Entropy on noncompact sets

Abstract In this paper we review and explore the notion of topological entropy for continuous maps defined on non compact topological spaces which need not be metrizable. We survey the different notions, analyze their relationship and study their properties. Some questions remain open along the paper.


Introduction
Topological entropy [1] is one of the most accepted tools to characterize complicated or chaotic behavior in discrete dynamical systems. It is commonly accepted that positive topological entropy implies the existence of dynamics that we identify as unpredictable. From the theoretical point of view is very useful, e.g. it is invariant by conjugacy. On the other hand, although it is quite di cult to compute in an e ective way, it gives you a very valuable information when you are able to compute it when, for instance, one is analyzing models depending on one or several parameters because it gives you an analytical proof on then existence of chaotic behavior, or more precisely, for what parameter values it is possible to nd chaotic motions. However, the above paragraph holds when the phase space is compact, and usually metric. In this paper we are interested in analyzing the notion of topological entropy when the topological space need not be neither compact nor metrizable. As we will point out, there exist several possible entropy notions which are not equivalent, some well-known properties in the compact case are not true when we remove the compactness, and extremely dynamically simple maps may exhibit positive entropy.
The aim of this paper is to survey what notions of topological entropy can be found in the literature, trying to show their advantages and disadvantages, and exploring the link among them. As we are going to work with several de nitions of topological entropy, to distinguish them, we denote it with the classical letter h plus a subscript with the initials of the authors who introduced or inspired the notion. For instance, we denote by h AKM (f ) the Adler, Konheim and McAndrew's de nition of topological entropy for maps f : X → X de ned on a compact space X [1]. In case some author introduces more than one de nition, we may add some characteristic to distinguish them. For instance, Hofer's de nition on compacti cations will be denoted by h * H (f ) [10]. Now, let f : X → X be a continuous map on a non necessarily compact space. Fixed x ∈ X, we denote by Orb f (x) the forward orbit of x under f , that is, the sequence f n (x) for n ∈ N. The set of accumulation points of Orb f (x) is the ω-limit set ω f (x). This ω-limit set is closed and strictly invariant by f , and then we can de ne the following subsets of X by K f = {x ∈ X : ω f (x) is compact and non empty}, is non empty and non compact}, Since ω f (x) is strictly invariant by f , we have that K f and C f are invariant by f . It is easy to see that E f is invariant as well. The above sets allow us to consider the following decomposition of X by X = K f ∪ C f ∪ E f . As we will show, the several de nitions that we will consider can be invisible for them. Although K f need not be compact, the restriction of the map to K f gives us the dynamics of f which is intrinsically compact, while f | C f and f | E f is related to the dynamics of f that are essentially non compact. Now, let us start by introducing the notions we are going to deal with. We start by recalling the wellknown notion of entropy for compact spaces.

The compact case
Before introducing the non compact case, we recall brie y the well-known Adler, Konheim and McAndrew's de nition of topological entropy for a continuous map f : X → X on a compact topological Hausdor space X is stated from its open covers [1]. Namely, if A = {A i } is an open cover of X, and N(A) is the minimum number of elements from A to cover X, then the topological entropy of f relative to A is given by the non negative number f is the identity on X. The limit exists due to the subadditivity properties of n log N ∨ n− i= f −i (A) given by the fact that The topological entropy of f is then It is a simple observation that, due to the compactness of X, the open cover A can be taken nite. The basic properties of h AKM (f ) are introduced below. Let f : X → X and g : Y → Y be continuous maps on the topological spaces X and Y, respectively and let φ : X → Y be continuous and such that φ • f = g • φ. The product map f × g : X × Y → X × Y is given by (f × g)(x, y) = (f (x), g(y)) for all (x, y) ∈ X × Y. The basic properties of topological entropy on compact topological spaces are the following: (TE1)If φ is surjective, then h AKM (f ) ≥ h AKM (g). If φ is injective, then h AKM (f ) ≤ h AKM (g). If φ the map φ is an homeomorphism, then h AKM (f ) = h AKM (g) (see [1] or [2,Chapter 4]).
.., k} (see [1] or [2,Chapter 4]). [8]. In addition, compact subsets Y of X with the property that h AKM (f | Y ) = h AKM (f ) have a special interest. In particular, the nonwandering set Ω(f ) given by the points x ∈ X for which any open neighborhood U, with x ∈ U, satis es that f n (U) ∩ U ≠ ∅ for some n ∈ N, is the smallest dynamical subset such that h AKM (f | Ω(f ) ) = h AKM (f ) [3] or [2,Chapter 4].
In this paper, we will try to study in detail when all the above properties (TE1)-(TE4) are satis ed in the non compact case, but is some cases we are unable to establish that the property is ful lled. We will point out when some property is not ful lled and make comments on the validity of the above properties when it is known.

