The Entropy of Co-Compact Open Covers

Zheng Wei <sup>1</sup>, Yangeng Wang <sup>2</sup>, Guo Wei <sup>3</sup>, \*, Tonghui Wang <sup>1</sup> and Steven Bourquin <sup>3</sup>


*Received: 3 April 2013; in revised form: 8 June 2013 / Accepted: 18 June 2013 / Published: 24 June 2013*

Abstract: Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space (compactness and metrizability are not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: (1) it does not require the space to be compact and, thus, generalizes Adler, Konheim and McAndrew's topological entropy of continuous mappings on compact dynamical systems; and (2) it is an invariant of topological conjugation, compared to Bowen's entropy, which is metric-dependent. Other properties of co-compact entropy are investigated, e.g., the co-compact entropy of a subsystem does not exceed that of the whole system. For the linear system, (R, f), defined by f(x)=2x, the co-compact entropy is zero, while Bowen's entropy for this system is at least log 2. More generally, it is found that co-compact entropy is a lower bound of Bowen's entropies, and the proof of this result also generates the Lebesgue Covering Theorem to co-compact open covers of non-compact metric spaces.

Keywords: topological dynamical system; perfect mapping; co-compact open cover; topological entropy; topological conjugation; Lebesgue number

Classification: MSC 54H20; 37B40

#### 1. Introduction

#### *1.1. Measure-Theoretic Entropy*

The concept of entropy per unit time was introduced by Shannon [1], by analogy with the standard Boltzmann entropy measuring a spatial disorder in a thermodynamic system. In the 1950s, Kolmogorov [2] and Sinai established a rigorous definition of K-S entropy per unit time for dynamical systems and other random processes [3]. Kolmogorov imported Shannon's probabilistic notion of entropy into the theory of dynamical systems, and the idea was vindicated later by Ornstein, who showed that metric entropy suffices to completely classify two-sided Bernoulli processes [4], a basic problem, which for many decades, appeared completely intractable. Kolmogorov's metric entropy is an invariant of measure theoretical dynamical systems and is closely related to Shannon's source entropy. The K-S entropy is a powerful concept, because it controls the top of the hierarchy of ergodic properties: K-S property ⇒ multiple mixing ⇒ mixing ⇒ weak mixing ⇒ ergodicity [3]. The K-S property holds if there exists a subalgebra of measurable sets in phase space, which generates the whole algebra by application of the flow [3]. The dynamical randomness of a deterministic system finds its origin in the dynamical instability and the sensitivity to initial conditions. In fact, the K-S entropy is related to the Lyapunov exponents, according to a generalization of Pesin's theorem [5,6]. A deterministic system with a finite number of degrees of freedom is chaotic if its K-S entropy per unit time is positive. More properties about K-S entropy can be found in papers [3,5,7]. The concept of space-time entropy or entropy per unit time and unit volume was later introduced by Sinai and Chernov [8]. A spatially extended system with a probability measure being invariant under space and time translations can be said to be chaotic if its space-time entropy is positive.

#### *1.2. Topological Entropy and Its Relation to Measure-Theoretic Entropy*

In 1965, Adler, Konheim and McAndrew introduced the concept of topological entropy for continuous mappings defined on compact spaces [9], which is an analogous invariant under conjugacy of topological dynamical systems and can be obtained by maximizing the metric entropy over a suitable class of measures defined on a dynamical system, implying that topological entropy and measure-theoretic entropy are closely related. Goodwyn in 1969 and 1971, motivated by a conjecture of Adler, Konheim and McAndrew [9], compared topological entropy and measure-theoretic entropy and concluded that topological entropy bounds measure-theoretic entropy [10,11]. In 1971, Bowen generalized the concept of topological entropy to continuous mappings defined on metric spaces and proved that the new definition coincides with that of Adler, Konheim and McAndrew's within the class of compact spaces [12]. However, the entropy according to Bowen's definition is metric-dependent [13] and can be positive even for a linear function (Example 5.1 or Walters' book, pp.176). In 1973, along with a study of measure-theoretic entropy, Bowen [12] gave another definition of topological entropy resembling Hausdorff dimension, which also equals to the topological entropy defined by Adler, Konheim and McAndrew when the space is compact. Recently, Cánovas and Rodríguez, and Malziri and Molaci proposed other definitions of topological entropy for continuous mappings defined on non-compact metric spaces [14,15].

