [7] is a classical result in undergraduate-level analysis. It states that if (u n ) n≥1 is a subadditive sequence of real numbers then the sequence u n n n≥1 has a limit λ ∈ R ∪ {−∞} as n tends to infinity (see for example [12,Proposition 9.6.4]). The goal of the present paper is to give an analogue of Fekete's lemma for subadditive and rightsubinvariant functions defined on the set of all finite subsets of a cancellative left-amenable semigroup. In order to state our main result, let us first recall some basic definitions and introduce notation. Let S be a semigroup, i.e., a set equipped with an associative binary operation. We denote by P(S) the set of all subsets of S. One says that S is left-amenable if there exists a finitely additive left-invariant probability measure defined on P(S), that is, a map µ : P(S) → [0, 1] satisfying the following conditions: (A1) µ(A ∪ B) = µ(A) + µ(B) for all A, B ∈ P(S) such that A ∩ B = ∅; (A2) µ(S) = 1; (A3) µ(L −1 s (A)) = µ(A) for all s ∈ S and A ∈ P(S), where L s : S → S denotes the left-multiplication by s, that is, the map defined by L s (t) = st for all t ∈ S.
One says that S is right-amenable if its opposite semigroup is left-amenable. This is equivalent to the existence of a finitely additive right-invariant probability measure defined on P(S), that is, a map µ : P(S) → [0, 1] satisfying (A1), (A2) and (A3') µ(R −1 s (A)) = µ(A) for all s ∈ S and A ∈ P(S), where R s : S → S is the right-multiplication by s, that is, the map defined by R s (t) = ts for all t ∈ S. A semigroup is called amenable if it is both left-amenable and right-amenable. The notion of an amenable group was introduced in 1929 by J. von Neumann [17]. His original motivation was the study of the Banach-Tarski paradox. The theory of amenable semigroups was subsequently developed in the 1940s and 1950s by M. Day (see [4], [5], [6], and the references therein). Day [4] showed in particular that every commutative semigroup is amenable, thus extending a result previously obtained by von Neumann for groups. Actually, when passing from groups to semigroups, one encounters many new phenomena. For example, left-amenability and right-amenability are equivalent for groups, every finite group is amenable, and every subgroup of an amenable group is itself amenable. On the other hand, in contrast with the group case, there exist finite semigroups that are leftamenable but not right-amenable and finite amenable semigroups containing semigroups that are neither left-amenable nor right-amenable.
In the group setting, E. Følner [8] gave a remarkable combinatorial characterization of amenability by showing that a group S is amenable if and only if it satisfies the following condition: (FC) for every finite subset K ⊂ S and every real number ε > 0, there exists a non-empty finite subset F ⊂ S such that (1.1) |kF \ F | ≤ ε|F | for all k ∈ K.
(Here and in the sequel, we use | · | to denote cardinality of finite sets.) Condition (FC) is known as the Følner condition.
In his thesis, A. Frey [9] adapted Følner arguments to semigroups and showed that every left-amenable semigroup S satisfies condition (FC) (see [16,Theorem 3.5] for a simpler proof). However, (FC) is a necessary but not sufficient condition for left-amenability of semigroups. Examples of semigroups that are not left-amenable but satisfy (FC) are provided by finite semigroups that are not left-amenable (observe that any finite semigroup S trivially satisfies (FC) by taking F = S).
Condition (FC) is equivalent to the existence of a directed net (F i ) i∈I of non-empty finite subsets of S such that Indeed, if S satisfies (FC), we can construct a directed net (F i ) i∈I satisfying (1.2) in the following way. We first take as I the directed set consisting of all pairs (K, ε), where K is a finite subset of S and ε > 0, with the partial ordering on I defined by (K 1 , ε 1 ) ≤ (K 2 , ε 2 ) if and only if K 1 ⊂ K 2 and ε 2 ≤ ε 1 . Then, for each i = (K, ε) ∈ I, we take as F i one of the non-empty finite subsets F ⊂ S satisfying (1.1). Conversely, suppose that (F i ) i∈I is a directed net of non-empty finite subsets of S satisfying (1.2). Let K ⊂ S be a finite subset and ε > 0. Then, for every k ∈ K, we can find i k ∈ I such that |kF i \ F i | ≤ ε|F i | for all i ≥ i k . To get a non-empty finite subset F ⊂ S satisfying (1.1), it suffices to take F := F i , where i ∈ I is such that i ≥ i k for every k ∈ K (the existence of such an index i follows from the fact that I is directed and K is finite). A directed net (F i ) i∈I of non-empty finite subsets of S satisfying (1.2) is called a left-Følner net of S.
