Metric Versus Topological Receptive Entropy of Semigroup Actions

We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov (Adv Math 115:54–98, 1995). We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowen generalized in the classical setting of amenable group actions by Zheng and Chen, and we prove partial versions of the Brin–Katok Formula and the local Variational Principle.


Introduction
In [29] a general notion of receptive topological entropy (the term "receptive" was coined later on) was introduced and studied for a uniformly continous action T : G × X → X of a locally compact semigroup G on a metric space (X , d). The concept defined there depends on a system = (N 0 , N 1 , . . . , N n , . . .) of compact subsets of the acting semigroup G satisfying N k N m ⊆ N k+m for every k, m ∈ Z + ; such is called a regular system in G. The receptive topological entropy of T with respect to is defined bỹ h(T , ) = sup lim where s n ( , K , T ) is the maximum size of an (n, )-separated set of K (i.e., a set E ⊆ K such that for any distinct x, y ∈ E there exists g ∈ N n with d(gx, gy) > ).
In case G = Z + , the action T is generated by the uniformly continuous selfmap f = T (1, −) : X → X , and taking N n = [0, n] for every n ∈ Z + , the receptive topological entropyh(T , ) coincides with the classical Bowen-Dinaburg's topological entropy h( f ). As in that situation, in (1.1) one can use the so called (n, )-spanning sets and r n ( , K , T ) (see Sect. 4.1).
In the present paper we study a natural similar definition of a receptive metric entropyh μ (T , ) of a measure-preserving action T : G × X → X of a discrete semigroup G on a probability space (X , μ) with respect to a regular system (N n ) n∈N in G. Also in this case, for G = Z + , f = T (1, −) and = ([0, n]) n∈N + , the receptive metric entropyh μ (T , ) coincides with the classical metric entropy h μ ( f ) by Kolmogorov and Sinai.
The dependence of the receptive entropies on the regular system is present starting from the classical case of actions of Z + . Indeed the regular system = ([0, n]) n∈Z + is accepted as natural and is very rarely mentioned explicitly, however, in general h(T , ) varies with . Kushnirenko [32] pointed this out for the case of the metric entropy of measure-preserving selfmaps f . He defined the concept of a metric Aentropy of f , where A = (t n ) n∈Z + is a sequence of positive integers. In particular, he proved that for A = (2 n ) n∈Z + , the metric A-entropy is different from the classical metric entropy, which corresponds to the choice A = (n) n∈Z + . A similar topological concept, called topological sequence entropy, was introduced and studied several years later by Goodman [25], who also investigated the relationship between his concept and the measure-theoretic analogue defined by Kushnirenko. After that there has been a significant activity in studying both kinds of sequence entropies -see for example [9,11,22,33].
On the other hand, there exist well-known definitions of metric entropy h μ and topological entropy h for amenable group and semigroup actions (see for example [10,18,20,36,38,42,43] for these notions and their main properties) and in this case the value of the entropy does not depend on the choice of the Følner sequence used in the definition. However, with these classical definitions, both the metric and the topological entropy are frequently zero (see Sects. 2.3 and 4.2 below). For example this is the case for smooth actions T of Z k + , with k > 1, on manifolds: as mentioned in [29], while h(T ) = 0, the receptive topological entropyh(T ) is not trivially zero. Other examples are given below whereh μ (T ) andh(T ) are non-zero, while the classical entropies h μ (T ) and h(T ) are both zero. Conversely, as one can see from the definitions, when h μ (T ) or h(T ) is non-zero our definitions produce ∞.
In this paper first of all, in Sect. 2 we study the basic properties of the receptive metric entropyh μ (T , ) of a measure-preserving action T : G × X → X on a probability space (X , μ), where is a regular system in G. We compareh μ (T , ) with the classical metric entropy h μ (T ) in case is also a Følner sequence (see Sect. 2.3), and we point out that the receptive metric entropy does not vanish in many of the cases when the classical one does (see Proposition 2.11 and Example 2.12). Further properties concerning restriction actions and products of actions are discussed in Sect. 3.
The receptive topological entropyh(T , ) is recalled in Sect. 4, together with some results from [29]. Also in this case we recall that several natural actions where the receptive topological entropy does not vanish while the classical one does (see [29] and also [44]).
In Sect. 5 we propose several different versions of the receptive topological entropy. We start considering a version using open covers for continuous actions on compact topological spaces, following the classical idea of Adler, Konhein and McAndrew [1].
We generalize a well-known notion by Bowen [7] defining a receptive-like entropy b(T , ) for a continuous action T : G × X → X of a semigroup G on a compact metric space X , where is a regular system in G. Then b(T , ) ≤h(T , ), and the equality b(T , ) =h(T , ) holds when G is commutative and finitely generated and is standard (i.e., N 1 generates G and N n = N n 1 for every n ∈ Z + ). This is proved in Theorem 5.5, which very roughly follows the line of [7], but its proof is more subtle and requires much more effort and computations.
It turns out that this receptive entropy b(T , ) coincides with another entropy-like quantity c(T , ) defined by using some ideas of Pesin [40] (see Theorem 5.8).
In Sect. 6, we discuss the relation of the receptive topological entropy with the receptive metric entropy, looking for some Variational Principle. Following [49,Chapter 6], for a compact metric space X we consider Borel probability measures on X having as σ -algebra of measurable sets precisely the σ -algebra of Borel sets; any such measure is necessarily inner and outer regular [49,Theorem 6.1]. The classical Variational Principle due to Goodwyn [27] and Goodman [24] states that, in case G = Z + and f = T (1, −) : X → X is a continuous selfmap of the compact metric space X , then It is well-known that the Variational Principle holds also for continuous actions of amenable groups or of countable cancellative semigroups on compact metric spaces, for the classical topological and metric entropy defined by means of a Følner sequence (see [38,39] and [36,46] respectively).
We conjecture that the Variational Principle holds for the receptive topological entropy and the receptive metric entropy (see Conjecture 6.3): if G is a semigroup, = (N n ) n∈Z + a regular system in G and T : G × X → X a continuous action on a compact metric space X , denote by M(X , T ) the family of all T -invariant Borel probability measure on X and assume that M(X , T ) is non-empty; theñ This remains an open problem, nevertheless we prove some partial results. We give also a brief comment on the algebraic entropy and its connection to the topological entropy both in the classical and in the "receptive" cases.
To further clarify the relationships between the concepts considered in this paper, for a semigroup action T : G × X → X on a compact metric space X and a regular system in G, in Sect. 7 we introduce a receptive local metric entropy h loc μ (T , ), where μ is a Borel probability measure on X ; this is defined by where h loc μ (x) is the receptive local metric entropy at x ∈ X . The latter entropy was defined in [3] in the case when the acting semigroup is a finitely generated group and the regular system is standard, while an analogous local metric entropy was defined in [51] for actions of amenable groups. All these authors were inspired by a similar notion by Brin and Katok [8].
We prove that, when μ is a T -invariant Borel probability measure on X ,  [51]. A positive response to this problem would ensure one of the inequalities in the Variational Principle (1.2) in view of the following argument.
In the classical setting of countable amenable group actions a local version of the Variational Principle was established by Zheng and Chen [51], by using a version of topological entropy inspired by the one by Bowen [7]. So, we conjecture that also in our setting, possibly under suitable hypotheses, a "local Variational Principle" holds (see Conjecture 7.7): if G is a semigroup, = (N n ) n∈Z + a regular system in G,  [31] for the so-called slow entropy. Indeed, starting from the above-mentioned problem that the classical metric and topological entropy vanish very often, Katok and Thouvenot [30] proposed a notion of slow metric entropy for Z d -actions (see also Hochman [28]. A slow topological entropy inspired by Bowen's topological entropy from [7] was introduced by Kong and Chen [31] for Z d -actions on compact metric spaces (note that the slow topological entropy does not coincide with the classical one even in the case d = 1). In the same setting, Kong and Chen introduced also a notion of slow lower local metric entropy (which does not coincide with that in [30]) and proved that the counterpart of the local Variational Principle for the slow entropies holds.
We warmly thank the referees for their careful reading and useful comments and suggestions.