Hofer's approaches . Hofer's de nition #1
If the space X is not compact, then its open covers need not have nite re nements. Hence, in [10], the topological entropy is introduced in an obvious way; just consider nite open covers of X, and then It is also mentioned in [10] there that (TE1) and (TE2) are true for h H (f ). Clearly, for the rst part of (TE2) the proof is analogous to that of [1], based in the equality which gives us the second inequality This property was proved in [5], jointly with the formula h H (f ) = h H (f − ) for homeomorphisms. For property (TE1), again in [5] was proved that if φ is injective and implying the equality if f and g are conjugate. It is unclear if property (TE3) holds in general. In [5] it is proved that if Y is closed in X, then and that if X = ∪ k i= Y i with all the sets Y i closed in X then we get the equality Finally, it is unclear whether the property (TE4) remains true for Hofer's de nition #1. It is clear that if A is a nite open cover of X and B is a nite open cover of Y, then the inequality and hence we have that but it is unclear whether this inequality can be replaced by an equality.

. Hofer's de nition #2
Also in [10] is given a second de nition of topological entropy for non compact spaces. We assume that X is a completely regular space (Tychono ) and given the continuous map f , we consider f * : X * → X * , the unique continuous extension of f to the Stone-Čech compacti cation X * of X. Then Both de nitions are equivalent when the space X is normal (see [10]). In addition, in [6], it was proved that the above result is false in general when X is not normal, disproving a conjecture from [10]. Additionally, in [6], it is proved that for completely regular spaces: continuous. In addition the inequality h * H (f ) ≤ h H (f ) was proved when the space is completely regular. Concerning the related properties (TE1)-(TE4), the injective case in (TE1), as well as the inequality [6]). It is proved in [5] that the surjectivity in (TE1) holds when φ(X) is dense in Y and when φ is an homeomorphism then we get the equality h * H (f ) = h * H (g). Property (TE2) follows since ∨ n− i= A is a nite cozero cover of X whenever A has this property and it is also proved in [5]. It is unclear whether property (TE4) holds, although it is clear that the Anyway, from now on we are mainly interested in normal spaces where Hofer's de nitions are equivalent.

De nition based on extensions . In mum of extensions
In what follows we will assume that all the spaces are normal. The Hofer's approach makes use of compactications of a normal topological space and the extensions of continuous maps. However, extending the map implies to embed the space into a compact space, which roughly speaking implies to add points to the dynamics of f , of course changing the topology. The following result from [9] shows that Hofer topological entropy is not a good tool to analyze a class of maps with essentially non compact dynamics (see [9]).
Moreover, the following example from [10] shows that other possibilities should be taken into account in order to de ne topological entropy on noncompact spaces.

Example 2.
We consider the map f : Z → Z given by f (n) = n + for all n ∈ Z holds that h * H (f ) = h H (f ) = ∞. Note that the dynamics of f is quite simple, although E f is the whole space. In addition, clearly f can be compacti ed by just one point and then, the extended map f * has zero topological entropy. This is then another problem in the compacti cation ideas, that is, it may not be unique and therefore, different compacti cations may give di erent values of entropy, as the above example shows. Then, a possible solution to this problem is to consider With this de nition it is easy to see that the topological entropy of the map f (n) = n + de ned above is zero. Additionally However, it is unclear whether properties (TE1)-(TE4) hold. For (TE2) it is clear that (f * ) is an extension of f for a suitable compacti cation of X and hence it is easy to see that h * (f n ) ≤ nh * (f ). For (TE4), note that f * × g * are extensions on suitable compacti cations of f × g and therefore h * (f × g) ≤ h * (f ) × h * (g).

. Friedland's like approach
Friedland's approach was made for totally bounded metric spaces and makes use of embeddings. We adopt a similar approach when X is not metric as follows. Let f : X → X be continuous and assume that X * is a compacti cation of X. Then the product space (X * ) N is also compact. Consider the shift map σ : X N → X N , given by σ(xn) = (x n+ ), which is continuous. Let X f be the subset of X N given by Then σ(X f ) ⊂ X f and Friedland topological entropy [7] is de ned as It should be emphasized that Friedland's approach was for invariant subsets of compact metric spaces, when there is no ambiguity in the selection of the compacti cation. Clearly, we can de ne as above In the compact case, that is, when X is compact, Goodwyn proved in [8] that h AKM (f ) = h AKM (σ| X f ), and therefore Friedland's approach makes sense. It is unclear how this de nition is related with the previous ones as well as the validity of properties (TE1)-(TE4).

Compact subsets de nition . Non invariant case de nition
There is a possibility of de ning topological entropy by following the de nition based on Misiurewicz's conditional topological entropy [12]. Let K ⊂ X be a compact subspace of X. For an open cover A of X, we de ne the number as the smallest cardinality of a subcover of ∨ n− i= f −i (A) such that the union of its elements contains K. Then exists, notice that subadditivity is not guaranteed, and then we can take where K(X) denotes the family of compact subsets of X.
Concerning property (TE1), we can state the following result.