#### *1.3. The Importance of Entropy*

The concepts of entropy are useful for studying topological and measure-theoretic structures of dynamical systems. For instance, two conjugated systems have the same entropy, and thus, entropy is a numerical invariant of the class of conjugated dynamical systems. Upper bounds on the topological entropy of expansive dynamical systems are given in terms of the −entropy, which was introduced by Kolmogorov-Tikhomirov [2]. The theory of expansive dynamical systems has been closely related to the theory of topological entropy [16–18]. Entropy and chaos are closely related, e.g., a continuous mapping, f : I → I, is chaotic if and only if it has a positive topological entropy [19]. But this result may fail when the entropy is zero, because of the existence of minimum chaotic (transitive) systems [20,21]. A remarkable result is that a deterministic system together with an invariant probability measure defines a random process. As a consequence, a deterministic system can generate dynamical randomness, which is characterized by an entropy per unit time that measures the disorder of the trajectories along the time axis. Entropy has many applications, e.g., transport properties in escape-rate theory [22–26], where an escape of trajectories is introduced by absorbing conditions at the boundaries of a system. These absorbing boundary conditions select a set of phase-space trajectories, forming a chaotic and fractal repeller, which is related to an equation for K-S entropy. The escape-rate formalism has applications in diffusion [27], reaction-diffusion [28] and, recently, viscosity [29]. Another application is the classification of quantum dynamical systems, which is given by Ohya [30]. Symbolic dynamical systems ( (p), σ) have various representative and complicated dynamical properties and characteristics, with an entropy log p. When determining whether or not a given topological dynamical system has certain dynamical complexity, it is often compared with a symbolic dynamical system [21,31]. For the topological conjugation with symbolic dynamical systems, we refer to Ornstein [4], Sinai [32], Akashi [33] and Wang and Wei [34,35].

#### *1.4. The Purpose, the Approach and the Outlines*

The main purpose of this article is to introduce a topological entropy for perfect mappings defined on arbitrary Hausdorff spaces (compactness and metrizability are not necessarily required) and investigate fundamental properties of such an entropy.

Instead of using all open covers of the space to define entropy, we consider the open covers consisting of the co-compact open sets (open sets whose complements are compact).

Various definitions of entropy and historical notes are mentioned previously in this section. Section 2 investigates the topological properties of co-compact open covers of a space. Section 3 introduces the new topological entropy defined through co-compact covers of the space, which is called co-compact entropy in the paper, and further explores the properties of the co-compact entropy and compares it with Adler, Konheim and McAndrew's topological entropy for compact spaces. Sections 4 investigates the relation between the co-compact entropy and Bowen's entropy. More precisely, Section 4 compares the co-compact entropy with that given by Bowen for systems defined on metric spaces. Because the spaces under consideration include non-compact metric spaces, the traditional Lebesgue Covering Theorem does not apply. Thus, we generalize this theorem to co-compact open covers of non-compact metric spaces. Based on the generalized Lebesgue Covering Theorem, we show that the co-compact entropy is a lower bound for Bowen's entropies. In Section 4.2, a linear dynamical system is studied. For this simple system, its co-compact entropy is zero, which is appropriate, but Bowen's entropy is positive.

#### 2. Basic Concepts and Definitions

Let (X, f) be a topological dynamical system, where X is a Hausdorff and f : X → X is a continuous mapping. We introduce the concept of co-compact open covers as follows.

Definition 2.1. Let X be a Hausdorff space. For an open subset, U of X, if X\U is a compact subset of X, then U is called a co-compact open subset. If every element of an open cover U of X is co-compact, then U is called a co-compact open cover of X.