Notation and Terminology
For a set X we denote by cov(X ) the set of covers of X and by par(X ) the set of For a topological space X we denote by cov(X ) the subfamily of cov(X ) consisting of open covers of X . If X is a measure space with a probability measure we denote by par(X ) the subfamily of par(X ) consisting of finite partitions of X by means of measurable subsets.
The set cov(X ) carries a natural preorder: if E and A are families of subsets of a set X we write A ≺ E if E refines A, i.e., every E ∈ E is contained in some element of A. If E and A are partitions of X , then A ≺ E implies that every A ∈ A is a union of elements of E.
If T : G × X → X , T (g, x) = g x, is an action of a semigroup G on a set X , for g ∈ G we use the notation g also for the selfmap g : x → gx of X and in such a sense we write g −1 A to denote the inverse image of a subset A of X under this selfmap. For g ∈ G and A ∈ cov(X ) we let Remark 1.1 In this sense, when G acts on a topological space X by continuous selfmaps, this action induces also an action of G on the lattice of open sets of X by U → g −1 U and consequently also on cov(X ).
Furthermore, when G acts on a probability space X by measure-preserving selfmaps, then G acts also on the Boolean σ -algebra of measurable sets of X , by B → g −1 B, and consequently also on par(X ).
We consider only semigroups G with identity e and actions T : G × X → X such that T (e, −) = id X .

Regular Systems
Definition 1.2 A regular system in a semigroup G is by definition a sequence of subsets of G of the form = (N 0 , N 1 , . . . , N n , . . .) such that e ∈ N 0 and N i N j ⊆ N i+ j for all i, j ∈ Z + . (1.5) Throughout this paper we assume that all sets N n are finite in a regular system = (N n ) n∈Z + .

Example 1.3
A prominent example of regular system is obtained by taking in (1.5) a finite set N 1 of generators of a finitely generated semigroup G with N 0 ⊆ N 1 (and so e ∈ N 1 ) and N m = N m 1 for all m ≥ 1. Following [29], we call such regular systems standard. Clearly, only finitely generated semigroups may have standard regular systems, in case G = ∞ i=0 N i is imposed. Our main example is G = Z k (or G = Z k + ) with a standard regular system given, for n ∈ Z + , by N n = [−n, n] k (respectively, N n = [0, n] k ).
When T : G × X → X is an action of a semigroup G on a set X , given a regular system = (N n ) n∈Z + in G and a cover A of X , define for every n ≥ 1 a cover of X by Clearly, A n is a partition whenever A is a partition. If the regular system is fixed we briefly write A n instead of A n .

Definition of the Receptive Metric Entropy
Assume that (X , M, μ) is a probability space with a probability measure μ defined on the σ -algebra M of subsets of X , and let be a measure-preserving action of G on X , i.e., the map g : X → X , x → gx, is measure-preserving for all g ∈ G. For a fixed regular system = (N n ) n∈Z + in G, we define the metric entropyh μ (T ) =h μ (T , ) of T with respect to essentially using the approach in [29]. Such definition in the case G = Z 2 was given in the honours thesis (diploma work) [47] of Leon Todorovich, however without much follow up.
Recall (see e.g. [49,Chapter 4]) that if A = {A 1 , A 2 , . . . , A k } ∈ par(X ), then the entropy of A is defined by with the usual agreement that 0 log 0 = 0. Since the measure μ is T -invariant, it follows that H μ (g −1 A) = H μ (A) for all g ∈ G.
Definition 2.1 Let G be a semigroup, = (N n ) n∈Z + a regular system in G, and (X , M, μ) a probability space. The receptive metric entropy of the measure-preserving action T : G × X → X with respect to A ∈ par(X ) and is The receptive metric entropy of T with respect to ish μ (T , ) = sup{h μ (T , A, ): A ∈ par(X )}.
When the system is clear from the context and no confusion is possible, we shall omit .

Remark 2.2
In the classical case G = Z + , consider the measure-preserving action T : G × X → X on the probability space (X , M, μ) as a single measure-preserving transformation T : X → X (i.e., T (1, −) = T ). With respect to the standard regular system = ([0, n]) n∈Z + , soh μ (T ) coincides with the Kolmogorov-Sinai metric entropy h μ (T ) of T .

Example 2.3
The trivial action T : G×X → X of a semigroup G on a probability space (X , M, μ) has alwaysh μ (T ) = 0 with respect to any regular system = (N n ) n∈Z + in G. Indeed, for every A ∈ par(X ) one has A n = A, so the limit in (2.2) is 0, hencẽ h μ (T , A) = 0 regardless of . This givesh μ (T ) = 0. Similarly,h μ (T ) = 0 when G is finite.

Remark 2.4
The computation of the receptive measure entropy can be reduced to the case of faithful actions. Indeed, the measure-preserving action T : G × X → X of a semigroup G on a measure probability space (X , M, μ) can be factorized in an obvious way through an action T : G × X → X of a quotient G of G that faithfully acts on X (i.e., if gx = x for all x ∈ X , then g = e G ). Since, given a regular system = (N n ) n∈Z + in G, for every n ≥ 1, A n coincides for both actions and for every A ∈ par(X ), we deduce thath μ (T , A) =h μ (T , A).
It follows immediately from the definition that the receptive metric entropy is a conjugacy invariant. This comes also from the following more general property of monotonicity (obtained with H = G and η = id G ): Proposition 2.5 Let G be a semigroup, = (N n ) n∈Z + a regular system in G and (X , M, μ) a probability space. Let (Y , N , ν) be another probability space, let S : H × Y → Y be a measure-preserving action and let there exist a measure-preserving surjective map ϕ : X → Y and a homomorphism η : G → H with S(η(g), ϕ(x)) = ϕ (T (g, x)) for all g ∈ G and x ∈ X (briefly, η(g) • ϕ = ϕ • g for every g ∈ G).