Recall that an element s in a semigroup S is called left-cancellable (resp. right-cancellable) if the map L s (resp. R s ) is injective. One says that s is cancellable if it is both leftcancellable and right-cancellable. The semigroup S is called left-cancellative (resp. rightcancellative, resp. cancellative) if every element in S is left-cancellable (resp. rightcancellable, resp. cancellable).
When S is a left-cancellative semigroup, it is known that the left-amenability of S is equivalent to the Følner condition (FC), and hence to the existence of a left-Følner net (see [16,Corollary 4.3]).
The purpose of the present paper is to establish the following result. Then there exists a real number λ ≥ 0, depending only on h, such that the net h(F i ) |F i | i∈I converges to λ for every left-Følner net (F i ) i∈I of S.
Observe that conditions (H3) is implied by (H2) when S admits an element s 0 such that the map L s 0 is onto. Indeed, in this case, (H2) gives us h({s}) ≤ h({s 0 }) for all s ∈ S. This happens for example when S is a monoid (i.e., S admits an identity element) since we can then take s 0 = 1 S . Thus, an immediate consequence of Theorem 1.1 is the following result.
Corollary 1.2. Let S be a cancellative left-amenable monoid and let h : P f in (S) → R be a subadditive and right-subinvariant map. Then there exists a real number λ ≥ 0, depending only on h, such that the net h(F i ) |F i | i∈I converges to λ for every left Følner net (F i ) i∈I of

S.
As far as we know, the above result is new even in the case when S is the additive monoid N of non-negative integers. However, when S is an amenable group, it was previously established by E. Lindenstrauss  Let us note that, in the case when S is a group, condition (H2) implies that h is rightinvariant since we then have h(A) = h(Ass −1 ) ≤ h(As) and hence h(As) = h(A) for all s ∈ S and A ∈ P f in (S). This is no more true in general for semigroups. For example, if S is the additive monoid N, then the map h : P f in (N) → R defined by h(A) = (1+max(A)) −1 |A| clearly satisfies (H1) and (H2) but h(A + s) < h(A) for A = {0} and s = 1.
The proof of Lindenstrauss and Weiss is based on the Ornstein-Weiss machinery of quasi-tiles (cf. [18,Section I.2: Theorem 6]). This is the reason why the group version of Theorem 1.1 is sometimes called the Ornstein-Weiss lemma although it does not appear explicitly in [18]. A detailed exposition of Gromov's proof of the Ornstein-Weiss lemma may be found in [14].
In the theory of dynamical systems, Theorem 1.1 is important in defining numerical invariants such as topological entropy, measure-theoretic entropy, and mean topological dimension. These invariants are obtained by taking limits of quantities defined from a left-Følner net and one can deduce from Theorem 1.1 that the choice of the left-Følner net is actually irrelevant for actions of cancellative left-amenable semigroups.
Our proof of Theorem 1.1 is entirely self-contained. At several points, it is inspired by some of the ideas developed by Gromov in [10, Section 1.3.1]. However, here again, the passage from groups to semigroups inevitably imposes significant modifications in the arguments.
We do not know to what extend Theorem 1.1 remains valid for non-cancellative leftamenable semigroups.
The paper is organized as follows. In Section 2, we establish some general properties of boundary sets in semigroups that are needed for the proof of our main result. We give in particular a characterization of Følner nets for cancellative semigroups in terms of relative amenability. In Section 3, we introduce a notion of ε-filling pattern for finite subsets of semigroups and prove a theorem about the existence of certain fillings by finite systems of tiles with small relative amenability (Theorem 3.8). This filling theorem is a key tool in the proof of Theorem 1.1 given in Section 4. In Section 5, which is mostly expository, we discuss the above-mentioned applications of Theorem 1.1 to the definition of numerical invariants of dynamical systems.