Theorem 2.1. *The intersection of finitely many co-compact open subsets is co-compact, and the union of any collection of co-compact open subsets is co-compact open.*

Proof. Suppose that <sup>U</sup>1, U2, ..., U<sup>n</sup> are co-compact open. Let <sup>U</sup> <sup>=</sup> <sup>6</sup><sup>n</sup> i=1 Ui. As X \ Ui, i = 1, 2, ...n are compact, <sup>X</sup> \ <sup>U</sup> <sup>=</sup> <sup>7</sup><sup>n</sup> (X \ Ui) is compact, and hence, U is co-compact open.

i=1 Suppose that {Uλ}<sup>λ</sup>∈<sup>Λ</sup> is a family of co-compact sets. Let U = 7 λ∈Λ Uλ. As any λ ∈ Λ X\U<sup>λ</sup> is compact, X\U = 6 λ∈Λ (X\Uλ) is compact. Hence, U is co-compact open.

Theorem 2.2. *Let* X *be Hausdorff. Then, any co-compact open cover has a finite subcover.*

Proof. Let U be a co-compact open cover. For any U ∈ U, X\U is compact. Noting that U is also an open cover of X \ U, there exists a finite subcover, V, of X \ U. Now, V∪{U} is a finite subcover of U.

Definition 2.2. Let X and Y be Hausdorff spaces and let f : X → Y be a continuous mapping. If f is a closed mapping and all fibers, <sup>f</sup> <sup>−</sup><sup>1</sup>(x), x <sup>∈</sup> <sup>Y</sup> , are compact, then <sup>f</sup> is called a perfect mapping.

In particular, if X is compact Hausdorff and Y is Hausdorff, every continuous mapping from <sup>X</sup> into <sup>Y</sup> is perfect. If <sup>f</sup> : <sup>X</sup> <sup>→</sup> <sup>Y</sup> is perfect, then <sup>f</sup> <sup>−</sup><sup>1</sup>(F) is compact for each compact subset, F ⊆ Y [36].

Theorem 2.3. *Let* X *and* Y *be two Hausdorff spaces and let* f : X → Y *be a perfect mapping. If* <sup>U</sup> *is co-compact open in* <sup>Y</sup> *, then* <sup>f</sup> <sup>−</sup><sup>1</sup>(U) *is co-compact open in* <sup>X</sup>*. Moreover, if* <sup>U</sup> *is a co-compact open cover of* <sup>Y</sup> *, then* <sup>f</sup> <sup>−</sup><sup>1</sup>(U) *is a co-compact Open Cover of* <sup>X</sup>*.*

Proof. It suffices to show that the pre-image of any co-compact set is co-compact. Let U be co-compact open in <sup>Y</sup> . Then, <sup>F</sup> <sup>=</sup> <sup>Y</sup> \ <sup>U</sup> is compact in <sup>Y</sup> . As <sup>f</sup> is perfect, <sup>f</sup> <sup>−</sup><sup>1</sup>(F) is compact in <sup>X</sup>. Hence, <sup>f</sup> <sup>−</sup><sup>1</sup>(U) = <sup>X</sup> \ <sup>f</sup> <sup>−</sup><sup>1</sup>(F) is co-compact open in <sup>X</sup>.

#### 3. The Entropy of Co-Compact Open Covers

For compact topological systems, Adler, Konheim and McAndrew introduced the concept of topological entropy and studied its properties [9]. Their definition is as follows: Let X be a compact topological space and f : X → X a continuous mapping. For any open cover, U of X, let NX(U) denote the smallest cardinality of all subcovers of U, *i.e*.,

$$N\_X(\mathcal{U}) = \min\{card(\mathcal{V}) : \mathcal{V} \text{ is a subcover of } \mathcal{U}\}$$

It is obvious that NX(U) is a positive integer. Let HX(U) = log NX(U). Then, ent(f, U, X) = lim<sup>n</sup>→∞ 1 <sup>n</sup>HX( n8−1 i=0 f <sup>−</sup><sup>i</sup> (U)) is called the topological entropy of f relative to U, and ent(f,X) = sup {ent(f, U, X)} is called the topological entropy of f.

U Now, we will generalize Adler, Konheim and McAndrew's entropy to any Hausdorff space for perfect mappings. Therefore, in the remainder of the paper, a space is assumed to be Hausdorff and a mapping is assumed to be perfect.

Let X be Hausdorff. By Theorem 2.2, when U is a co-compact open cover of X, U has a finite subcover. Hence, NX(U), abbreviated as N(U), is a positive integer. Let HX(U) = log N(U), abbreviated as H(U).