First Properties of the Receptive Metric Entropy
Proposition 2.6 Let G be a semigroup, = (N n ) n∈Z + a regular system in G, (X , M, μ) a probability space and T : G × X → X be a measure-preserving action. Let p ≥ 1 be an integer and = (N pn ) n∈Z + . Thenh μ (T , A, ) = p ·h μ (T , A, ) for every A ∈ par(X ), and thush μ (T , ) = p ·h μ (T , ).
Proof Given A ∈ par(X ), it is enough to prove that To prove the converse inequality, note that there exists an increasing sequence of integers (n r ) r ∈Z + with n r → ∞ and such that lim sup For every r ≥ p put m r := n r p ≥ 1, so that m r p ≤ n r < (m r + 1) p.
For a slight generalization of this property see [4,Remark 2.3(b)].
For G = Z + the equalityh μ (T , ) = p ·h μ (T , ) is known as "logarithmic law", since it becomes h μ ( It is natural to compare the metric entropy of each individual map g ∈ G with h μ (T ). It turns out that for elements g ∈ N 1 the metric entropy of g is bounded above byh μ (T ).

Proposition 2.7 Let G be a semigroup,
= (N n ) n∈Z + a regular system in G, (X , M, μ) a probability space and T : G × X → X a measure-preserving action. If g ∈ N m for some integer m ≥ 1, then h μ (g) ≤ m ·h μ (T ). Thus, By well-known properties of partitions (see e.g. [49,Theorem 4 . Thus, letting = (N mn ) n∈Z + and applying Proposition 2.6 in the last equality, we deduce that Therefore h μ (g) ≤ m ·h μ (T ).

Remark 2.8
The receptive metric entropy of a measure-preserving action T : G × X → X as above depends on the choice of the regular system . One can be tempted to get rid of this dependence by taking sup{h μ (T , ): is a regular system for G}.
Proposition 2.6 shows that (2.7) cannot be a good remedy, as it can take only the values 0 and ∞.

Classical Metric Entropy
Recall that a discrete countably infinite cancellative semigroup G is called left amenable if it admits a Følner sequence, i.e., a sequence (F n ) n∈Z + of finite subsets of G such that lim n→∞ |F n gF n | |F n | = 0.
For countably infinite groups left and right amenability coincide. Finite semigroups are obviously amenable. The classical definition of metric entropy of a measure-preserving amenable semigroup action T : See [10,18,42,43]. In this case the limit is known to exist (see e.g. [10, Theorem 2.1]).

Example 2.9
When G is a finite semigroup, for a measure-preserving action T : For G = Z k + every standard regular system gives rise to a Følner sequence (see [4]), while in G = Z + , (N n ) n∈Z + with N n = {0, 2, 4, . . . , 2n} for every n ∈ Z + , is a regular system which is not a Følner sequence.

Remark 2.10
Let G be a semigroup and suppose that the regular system = (N n ) n∈Z + in G is a Følner sequence and satisfies |N n | ≥ cn 2 for some constant c > 0 and all n ≥ 1. (2.8) Let T : G × X → X be a measure-preserving action on a probability space (X , M, μ).

It then follows from the definitions that if for some
The condition (2.8) is available whenever G is a finitely generated group that is not commensurable with a cyclic group (e.g., G = Z k for some k ≥ 2, see [4] for more detail). Recall that two subgroups H 1 and H 2 of a group G are said to be commensurable if the subgroup H 1 ∩ H 2 has finite index both in H 1 and H 2 .
In contrast with Proposition 2.7, in the classical case, as mentioned for example in [10], h μ (T ) ≤ h μ (g) for any g ∈ G. More importantly, we have the following fact proved by Conze [10, Theorem 2.3] (where the case k = 2 is dealt with, the general case is proved in a similar way).
. . , g k be generators of G, and let T : G × X → X be a measure-preserving action of G on a probability space The next example shows that unlike the classical metric entropy the receptive one does not vanish in this situation.

Example 2.12
Let f : X → X be a measure-preserving transformation of the probability space (X , M, μ), and let G = Z k + with k > 1 and the standard regular system Assume h μ ( f ) < ∞. It then follows from Proposition 2.11 that h μ (T ) = 0. On the other hand, we compute thath μ (T ) = k · h μ ( f ). Indeed, given A ∈ par(X ), for every n ≥ 1, and taking limits as n → ∞ (the limit exists in the right-hand side by Remark 2.2, so it exists in the left-hand side as well), we geth μ (T , To get a particular example, let X = ∞ −∞ {1, 2, . . . , r }, let f = σ : X → X be the Bernoulli shift on r symbols with probabilities p 1 , p 2 , . . . , p r > 0 and r i=1 p i = 1, and let μ be the corresponding invariant (product) measure on X . [49,Theorem 4.26]). Then for the action T defined above it follows thath μ (T ) = −k r i=1 p i log p i < ∞.

Further Properties of the Receptive Metric Entropy
We begin with some facts about subactions. For T : G × X → X a measure-preserving action of the semigroup G on the probability space (X , M, μ), and H a subsemigroup of G, let T H : H × X → X be the action of H on X induced by T .
Proof (a) Clearly, for A ∈ par(X ) and for all n ≥ 1, 1 The first inequality is item (a). Without loss of generality we may assume that Let n ≥ p k . Then N n = M n + F. Indeed, for any g = (m 1 , m 2 , . . . , m k ) ∈ N n and for every i ∈ {1, 2, . . . , k} we can write m i = t i p i + s i for some integers t i ≥ 1 and s i ∈ {0, 1, . . . , p i − 1}. Thus, g = u + s, where u = (t 1 p 1 , t 2 p 2 , . . . , t k p k ) ∈ M n and s = (s 1 , s 2 , . . . , s k ) ∈ F. The above argument implies that, for A ∈ par(X ) and an integer n ≥ 1, and therefore the last equality due to the fact that each s ∈ F is measure-preserving. Thus, for all n ≥ 1, Now we make a different choice of the regular system in a specific case.
Proof Let A ∈ par(X ). For every n ≥ 1 and every g ∈ N n we have pg ∈ pN n ⊆ N pn ∩ H , so Thus, for all n ≥ 1, Obviously, lim sup Combining this with the above gives By the final assertion of that proposition,h μ (S, p ) =h μ (T , ), provided η is an isomorphism (equivalently, injective). Let us apply this observation to the monoid G = Z k + with k ≥ 1 and standard regular system = (N n ) n∈Z + , with N n = [0, n] k for every n ∈ Z + . Now for every p ≥ 1 the homomorphism η : G → H , defined above, is injective. So for the subsemigroup H = pG of G and for every measure-preserving action T : G × X → X on a probability measure space (X , M, μ), the above argument givesh μ (S, p ) =h μ (T , ) for the restriction S of T on H . Since, obviously p = H = ( pN n ) n∈Z + in this case, we can add to the above equality alsoh This should be compared with the much less sharp inequalities from Proposition 3.1(b) and from Lemma 3.2.
Next we deal with products of measure-preserving actions. The proof of the following proposition is similar to that of the corresponding result about the case G = Z + (see e.g. the proof of [49,Theorem 4.23]). To prove Proposition 3.5 we need the following lemma.