Boundaries and relative amenability
Let S be a semigroup. Let K and A be subsets of S. The right K-interior of A is the set consisting of all the elements s in A such that the right-translate of K by s is entirely contained in A.
The right K-boundary of A is the set ∂ K (A) ⊂ A defined by Thus, an element s ∈ S is in ∂ K (A) if and only if s is in A and Ks meets the complement of A in S.
Proposition 2.1. Let S be a semigroup. Let K, A, and B be subsets of S. Then one has , then Ks ∩ A = ∅ and hence s ∈ k∈K L −1 k (A ∩ kS). As s / ∈ A, inclusion (iv) immediately follows. (v) Suppose that s ∈ S is right-cancellable and let g ∈ (Int K (A)) s. This means that there exists a ∈ Int K (A) such that g = as. Hence g ∈ As and Kg = K(as) = (Ka)s ⊂ As since a ∈ Int K (A). Thus g ∈ Int K (As). This gives the inclusion (Int K (A)) s ⊂ Int K (As).
Conversely, suppose now that g ∈ Int K (As). Then g ∈ As and Kg ⊂ As. Thus, there exists a ∈ A such that g = as and (Ka)s = K(as) ⊂ As. Remark that the inclusion (Ka)s ⊂ As is equivalent to the inclusion Ka ⊂ A by injectivity of R s . This proves that a ∈ Int K (A) so that g ∈ (Int K (A)) s. Hence Int K (As) ⊂ (Int K (A)) s. This completes the proof of (v).
(vi) If s ∈ S is right-cancellable, we have This shows (vi).
Lemma 2.2. Let S be a semigroup. Suppose that K and A are finite subsets of S and that every element of K is left-cancellable. Then one has Proof. It follows from Proposition 2.1.(ii) that

This implies
As |L k −1 (kA \ A)| = |kA \ A| for all k ∈ K by injectivity of L k , this gives us (2.1). On the other hand, given k ∈ K, we deduce from (2.3) that Let A and K be subsets of S with A finite and non-empty. Then ∂ K (A) is also finite since ∂ K (A) ⊂ A. We define the amenability constant of A with respect to K by Note that α(A, K) is rational and that one has 0 ≤ α(A, K) ≤ 1. For left-cancellative semigroups, left-amenability is equivalent to the existence of finite subsets with arbitrary small relative amenability. More precisely, we have the following result. Proposition 2.3. Let S be a left-cancellative semigroup. Then the following conditions are equivalent: (a) S is left-amenable; (b) for every finite subset K of S and every real number ε > 0, there exists a non-empty We deduce that the Følner condition (FC) is equivalent to condition (b) of the statement. On the other hand, as S is left-cancellative, we know from the result mentioned in the Introduction that S is left-amenable if and only if it satisfies (FC). This shows the equivalence between conditions (a) and (b).
Similarly, we have the following characterization of Følner nets in left-cancellative and left-amenable semigroups.
Proposition 2.4. Let S be a left-cancellative and left-amenable semigroup. Let (F i ) i∈I be a directed net of non-empty finite subsets of S. Then the following conditions are equivalent: Proof. Let s ∈ S and take K = {s}. Then one has |sF Conversely, suppose that (F i ) i∈I is a left-Følner net for S. Let K be a finite subset of S and ε > 0. Then there exists i k ∈ I such that |kF

Fillings
The goal of this section is to establish Theorem 3.8 which is a key tool in the proof of Theorem 1.1 that will be given in the next section.