If φ is an homeomorphism, then h M (f ) = h M (g).
Proof. We prove 1. Let K be a compact subset of X.
This idea is taken from [5] and the open cover B is given by sets B which gives us the inequality h M (g, K) ≤ h M (f ), which obviously implies 2. Then, 3 follows directly from 1 and 2. Now, we consider property (TE2), stating the following.

Proposition 4. Let assumptions of (TE2) hold for noncompact sets. Then h M (f n ) = n · h M (f ).
Proof. We take ideas from [2]. Since we obtain that for any compact subset K of X For the converse inequality, we simply realize that which concludes the proof.

Note that it remains open the property h M
Regarding property (TE3) we show the following.

. Invariant compact subsets de nition
We can consider a subfamily within the compact subsets of X as follows. Let K(X, f ) be the family of compact subsets K of X which are invariant by f , that is, f (K) ⊆ K. The restricted map f | K is the properly de ned in a compact space, and therefore we can compute its topological entropy h(f | K ). Then, the topological entropy of f is de ned by h CR (f ) = sup{h AKM (f | K ) : K ∈ K(X, f )}.
If K(X, f ) is empty, then h CR (f ) = . This de nition was introduced by Cánovas and Rodríguez in [4], and obviously h CR (f ) ≤ h M (f ). In addition, since the compact subsets of X are compact subsets of any compactication X * , we immediately have that h CR (f ) ≤ h * (f ), and then for the map f (n) = n + de ned in Example 2, we get that h CR (f ) = h * (f ) = . In addition, it is clear from the de nition of h CR (f ) that if a map f holds that K f = ∅, then h CR (f ) = ,that is, this de nition is blind for the dynamics which is not essentially compact. One of the most interesting problems in this setting is when the inequality h CR (f ) ≤ h * (f ) can be replaced by an equality. Some examples will be introduced along this paper. It ful lls all the properties (TE1)-(TE4) except for the surjectivity case of (TE1) which is satis ed when φ is a proper map, that is, φ − (K) is compact for all K ∈ K(X). If φ is not proper, the following example can be found in [4].
Example 6. Let f : R → R de ned by f (x) = x for all x ∈ R and let g : S → S de ned also by g(x) = x for all x ∈ S . Then g is a factor of f and h CR (f ) = while h CR (g) = h AKM (g) = log (see [1,Chapter 4]).
It is also remarkable that the formula h CR (f ) = max{h CR (f | X i ) : i = , ..., k},where X i are invariant subsets of X whose union is X, is not true in general. Again the following example can be seen in [4].
Example 7. Let f : X → X be a minimal homeomorphism de ned on a compact set with positive topological entropy (see [14]). Let x ∈ X and we de ne X := FullOrb f (x) = {f n (x) : n ∈ Z} and X := X \ X . It is clear that both sets are invariant by f and that they have no compact invariant subsets (in case of that they have compact invariant subsets, the map f is not minimal). Therefore h CR It is clear that following the steps of the proof of Proposition 5 we can prove the equality h CR (f ) = max{h CR (f | X i ) : i = , ..., k} when X i are closed subsets.of X.

Proper maps
Now, we introduce the so called co-compact topological entropy introduced rst by Patrao in [13] (see also [15]) based on open covers with special properties. Namely, an open cover A = {A i : i ∈ I} of a normal topological space X is co-compact if for any A i ∈ A the subset X \ A i is compact. Co-compact covers have nite subcovers. We assume that f : X → X is proper. Then f − (A) is a co-compact cover of X and then Adler, Konheim and McAndrew's de nitions makes sense in this setting, that is, the co-compact topological entropy of f relative to the co-compact cover A is and the co-compact topological entropy of f is then It is easy to see that h P (f ) ≤ h H (f ). The following result establishes the relationship between h P (f ) and h CR (f ).

Final remarks and conclusion
The product map formula (TE4) deserves a comment. It was proved in [8] using inverse limits, close to Friedland's approach included in this paper. It is unclear whether it works for most of the de nitions. Even for Bowen's like de nitions of entropy, in the case of metric spaces, a counterexample disproving it was recently found (see [11]). Although some preliminary results have been stated, it is unclear in general what is the in uence of sets K f , C f and E f in the entropy notions that we have considered. We know that h H (f ) = +∞ when E f ≠ ∅ and that h CR (f ) = when K f = ∅, but we do not know how the emptyness or not of such subsets in uence the remaining entropy notions.
This author knows that this paper is in some sense minimalist, in the sense that if we add metric or uniform structures we can obtain richer results. However, the list of possible de nitions increases a lot. In addition, we x here the basic ideas for possible de nitions of topological entropy which are extending the space to a compact space, taking supremum over compact subsets and xing open covers with special properties. We have studied the basic properties with the only use of open covers following the seminal ideas from [1]. The reader should have noticed that there are many holes or open questions in this review, the most important one is to establish a connection between all the introduced notions or disprove that connections.