Lemma 3.4 For finite partitions
Proposition 3.5 For i = 1, 2, let G i be a commutative finitely generated semigroup with a regular system (i) n for all n ∈ Z + . Let T : G × X → X be the action defined by Then where A ∈ par(X 1 ) and B ∈ par(X 2 ). For every n ≥ 1, By Lemma 3.4, (2) ).
for some increasing sequence of integers (n k ) k∈Z + with n k → ∞. Then the above equalities show that This implies the second inequality in (3.1). The first inequality in (3.1) follows immediately from (3.2).
The above is easily generalised to arbitrary finite products, so that it can be applied in the next example.
Consider the standard regular system = ([0, n] k ) n∈Z + in Z k + and define the action Then Proposition 3.5 gives As mentioned in [29], if the components f i in Example 3.6 are smooth maps on smooth compact manifolds with topological entropy h( f i ) > 0, then the smooth action T of Z k + has positive receptive topological entropy (to be defined in the next section). The above shows that the same kind of examples exist with respect to the receptive metric entropyh μ .

Receptive Topological Entropy
Let (X , d) be a metric space, let T : G × X → X be an action of the semigroup G on X such that each of the maps g : x → gx is uniformly continuous (briefly, T is a uniformly continuous action).
Let = (N n ) n∈Z + be a regular system in G. The topological entropy that we will consider here was defined and studied in [29] under more general assumptions (in [29] G is a locally compact group and each N n is just a compact subset of G), using spanning and separated sets, following R. Bowen's definition [6] in the classical case of actions of Z + .
= (N n ) n∈Z + a regular system in G and T : G × X → X a uniformly continuous action on a metric space X . Let K be a compact subset of X , let > 0 and let n ≥ 1.
A subset F of K is called an (n, )-spanning set for K (with respect to ) if for every x ∈ K there exists y ∈ F such that d(gx, gy) ≤ for all g ∈ N n .
A subset E of K is called (n, )-separated (with respect to ) if for any x, y ∈ E, x = y, there exists g ∈ N n such that d(gx, gy) > .
When it is clear from the context we shall omit the regular system . Namely, F is an (n, )-spanning set for the compact subset K of X if K ⊆ x∈F D n (x, ), and E is (n, )-separated if for any x, y ∈ E with x = y, D n (x, ) ∩ D n (y, ) = ∅.
By the compactness of K , the minimal (n, )-spanning sets and the maximal (n, )separated sets are finite, so we can define

Definition 4.2
Let G be a semigroup, = (N n ) n∈Z + a regular system in G and T : G × X → X a uniformly continuous action on a metric space X . Let K be a compact subset of X . We callh (T , K , ) the receptive topological entropy of T with respect to K and the given regular system in G. The receptive topological entropy of T with respect to is When the system is clear from the context and no confusion is possible, we shall omit and write simplyh(T , K ) andh(T ).
When X is compact, it is easy to see thath(T ) =h(T , X ). In this case we will also use the shorter notation r n ( , T ) = r n ( , T , X ),r ( , T ) =r ( , T , X ), etc. It follows from [29, Proposition 2.6] that for the uniform continuous action T : G × X → X of the semigroup G on the metric space X and the regular system where h(g) is the topological entropy of the map g : x → gx (a proof can be given in the line of Proposition 2.7 above).
For the sake of completeness we mention the following special case of [29, Proposition 2.7], which is an analog of Bowen-Kushnirenko Theorem in the classical case (e.g., see [49,Theorem 7.15]).

Proposition 4.4 Let X be a Riemannian manifold and let the continuous action
T : G × X → X of the semigroup G be such that each of the maps g : x → gx is smooth. Consider the standard regular system = (N n ) n∈Z + in G where N 1 is a finite generating subset of G. Consider X with the metric generated by the Riemannian metric of X , and let k = dim X . Theñ where a = sup x∈X max g∈N 1 d x g and d x g is the differential of g at x ∈ X.

Classical Topological Entropy
Let G be a semigroup and T : G × X → X a uniformly continuous action on a metric space X . Assume in addition that G is amenable and let = (N n ) n∈Z + be a Følner sequence in G. For a compact subset K of the metric space X , define r n ( , T , K ) and s n ( , T , K ) as in (4.1) and set The topological entropy of T with respect to K is then while the topological entropy is This is the definition of the classical topological entropy using spanning and separated sets (see e.g. [42]). An equivalent definition for actions on compact spaces using open covers was introduced in [18,38]. In fact, a more general concept, the so-called topological pressure, has been defined similarly and studied extensively (see [20,36,38,39,42,43]).
As in the case of metric entropy (see Proposition 2.11), the classical topological entropy is zero for some actions of Z k + , with k > 1):

Receptive Topological Entropy via Open Covers
First we define the receptive topological entropy adapting the initial approach of Adler, Konheim and McAndrew [1]. Let X be a compact topological space and for any A ∈ cov(X ) we let N (A) be the number of elements of a subcover of A of the smallest possible cardinality.
Let T : G× X → X be an action of the semigroup G on X such that each of the maps g : x → gx is continuous (briefly, T is a continuous action), and let = (N n ) n∈Z + be a regular system in G.
Given an integer n ≥ 1 and A ∈ cov(X ), set In case h(T ) = ∞, let a > 0 and let γ ∈ cov(X ) with h(T , γ ) > a. Then proceed as in the preceding case.

Theorem 5.3 Let G be a semigroup, = (N n ) n∈Z + a regular system in G, and T : G × X → X a continuous action of G on a compact metric space X . Then h(T ) =h(T ).
Proof For any integer m ≥ 1 fix A m ∈ cov(X ) consisting of open balls of radius 2/m and γ m ∈ cov(X ) consisting of open balls of radius 1/(2m). By Lemma 5.1 we obtain, for all integers n ≥ 1, Taking lim sup n→∞ , gives

Receptive Topological Entropy Following Bowen's Definition
Next, we give another equivalent definition of the receptive topological entropy similar to the one by Bowen in [7]. Although this definition works for non-compact spaces as well, here we concentrate on the case of a compact metric space X . Denote by B(X ) the family of Borel subsets of X and by [X ] ≤ω the family of all non-empty finite or countable subsets of X . Let G be a semigroup, = (N n ) n∈Z + a regular system in G and T : G × X → X a continuous action. Let also A = {A 1 , A 2 , . . . , A k } ∈ cov(X ). Modifying a definition in [7], for every non-empty subset E of X we denote We set n A (E) = 0 if E is not contained in any element of A. Then set Next, given a subset Y of X , > 0 and any λ ∈ R, consider the family F (Y ) of all finite or countable covers E of Y with diam A,T (E) < . Clearly, For a family E ⊆ [X ] ≤ω and any real λ > 0 define Therefore, the function increases when > 0 decreases. This allows us to define an outer measure m A,λ on X by setting The function m A,λ (Y ) is non-increasing in λ and if 0 < m A,λ (Y ) < ∞ for some λ > 0, then m A,λ (Y ) = 0 for λ > λ and m A,λ (Y ) = ∞ for λ < λ. Set Finally, set b(T , ) = b(T , X , ). When the system is clear from the context and no confusion is possible, we shall omit .