Lemma 3.2. Let X be a set and (A j ) j∈J a finite ε-disjoint family of finite subsets of X. Then one has Lemma 3.3. Let S be a semigroup. Let also K be a finite subset of S and 0 < ε < 1. Suppose that (A j ) j∈J is a finite ε-disjoint family of non-empty finite subsets of S. Then one has Proof. Let us set M := max j∈J α(A j , K). It follows from Proposition 2.1.(iii) that As the family (A j ) j∈J is ε-disjoint, we deduce from Lemma 3.2 that Lemma 3.4. Let S be a semigroup. Let K, A and Ω be finite subsets of S such that every element of K is left-cancellable and ∅ = A ⊂ Ω. Suppose that ε > 0 is a real number such that |Ω \ A| ≥ ε|Ω|. Then one has Proof. By Proposition 2.1.(iv), we have that This implies Now, for all k ∈ K, the injectivity of L k implies that where the last equality follows from the fact that A and kA have the same cardinality. Hence, by using inequality (2.2) in Lemma 2.2, we get From (3.1) and (3.2), we deduce that It follows that  If we fix a ∈ A and s ′ ∈ S, the injectivity of L a implies that there exists at most one element s ∈ S such that as = s ′ . It follows that, given s ′ ∈ S, there are at most |A| elements s ∈ S such that s ′ ∈ As. In other words, we have Remarks. 1) The argument used in the preceding proof shows that the inequality in Lemma 3.5 can be replaced by an equality if L a is bijective for every a ∈ A (e.g., if S is a group). In fact, when S is a group, the equality s∈S |As ∩ B| = |A||B| is obtained by taking f = χ A and g = χ B in the formula f * g 1 = f 1 g 1 , valid for all f, g ≥ 0 in the convolution Banach algebra ℓ 1 (S).
2) The inequality in Lemma 3.5 may be strict. Consider for example the additive monoid N of non-negative integers and two non-empty finite subsets A, B ⊂ N with max B < min A. Then one has s∈N |(A + s) ∩ B| = 0 but |A||B| ≥ 1. Note that the additive monoid N is commutative (and hence amenable) and cancellative.
3) Lemma 3.5 becomes false if we drop the hypothesis that every element of A is leftcancellable. Indeed, consider the monoid S = {s 0 , s 1 }, where s 0 is an identity element and s 1 = s 0 satisfies s 2 1 = s 1 . Then, by taking A = B = {s 1 }, we have s∈S |As ∩ B| = 2 but |A||B| = 1.
Definition 3.6. Let S be a semigroup. Let K and Ω be finite subsets of S. Given a real number ε > 0, a finite subset P ⊂ S is called an (ε, K)-filling pattern for Ω if the following conditions are satisfied: The following lemma will be used in the proof of Theorem 3.8 (compare with (+) ε in Section 1.3.1 of [10] in the group case). It can be viewed as a kind of analogue of Euclidean division for integers. Lemma 3.7 (Filling lemma). Let S be a cancellative semigroup. Let Ω and K be nonempty finite subsets of S. Then, for every ε ∈ (0, 1], there exists an (ε, K)-filling pattern P for Ω such that Proof. Let P denote the set consisting of all (ε, K)-filling patterns for Ω. Observe that P is not empty, since ∅ ∈ P, and that every element of P has cardinality bounded above by | Int K (Ω)|, since it is contained in Int K (Ω). Choose a pattern P ∈ P with maximal cardinality. Let us show that (3.5) is satisfied. To slightly simplify notation, let us set
If s ∈ P , then Ks ∩ B = Ks and (3.7) holds true since ε ≤ 1. Let now s ∈ Int K (Ω) \ P and suppose, by contradiction, that |Ks ∩ B| < ε|Ks|. Then, we have that which implies that P ∪{s} is an (ε, K)-filling pattern for Ω. This contradicts the maximality of the cardinality of P . This proves (3.7). Finally, we obtain  Then there exists an integer n 0 = n 0 (ε) ≥ 1 such that for each integer n ≥ n 0 the following holds.