Remark 5.4
Only real numbers λ ≥ 0 were considered above, as the negative values of λ are irrelevant for the definition of b (T , Y , ). Indeed, m A,λ (Y ) is non-increasing in λ and then we take an inf in the definition of b A (T , Y , ). Actually, Here is an alterative direct proof of (5.2) that works also for singleton Y when λ < 0.
In the next theorem the inequality b(T ) ≤h(T ) holds for any regular system and also when the semigroup is not commutative. We follow the line of [7], however some significant modifications will be necessary.

Proof In view of Theorem 5.3 we show that b(T ) = h(T ). First, we show that b(T ) ≤ h(T ). Let
. . , A k } ∈ cov(X ), let n ≥ 1 be an integer, and let E n = {E 1 , E 2 . . . , E m } be a subcover of A n with m = |E n | = N (A n ). By the definition of A n , any E i can be written in the form E i = g∈N n g −1 A g for some A g ∈ A. So, for any g ∈ N n we get gE i ⊆ A g . Hence for every E i we have n A (E i ) ≥ n. Therefore, given λ ∈ R ≥0 , If λ > h(T , A) = lim sup n→∞ 1 n log N (A n ), then taking δ > 0 so that λ > h(T , A) + δ, we get D A (E n , λ) < e −nδ for all sufficiently large n ≥ 1, so we must have m A,λ (X ) = 0. This demonstrates that b A (T , Next, we show that h(T ) ≤ b(T ). The proof is articulated in four claims.
Assume that b(T ) < ∞; otherwise there is nothing to prove. Fix a small constant > 0 and an arbitrary λ The definition of b(T ) now implies b B (T , X ) < λ and so m B,λ (X ) = 0 for any finite B ∈ cov(X ). Our aim will be to deduce from this that h(T ) ≤ b(T ) + 2 . Since was chosen arbitrarily small, this will imply that h(T ) ≤ b(T ). Let us fix a finite set of generators f 1 , f 2 , . . . , f of G, so that = (N n ) n∈Z + is the standard regular system defined by N 1 = {0, f 1 , f 2 , . . . , f }. Our next step is to Let δ > 0 be a Lebesgue number for A. Since N 1 is finite and f i : X → X is uniformly continuous for every i ∈ {1, . . . , }, there exists δ 1 ∈ (0, δ/2) such that g B δ 1 (x) ⊆ B δ (gx) for every g ∈ N 1 and every x ∈ X . That is, for any g ∈ N 1 and any There exists a finite cover B = {B 1 , B 2 , . . . , B p } of X consisting of open balls of radius δ 1 . By what we said above, Let δ 2 ∈ (0, δ 1 ) be a Lebesgue number for B. Hence, B ≺ {E} for every subset E of X with diam(E) < δ 2 .
As mentioned above, the choice of λ implies m B,λ (X ) = 0, so in particular there exists a finite or countable cover E = {E i : i ∈ I } of X with diam(E i ) < δ 2 for all i ∈ I and D B (E , λ) < 1. Next, for every i ∈ I let n i := n B (E i ) and for every g ∈ N n i fix a member B g ∈ B, such that gE i ⊆ B g (we noted above that B ≺ {gE i }). Now put, for every i ∈ I , By Note that a j ≥ 1 for all j ∈ J . This implies series converges as D B (E, λ) < 1).
For the sake of brevity set f = f and let C(V j ) := V j for every j ∈ J . For s > 1 and j 1 , . . . , j s ∈ J , let For any pair of integers s ≥ 1 and n ≥ 1 let J n,s := {j = ( j 1 , . . . , j s ) ∈ J s : n ≤ a j 1 + . . . + a j s ≤ n + M}.
As a j ≥ 1 for all j ∈ J , it is clear that J 1,1 = J , while J n,s = ∅ implies s ≤ n + M, hence s≥2 j ∈J n,s 1 < ∞ for all n ≥ 1.

Claim 1 For any integer n ≥ 2M, the family
is an open cover of X .
is strictly increasing and the gaps between two adjacent members are ≤ M. As n ≥ 2M > a j 1 , there exists an s ≥ 2 such that a j 1 + . . . + a j s−1 < n ≤ a j 1 + . . . + a j s−1 + a j s , then obviously, a j 1 +. . .+a j s−1 +a j s ≤ n + M, as a j s ≤ M. Hence, C(V j 1 , . . . , V j r ) ∈ C n . This ends up the proof, since x ∈ C(V j 1 , . . . , V j s ), as mentioned above.
Recalling that f = f , for any integer n ≥ 1 set Claim 2 For all n ≥ 2M, A n, ≺ C n .
Proof It is enough to prove the following statement for all integers s ≥ 1: P(s) : If a set of the form C = C(V j 1 , . . . , V j s ) belongs to C n for an integer n ≥ 2M, then A n, ≺ {C}.
We argue by induction on s ≥ 1. To prove P(1), let C = C(V j ) = V j ∈ C n for some integer n ≥ 1. Then n ≤ a j . So A n, ≺ A a j , , hence it suffices to check that A a j , ≺ {C}, or equivalently, gC is contained in some element of A for every g = f p , with 0 ≤ p < a j . To this end note that C = V j and diam(V j ) < δ 2 implies that C is contained in some element B i of B. Now a j = n B (V j ) means that gC is contained in some element of B, and therefore in some element of A (since A ≺ B) for all g ∈ N a j . Thus, f p C is contained in some element of A.
Assume that s > 1 and the statement P(r ) is true for all 1 ≤ r ≤ s − 1. We will prove P(s). Let C = C(V j 1 , . . . , V j s ) be such that C ∈ C n for some n ≥ 1. Then n ≤ a j 1 + . . . + a j s ≤ n + M. Set u = a j 1 , n = n − u and C = C(V j 2 , . . . , V j s ). It is easy to see that Since n ≤ u +(a j 2 +. . .+a j s ) ≤ n + M implies that n ≤ a j 2 +. . .+a j s ≤ n + M and consequently C ∈ C n , our inductive assumption gives A n , ≺ {C }. Now we show that A n, ≺ {C}, i.e., gC is contained in some A i for every g = f p with 0 ≤ p < n.
The case p < u follows from the argument proving P(1) above, so assume that g = f p for some integer p with u ≤ p < n − 1. As a i ≥ 1 for all i ∈ J , it is clear that n,s = ∅ implies s ≤ n + M, hence ∞ s=1 | n,s | < ∞ for all n ≥ 1.
Combining this with (5.5) proves the required inequality. for some constant K < ∞ independent of n. This yields, for every n ≥ 2M, Therefore

Combining this with (5.3) gives h(T
. This proves the theorem.

Remark 5.6
Notice that the first part of the proof of Theorem 5.5 works without assuming that the semigroup is commutative and finitely generated, and moreover without any specific assumptions about the regular system. Thus, b(T ) ≤h(T ) holds for any continuous action of a semigroup G on a metric space and with respect to any regular system in G.

An Alternative Definition Following Pesin
Let G be a semigroup, = (N n ) n∈Z + a regular system in G, and T : G × X → X a continuous action of G on a compact metric space X . In what follows we will give an equivalent definition of Bowen's entropy b(T , Y ) following some ideas of Pesin (see [40,Section 4.11]). Let Y be a non-empty subset of X and let λ ≥ 0. Given > 0 and an integer N ≥ 1 set The limits exist due to the monotonicity of m N , ,λ (Y ) with respect to both N and . Just as in the case of m A,λ (Y ), it is easy to see that there exists a critical point s ∈ R such that m λ (Y ) = 0 for λ > s and m λ (Y ) = ∞ for λ < s. Set c(T , Y , ) = s. Thus, we define a new entropy-like quantity For Y = X set c(T , ) = c(T , X , ). When the system is clear from the context and no confusion is possible, we shall omit .

Remark 5.7
In [5], following the line of [40, Sections 10 and 11], we defined the upper capacity for Pesin-Carathéodory structures, and we described the receptive topological entropyh(T , Y , ) as a limit of upper capacities for suitable Pesin-Carathéodory structures. The entropy c(T , Y , ) can be analogously obtained as a limit of upper capacities.
Theorem 5.8 Let G be a semigroup, = (N n ) n∈Z + a regular system in G, and let T : G × X → X be a continuous action of G on a compact metric space X . If Y is a non-empty subset of X , then b (T , Y , ) = c(T , Y , ).
Proof Let A = {A 1 , . . . , A k } ∈ cov(X ) and let > 0 be a Lebesgue number for A. Given an integer N ≥ 1 and a real number λ > 0, set (5.7) The map N → m N ,A,λ (Y ) is monotone, and it can be seen as above that m N ,A,λ (Y ) has similar properties to m N , ,λ (Y ).
Notice that if E = D n (x, ) for some x ∈ X and n ≥ 1, then n A (E) ≥ n. Indeed, given g ∈ N n , there exists j ∈ {1, . . . , k} with B (gx) ⊆ A j . For any y ∈ E we have d(gx, gy) < , so gy ∈ B (gx) ⊆ A j . Thus, gE ⊆ A j . This proves that n A (E) ≥ n.
Hence, taking N ≥ 1 sufficiently large so that e −N < , the set in the righthand side of (5.7) is contained in the set in the right-hand side of (5.1). Therefore m N ,A,λ (Y ) ≤ m N , ,λ (Y ). Moreover, the set in the right-hand side of (5.6) is contained in the set in the right-hand side of (5.7). Taking limits N → ∞, → 0, gives To prove the opposite inequality, take small constants δ > 0 and > 0, and let A = {A 1 , , . . . , A k } ∈ cov(X ) with diam(A i ) < for all i ∈ {1, . . . , k} and such that Then m A,λ (Y ) = 0 for all λ > λ 0 . Let λ > λ 0 and take a large N ≥ 1 so that e −N < and m N ,A,λ (Y ) < δ. Then there exists a cover E = {E i : i ∈ I } of Y as in the right-hand side of (5.7) with i∈I e −λn A (E i ) < δ.
For each i ∈ I , fix an arbitrary x i ∈ E i and set The following is a direct consequence of Theorem 5.5 and Theorem 5.8.

Corollary 5.9
Let G be a finitely generated commutative semigroup, = (N n ) n∈Z + a standard regular system in G, and T : G × X → X a continuous action of G on a compact metric space X . Thenh(T , ) = b(T , ) = c(T , ).

Comparisons Between Metric and Topological Entropy
Let again T : G × X → X be a continuous action of the semigroup G on a compact metric space X , and let = (N n ) n∈Z + be a regular system in G.
Denote by M(X ) the space of all Borel probability measures on X considered with the weak * topology (see e.g. [49,Chapter 6]). Let M(X , T ) be the closed subspace of M(X ) consisting of all T -invariant measures μ ∈ M(X ).

Remark 6.1
In general, even when the semigroup G is finitely generated, M(X , T ) could be empty (e.g., see [48, Example 4.1.1]). Nevertheless, it is known (see [48, pages 97-98]) that for two commuting homeomorphisms f , g : X → X there exists a Borel probability measure on X which is both f -invariant and g-invariant. The argument can easily be extended to the case of two, and so finitely many, pairwise commuting continuous selfmaps. Therefore, M(X , T ) is non-empty in case G is a finitely generated commutative semigroup.
The existence of a T -invariant Borel probability measure on X is ensured also when G is an amenable (not necessarily finitely generated) group (see [19,Theorem 8.10]).
The classical Variational Principle due to Goodwyn [27] and Goodman [24] states that, in case G = Z + and T = T (1, −) : X → X is a continuous selfmap, In fact there is a more general Variational Principle concerning the topological pressure. Proofs of the Variational Principle have been given by various authors in a variety of specific situations -see [42] (for G = Z k and topological pressure, under some conditions), [20,36,38,39,43] and others.

Remark 6.2
In particular, it is well-known (see [38,39]) that the Variational Principle (6.1) holds for continuous actions of amenable groups, where h(T ) and h μ (T ) are the classical topological and metric entropy defined by means of a Følner sequence (N n ) n∈Z + (see Sects. 2.3 and 4.2). Moreover, the same result was proved by Misiurewicz [36] for actions of Z n + , and for actions of countable cancellative semigroups in [46].
For the receptive topological entropyh(T ) considered here a few remarks follow (the general case is not done yet). What we aim to prove is the following Receptive Variational Principle. The main problem in all existing proofs of similar claims, including the one by Misiurewicz in [36], is that in doing some estimates the "error term" is multiplied by |N n |. Then using the classical definition of entropy, a division by |N n | follows which solves the problem. In our case we only divide by n and this cannot kill the extra factor. Since |N n |/n → ∞ as n → ∞, this impliesh μ (T , A) = lim sup n→∞ 1 n H μ (A n ) = ∞. Thus,h μ (T ) = ∞, so (6.2) holds.