If (K j ) 1≤j≤n is a finite sequence of non-empty finite subsets of S such that and D is a non-empty finite subset of S such that (3.9) α(D, K j ) ≤ ε 2n for all 1 ≤ j ≤ n, then there exists a finite sequence (P j ) 1≤j≤n of finite subsets of S satisfying the following conditions: (T1) the set P j is an (ε, K j )-filling pattern of D for every 1 ≤ j ≤ n; (T2) the subsets K j P j ⊂ D, 1 ≤ j ≤ n, are pairwise disjoint; (T3) the subset D ′ ⊂ D defined by Proof. Fix ε ∈ (0, 1 2 ] and a positive integer n. Let K j , 1 ≤ j ≤ n, and D be non-empty finite subsets of S satisfying conditions (3.8) and (3.9).
Let us first define, by induction, a finite process with at most n steps for constructing suitable finite subsets P n , P n−1 , . . . , P 1 of S. We will see that these subsets have the required properties when n is large enough, i.e., for n ≥ n 0 with n 0 = n 0 (ε) that will be made precise at the end of the proof.
Using Lemma 3.7 with Ω = D 0 = D and K = K n , we can find a finite subset P n ⊂ S such that (H(1;b)) P n is an (ε, K n )-filling pattern for D 0 and (H(1;c)) Setting D 1 := D 0 \ K n P n , we deduce from (3.10) that Step k. We continue this process by induction as follows. Suppose that the process has been applied k times, with 1 ≤ k ≤ n − 1. It is assumed that the induction hypotheses at step k are the following: (H(k;a)) D k−1 is a subset of D satisfying (H(k;b)) P n−k+1 ⊂ S is an (ε, K n−k+1 )-filling pattern for D k−1 ; (H(k;c)) setting Note that these induction hypotheses are satisfied for k = 1 by Step 1. Let us pass to Step k + 1.
Step k + 1. If |D k | ≤ ε|D k−1 | and hence |D k | ≤ ε|D|, then we take P j = ∅ for all 1 ≤ j ≤ n − k and stop the process. Otherwise, we have |D k | > ε|D k−1 |. Let us estimate from above, for all 1 ≤ j ≤ n − k, the relative amenability constants α(D k , K j ).
Suppose now that P n−k+1 = ∅. Then we can apply Lemma 3.4 with Ω := D k−1 and A := K n−k+1 P n−k+1 . This gives us (3.11) Proposition 2.1.(vi) and condition (3.8) imply that, for all s ∈ S, Since the family (K n−k+1 s) s∈P n−k+1 is ε-disjoint, the preceding inequality together with Lemma 3.3 give us From inequality (3.11) and the induction hypothesis (H(k;a)), we deduce that This shows (H(k+1;a)). Using Lemma 3.7 with Ω := D k and K := K n−k , we can find a finite subset P n−k ⊂ S such that P n−k is an (ε, K n−k )-filling pattern for D k , thus yielding (H(k+1;b)), and satisfying Setting D k+1 := D k \ K n−k P n−k , we deduce from (3.12) that Together with the inequality of the induction hypothesis (H(k;c)), this yields Thus condition (H(k+1;c)) is also satisfied. This finishes the construction of Step k + 1 and proves the induction step. Now, suppose that this process continues until Step n. Using (H(k;c)) for k = n, we obtain We will show that for n ≥ n 0 , with n 0 = n 0 (ε) only depending on ε, we get |D n | ≤ ε|D|.

Proof of the main result
In this section, we give the proof of Theorem 1. On the other hand, we deduce from (H1) that so that, by using (H3), we get Consider the quantity Note that 0 ≤ λ ≤ M by (4.1) and (4.2).
Recall that one says that a finite sequence (K j ) 1≤j≤n is extracted from the net (F i ) i∈I if there are indices Let ε > 0 and let n be a positive integer. By (4.3) and (4.4), it is clear that we can find, using induction on n, a finite sequence (K j ) 1≤j≤n extracted from the net (F i ) i∈I such that: Suppose now that 0 < ε ≤ 1 2 and that n ≥ n 0 , where n 0 = n 0 (ε) is as in Theorem 3.8.
Let D ⊂ S be a non-empty finite subset satisfying α(D, K j ) ≤ ε 2n for all 1 ≤ j ≤ n. By Theorem 3.8, we can find a sequence (P j ) 1≤j≤n of finite subsets of S satisfying the following conditions: (T1) the set P j is an (ε, K j )-filling pattern for D for every 1 ≤ j ≤ n; (T2) the subsets K j P j ⊂ D, 1 ≤ j ≤ n, are pairwise disjoint; (T3) the subset D ′ ⊂ D defined by By applying the subadditivity property (H1) of h, it follows that As |D ′ | ≤ ε|D| by (T3), we deduce from (4.2) that On the other hand, for all 1 ≤ j ≤ n, we have s∈P j |K j s| (by (4.5)).
As the family (K j s) s∈P j is ε-disjoint by (T1), we then deduce from Lemma 3.2 that This implies since the sets K j P j , 1 ≤ j ≤ n, are pairwise disjoint subsets of D by (T2). From (4.6), (4.7), and (4.8), we deduce that By (4.3), we can find i 0 ∈ I such that, for all i ≥ i 0 , Hence, by replacing D by F i for i ≥ i 0 in inequality (4.9), we obtain This implies Since the latter inequality is satisfied for all ε ∈ (0, 1 2 ], taking the limit as ε tends to 0, we obtain This shows that (4.4) is indeed a true limit.
It only remains to show that λ = lim i h(F i ) |F i | does not depend on the choice of the left-Følner net (F i ) i∈I . So suppose that (G j ) j∈J is another left-Følner net for S and let Take disjoint copies I ′ and J ′ of the sets I and J, i.e., sets I ′ and J ′ with I ∩ I ′ = ∅ and J ∩ J ′ = ∅ together with bijective maps ϕ : I → I ′ and ψ : J → J ′ . Consider the set T = (I × J) ∪ (I ′ × J ′ ) with the partial ordering defined as follows. Given t 1 , t 2 ∈ T , we write t 1 ≤ t 2 if and only if there exist indices i 1 , i 2 ∈ I and j 1 , j 2 ∈ J such that i 1 ≤ i 2 , j 1 ≤ j 2 , and (t 1 = (i 1 , j 1 ) or t 1 = (ϕ(i 1 ), ψ(j 1 ))) and (t 2 = (i 2 , j 2 ) or t 2 = (ϕ(i 2 ), ψ(j 2 ))).
Observe that (T, ≤) is a directed set since (I, ≤) and (J, ≤) are directed sets. Now we define a net (H t ) t∈T of non-empty finite subsets of S by setting Clearly (H t ) t∈T is a left-Følner net for S. By the first part of the proof, the net h(H t ) |H t | t∈T converges to some τ ≥ 0. Using the fact that for every t 1 in T , there exits t 2 in I × J (resp. in I ′ × J ′ ) such that t 1 ≤ t 2 , we conclude that τ = λ = ν. This completes the proof of Theorem 1.1.

Applications to dynamical systems
Topological entropy. (cf. [1]) Let X be a compact topological space.
An open cover of X is a family of open subsets of X whose union is X. Let U = (U j ) j∈J and V = (V k ) k∈K be two open covers of X. One says that V is finer than U, and one Since X is compact, every open cover of X admits a finite subcover. Given an open cover U of X, let N(U) denote the smallest integer n ≥ 0 such that U admits a subcover of cardinality n.
Lemma 5.1. Let X be a compact space. Let U = (U j ) j∈J and V = (V k ) k∈K be two open covers of X. Then one has Proof. These properties are all obvious (see for example [1]). Now suppose that the compact space X is endowed with a continuous action of a semigroup S. This means that we are given a map S × X → X, (s, x) → sx, satisfying the following conditions: (1) one has s 1 (s 2 x) = (s 1 s 2 )x for all s 1 , s 2 ∈ S and x ∈ X; (2) the map T s : X → X defined by T s (x) := sx is continuous for all s ∈ S. Suppose now that A and B are disjoint. Then we have U A∪B = U A ∨ U B and hence If A and B are arbitrary subsets of S, we can write this shows that h U is subadditive.
To prove right-subinvariance, we first observe that, for every s ∈ S and any finite subset A of S, we have . We then deduce that h U (As) = log N(U As ) = log N(T −1 s (U A )) ≤ log N(U A ) = h U (A) by using assertions (iii) and (iv) in Lemma 5.1. This shows that h U is right-subinvariant.