Conjecture 6.5 Let G be a semigroup,
= (N n ) n∈Z + a regular system in G and let T : G × X → X be a continuous action on a compact metric space X with M(X , T ) = ∅. Thenh We conclude with a special case, namely when X is a compact topological group with Haar measure μ and G acts on X by continuous surjective endomorphisms. By a well known Halmos paradigm, this ensures that the actions T is measure preserving. In , as established in [45]. The equality h(T ) = h μ (T ) is true also for Z d actions on compact groups for the classical entropy (see [34]). Clearly, this can be seen as a (strongly) positive answer to Conjecture 6.3. In this vein one can ask: Question 6.7 Under which conditions on the semigroup G the equalityh(T ) =h μ (T ) is true in the above situation?
The notion of algebraic entropy h alg ( f ) for an endomorphism f : A → A of an abelian group A was briefly introduced in [1] and further studied in [14,16]. A nice connection, named Bridge Theorem, was found between this entropy and the topological entropy h( f ) of the dual (continuous) endomorphism f of the (compact) Pontryagin dual A, namely h alg ( f ) = h( f ) (see [13]). This result extends earlier previous ones of M. Weiss [50] (for A torsion) and Peters [41] (for A countable and f an automorphism).
Extending the definition of h alg to the case of an action T of an amenable cancellative semigroup G on an abelian group A via endomorphisms, the algebraic entropy h alg (T ) was defined in [12]. Moreover, the Bridge Theorem was extended in this setting as well, in case the group A is torsion. Namely, if T is an action of an amenable cancellative semigroup G on a torsion abelian group A via endomorphisms, then the topological entropy h( T ) of the dual action T of G on the (compact and totally disconnected) Pontryagin dual A coincides with h alg (T ).
Recently, the authors [4] defined the receptive algebraic entropyh alg (T ) of an action T of a semigroup G, provided with a regular system = (N n ) n∈Z + , on an abelian group A by endomorphisms. In case the abelian group A is torsion, a Bridge Theorem between this entropyh alg (T ) and the receptive topological entropyh( T ) of the dual action T was established in [23].
There is an obvious analogy between the Variational Principle, connecting (receptive) topological entropy and (receptive) metric entropy of the same action T , and the Bridge Theorem, connecting the (receptive) topological entropy of an action T and the (receptive) algebraic entropy of the dual action T (for example see [15]). This motivated our choice to formulate Conjecture 6.3 in view of the positive evidence, in this sense, provided by the Bridge Theorem available in all these cases, as well as the results of Sect. 7 that are in the spirit of the Variational Principle.

Receptive Local Metric Entropy
For a continuous selfmap f : X → X of a metric space X and a Borel probability measure μ on X , Bowen [7] introduced a topological entropy h B of subsets inspired by Hausdorff dimension. Then Brin and Katok [8]  Later on, Ma and Wen [35] showed that the lower local metric entropy h loc μ with respect to any Borel probability measure μ on X is always smaller than Bowen topological entropy. This result was taken to a local Variational Principle by Feng and Huang [21], who proved in particular that, in case X is a compact metric space, More recently, Bowen's topological entropy inspired by Hausdorff dimension was extended to continuous actions of amenable groups on compact metric spaces by Zheng and Chen [51], and under suitable hypotheses they proved that this Bowen topological entropy coincides with the classical topological entropy. Moreover, they introduced upper and lower local metric entropy, extended the Brin-Katok Formula and prove a "local Variational Principle" between Bowen topological entropy and lower local metric entropy in the case of actions of countable amenable groups on compact metric spaces.
The definition of upper and lower receptive local metric entropy that we use here is the analogue of the one in [51] adapted to our definition of receptive topological entropy. The pointwise concepts h loc μ (x) and h loc μ (x, ) below are as in [3], where they were introduced in the particular case of actions of finitely generated groups on compact metric spaces with respect to the finite generating set of the group. Definition 7.1 Let T : G × X → X be a continuous action of the semigroup G on a compact metric space X , and let = (N n ) n∈Z + be a regular system in G. Let μ be a Borel probability measure on X . The lower and the upper receptive local metric entropy of T at x ∈ X are defined by  Clearly, h loc μ (x, ) and h loc μ (x) measure the decay of the dynamic ball D n (x, ) with respect to μ. We shall omit to write in h loc μ (T , ) when it is clear from the context.
The argument in the proof of the following folklore fact was kindly proposed to us by Hans Weber in the case of the trivial action. It is used in the above definition to apply the integral to the function h loc μ : X → R ≥0 .
Lemma 7.2 Let T : G × X → X be a continuous action of the semigroup G on a compact metric space X , let = (N n ) n∈Z + be a regular system in G and μ a Borel probability measure on X . The functions h loc μ (−, ): X → R ≥0 for > 0 and h loc μ : X → R ≥0 are measurable.
Proof We prove that, fixed n ∈ Z + and > 0, the function f n : X → [0, 1], x → μ(D n (x, )) is measurable. So also the function x → 1 n log μ(D n (x, )) is measurable. Then h loc μ (−, ) and h loc μ result to be measurable as limits of measurable functions.
Let f = f n . First we verify that f is lower semicontinuous. To this end, fix x 0 ∈ X and a ∈ [0, 1] with f (x 0 ) > a. Since Let¯ = − . For every g ∈ N n , g : X → X is (uniformly) continuous, so there exists δ g > 0 such that x ∈ B(x 0 , δ g ) implies gx ∈ B(gx 0 ,¯ ). Let δ = min g∈N n δ g and U = B(x 0 , δ). If x ∈ U , then (x 0 , )) > a. This shows that f is lower semicontinuous.
The fact that f is lower semicontinuous implies that f −1 ((γ , 1]) is open in X for every γ ∈ (0, 1), and this last property entails that f is measurable.
For reader's convenience, we give an auxiliary result based on the classical theorems of Lusin and Egorov adapted to our more specific situation. Lemma 7.3 Let X be a compact metric space and μ a Borel probability measure on X.