Finally, for all s ∈ S, we have h U ({s}) = log N(T −1 s (U)) ≤ log N(U) by Lemma 5.1.(iv). It follows that h U is uniformly bounded on singletons.
From Proposition 5.2 and Theorem 1.1, we deduce the following result.
Theorem 5.3. Let X be a compact space equipped with a continuous action of a cancellative left-amenable semigroup S and let U be an open cover of X. Then, for every left-Følner net (F i ) i∈I of S, the limit |F i | exists and is finite. Moreover, η U does not depend on the choice of the left-Følner net The quantity 0 ≤ η ≤ +∞ defined by η := sup U η U , where U runs over all open covers of X, is the topological entropy of the continuous dynamical system (X, S).
Let U = (U j ) j∈J be a finite open cover of X. The local order of U at a point x ∈ X is the integer ord(U,  (iv) if f : X → X is a continuous map then D(f −1 (U)) ≤ D(U).
Let X be a compact metrizable space equipped with a continuous action of a semigroup S. Let U be a finite open cover of X. Consider the map h dim The quantity 0 ≤ η dim ≤ +∞ defined by η dim := sup U η dim U , where U runs over all finite open covers of X, is the topological mean dimension of the continuous dynamical system (X, S).
A finite measurable partition of X is a finite family U = (U j ) j∈J of pairwise disjoint measurable subsets of X whose union is X (here, equalities for subsets of X are understood to hold up to null-measure sets). The join operation ∨, as well as the relations ≻ and ∼ =, can also be defined for finite measurable partitions. Moreover, if T : X → X is a measurable map and U = (U j ) j∈J is a finite measurable partition of X, then T −1 (U) := (T −1 (U j )) j∈J is also a finite measurable partition of X.
If U = (U j ) j∈J is a finite measurable partition of X, we define the real number E(U) ≥ 0 by E(U) := − j∈J p(U j ) log p(U j ), with the usual convention 0 log 0 = 0. A measurable map T : X → X is said to be measure-preserving if p(T −1 (B)) = p(B) for all B ∈ B.
Lemma 5.7. Let (X, B, p) be a probability space. Let U = (U j ) j∈J and V = (V k ) k∈K be two finite measurable partitions of X. Then one has (iv) if T : X → X is a measure-preserving map then E(T −1 (U)) = E(U).
Let (X, B, p) be a probability space. Suppose that X is equipped with a measurepreserving action of a semigroup S, that is, a family of measure-preserving maps T s : X → X, s ∈ S, such that T s 1 • T s 2 = T s 1 s 2 p − a.e.
for all s 1 , s 2 ∈ S. Let U be a finite measurable partition of X. Consider the map h KS U : P f in (S) → R defined by (5.4) h KS U (A) := E(U A ), where U A is defined by (5.2).
Proposition 5.8. Let (X, B, p) be a probability space equipped with a measure-preserving action of a semigroup S and let U be a finite measurable partition of X. Then the map h KS U : P f in (S) → R defined by (5.4) is non-decreasing, subadditive, right-invariant, and uniformly bounded on singletons.
Proof. Mutatis mutandis, the proof is that of Proposition 5.2 with Lemma 5.7 replacing Lemma 5.1. Note that h KS U is indeed right-invariant since in Lemma 5.7.(iv) an equality holds.
From Proposition 5.8 and Theorem 1.1, we deduce the following result.
Theorem 5.9. Let (X, B, p) be a probability space equipped with a measure-preserving action of a cancellative left-amenable semigroup S and let U be a finite measurable partition of X. Then, for every left-Følner net (F i ) i∈I of S, the limit exists and is finite. Moreover, η KS U does not depend on the choice of the left-Følner net (F i ) i∈I .
The quantity 0 ≤ η KS ≤ +∞ defined by η KS := sup U η KS U , where U runs over all finite measurable partitions of X, is the measure-theoretic entropy, or Kolmogoroff-Sinai entropy, of the measure-preserving dynamical system (X, S).