(a) A real valued function f on X is measurable if and only if for every > 0 there exists a compact subset K of X with μ(X \K ) < and such that f K is continuous. (b) If a sequence of measurable real valued functions f n on X converges to a real
valued measurable function f on X at each x ∈ X , then for each > 0 there exists a compact subset K of X such that μ(X \K ) < and f n converges to f uniformly on K .
We see that the lower receptive local metric entropy is always smaller than the receptive metric entropy. This can be seen as one half of the receptive version of the Brin-Katok Formula.
Clearly, g(x) = lim m→∞ g m (x) for all x ∈ X . Since each g m is measurable by Lemma 7.2, by iterated applications of Lemma 7.3, one can find a compact subset K of X such that all functions g m and g are continuous on K , g m converges uniformly to g on K and μ(X \K ) < . Consequently, X g(y) dμ ≤ K g(y) dμ + δ.
Take m 0 ≥ 1 so large that g(y) − g m (y) < δ for all y ∈ K and all m ≥ m 0 . Then, for all m ≥ m 0 , Next, fix an arbitrary integer m ≥ m 0 . Let A = {A 1 , A 2 , . . . , A k } be a μmeasurable partition of X such that diam(A i ) < 1/m for all i ∈ {1, 2, . . . , k} (this is straightforward, otherwise see for example [49,Lemma 8.5] for an even stronger statement). For any integer N ≥ 1 set This defines an increasing sequence (K N ) N ≥1 of subsets of K with N ≥1 K N = K . So there exists an integer N ≥ 1 such that Let n ≥ N . Then If Y ∈ A n is such that Y ∩K N = ∅, then for any y ∈ Y ∩K N we have Y ⊆ D n (y, 1/m), so μ(Y ) ≤ μ(D n (y, 1/m). Combining the above inequalities we obtain Letting n → ∞ we obtain that h loc μ (T ) ≤h μ (T , A) + 4δ, therefore h loc μ (T ) ≤ h μ (T ) + 4δ. This is true for any δ > 0, so h loc μ (T ) ≤h μ (T ).
Let M > 0 be an arbitrary (large) number. Using Lemmas 7.2 and 7.3 as above, there exists a compact subset K of X such that all functions g m are continuous on K , g m converges uniformly to g on K , and Fix an integer m 0 ≥ 1 so large that g m (x) ≥ g(x) − 1 for all x ∈ K and all m ≥ m 0 . Define the sets K N as above with δ = 1. Then again (K N ) N ≥1 is an increasing sequence of subsets of K with N ≥1 K N = K , so for a sufficiently large integer N ≥ 1, Now for n ≥ N and m ≥ m 0 , using estimates similar to those above, we get log μ(D n (y, 1/m)) dμ(y) + 3.
The following question arises in a natural way in view of the Brin-Katok Formula and Theorem 7.4. Question 7.5 Let G be a semigroup, = (N n ) n∈Z + a regular system in G, T : G × X → X a continuous action of G on a compact metric space X and μ ∈ M(X , T ). Does h loc μ (T ) =h μ (T ) hold under suitable conditions?
Let T : G × X → X be a continuous action of the semigroup G on a compact metric space X , let = (N n ) n∈Z + be a regular system in G and μ a T -invariant Borel probability measure on X . We denote by S μ (T , ) the essential supremum of h loc μ , that is, When there is no possibility of confusion we omit and write simply S μ (T ).

Remark 7.6
Let T : G × X → X be a continuous action of the semigroup G on a compact metric space X , and let = (N n ) n∈Z + be a regular system in G. Let μ be a Borel probability measure on X . Put s(Y ) := inf{h loc μ (x) : x ∈ Y } for Y ⊆ X . Since S μ (T ) > s(Y ) for every measurable subset Y of X with μ(Y ) > 0 and S μ (T ) is the smallest number with this property, an alternative description of S μ (T ) can be S μ (T ) = sup{s(Y ) : Y ⊆ X measurable, μ(Y ) > 0}.
We conjecture that a local Variational Principle in the spirit of the one mentioned above by Zheng and Chen holds: Conjecture 7.7 If G is a semigroup, = (N n ) n∈Z + a regular system in G, T : G × X → X a continuous action on a compact metric space X , theñ h(T , ) = sup{h loc μ (T , ): μ Borel probability measure on X }. (7.3) As mentioned above, the validity of the conjecture for Z + -actions was established by Feng and Huang [21]. The following result can be seen as one half of the local Variational Principle (7.3), namely, h loc μ (T ) ≤h(T ) for every Borel probability measure μ. Therefore, in case Question 7.5 has a positive answer, this would provide also one half of the Variational Principle (6.2), namely,h μ (T ) ≤h(T ) for every μ ∈ M(X , T ).

Theorem 7.8 Let G be a semigroup,
= (N n ) n∈Z + a regular system in G and T : G × X → X a continuous action of G on a compact metric space X . Then, for a Borel probability measure μ on X , S μ (T ) ≤ c(T ), and therefore S μ (T ) ≤h(T ). In particular, h loc μ (T ) ≤h(T ). Set F = K m 0 ,N 0 and assume for a contradiction that c(T , F) < S − δ. Fix an arbitrary λ ∈ R with c(T , F) < λ < S − δ. Then m λ (F) = 0 by the definition of c(T , F). This implies that there exist ∈ (0, 1/m 0 ), an integer N ≥ N 0 and a finite or countable cover {D n i (x i , ): i ∈ I } of F with x i ∈ F and n i ≥ N 0 for all i ∈ I , such that i∈I e −λn i < μ(F) 2 . (7.4) Then by the choice of F, for all i ∈ I we have − log μ(D n i (x i , )) > n i (S − δ), so μ(D n i (x i , )) < e −n i (S−δ) . Since F ⊆ i∈I D n i (x i , ), it now follows that i∈I e −λn i > i∈I e −(S−δ)n i > i∈I μ(D n i (x i , )) ≥ μ(F), and this is a contradiction with (7.4).

Proof
In this way we have proved that c(T , F) ≥ S − δ. Letting δ → 0 we obtain that c(T , F) ≥ S which in turn implies c(T ) ≥ S. By Theorem 5.8 and Remark 5.6, we conclude also that S ≤ c(T ) = b(T ) ≤h(T ).
The last assertion follows from the inequality h loc μ (T ) ≤ S μ (T ) since by definition h loc μ (T ) and S μ (T ) are respectively the integral and the essential supremum of the function h loc μ : X → R ≥0 . Remark 7.9 Theorem 7.8 is closely related to [3,Corollary 5.4], where G is a finitely generated group acting on a compact metric space X and the regular system is standard. In the notation of Theorem 7.8 and Remark 7.6, it is proved in [3,Corollary 5.4] that if E is a Borel subset of X with μ(E) > 0, then s(E) ≤h(T , E). Therefore, in view of Remark 7.6, S μ (T ) ≤h(T ).
The following question arises as very natural in view of Theorem 7.8.

Question 7.10
Let G be a semigroup, = (N n ) n∈Z + a regular system in G and T : G × X → X a continuous action of G on a compact metric space X . Does c(T ) = sup{S μ (T ) : μ Borel probability measure on X } hold? Doesh(T ) = sup{S μ (T ) : μ Borel probability measure on X } hold at least when G is commutative and finitely generated and is standard?
In view of the Variational Principle and of Conjecture 7.7 it makes sense to ask whether also the following version of the local Variational Principle holds.

Question 7.11
If G is a semigroup, = (N n ) n∈Z + a regular system in G, T : G× X → X a continuous action on a compact metric space X admitting some T -invariant Borel probability measure, does h(T , ) = sup{h loc μ (T , ): μ ∈ M(X , T )}? (7.5) In view of Theorem 7.4, this version (7.5) of the local Variational Principle would imply the "hard half" (i.e., the inequality ≤) of the Variational Principle (6.2), as (7.5) and Theorem 7.4 would giveh(T ) ≤ sup{h μ (T ) : μ ∈ M(X , T )}. So only the inequality ≥ in (6.2) (sometimes referred to as "easy half") of the Variational Principle would be missing, now stated as (6.3) in Conjecture 6.5.
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