Let U and V be two open covers of X. Define

$$\mathcal{U}\bigvee\mathcal{V} = \{ U \cap V : U \in \mathcal{U} \text{ and } V \in \mathcal{V} \}.$$

If for any U ∈ U, there exists V ∈ V, such that U ⊆ V , then U is said to be a refinement of V and is denoted by V≺U.

The following are some obvious facts:

Fact 1: For any open covers, U and V, of X, U≺U 8 V.


Fact 5: For any co-compact open covers, U and V, H(U 8 V) ≤ H(U) + H(V).

To prove Fact 5, let U<sup>0</sup> be a finite subcover of U, with the cardinality, N(U). Let V<sup>0</sup> be a finite subcover of V with the cardinality, H(V). Then, U<sup>0</sup> 8 V<sup>0</sup> is a subcover of U 8 V, and the cardinality of U<sup>0</sup> 8 V<sup>0</sup> is at most N(U)×N(V). Hence, N(U 8 V) ≤ N(U)×N(V), and therefore, H(U 8 V) ≤ H(U) + H(V).

Fact 6: For any co-compact open cover, <sup>U</sup>, of <sup>X</sup>, <sup>H</sup>(<sup>f</sup> <sup>−</sup><sup>1</sup>(U)) <sup>≤</sup> <sup>H</sup>(U), and if <sup>f</sup>(X) = <sup>X</sup>, the equality holds.

To prove Fact 6, let <sup>U</sup><sup>0</sup> be a finite subcover of <sup>U</sup>, with the cardinality, <sup>N</sup>(U). <sup>f</sup> <sup>−</sup><sup>1</sup>(U0) is a subcover of <sup>f</sup> <sup>−</sup><sup>1</sup>(U). Hence, we have <sup>H</sup>(<sup>f</sup> <sup>−</sup><sup>1</sup>(U)) <sup>≤</sup> <sup>H</sup>(U).

Now, assume <sup>f</sup>(X) = <sup>X</sup>. Let {<sup>f</sup> <sup>−</sup><sup>1</sup>(U1), f <sup>−</sup><sup>1</sup>(U2), ..., f <sup>−</sup><sup>1</sup>(Un)}, <sup>U</sup><sup>i</sup> ∈ U be a finite subcover of <sup>f</sup> <sup>−</sup><sup>1</sup>(U), with the cardinality, <sup>N</sup>(<sup>f</sup> <sup>−</sup><sup>1</sup>(U)). As <sup>X</sup> <sup>⊆</sup> <sup>7</sup><sup>n</sup> i=1 <sup>f</sup> <sup>−</sup><sup>1</sup>(Ui), we have <sup>X</sup> <sup>=</sup> <sup>f</sup>(X) <sup>⊆</sup> 7n i=1 <sup>f</sup>(<sup>f</sup> <sup>−</sup><sup>1</sup>(Ui)) = <sup>7</sup><sup>n</sup> i=1 Ui. Hence, U1, U2, ..., U<sup>n</sup> is a finite subcover of U. This shows H(U) ≤ <sup>H</sup>(<sup>f</sup> <sup>−</sup><sup>1</sup>(U)). This inequality and the previous inequality together imply the required equality.

Lemma 3.1. *Let* {an}<sup>∞</sup> <sup>n</sup>=1 *be a sequence of non-negative real numbers satisfying* an+<sup>p</sup> ≤ an+ap, n ≥ <sup>1</sup>, p <sup>≥</sup> <sup>1</sup>*. Then,* lim<sup>n</sup>→∞ a<sup>n</sup> <sup>n</sup> *exists and is equal to* inf <sup>a</sup><sup>n</sup> <sup>n</sup> *(see [13]).*

Let U be a co-compact open cover of X. By Theorem 2.3, for any positive integer, n, and perfect mapping, <sup>f</sup> : <sup>X</sup> <sup>→</sup> <sup>X</sup>, <sup>f</sup> <sup>−</sup><sup>n</sup>(U) is a co-compact open cover of <sup>X</sup>. On the other hand, by Theorem 2.1, <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (U) is a co-compact open cover of X. These two facts together lead to the following result:

Theorem 3.1. *Suppose that* X *is Hausdorff. Let* U *be a co-compact open cover of* X*, and* f : X → <sup>X</sup>*, a perfect mapping. Then,* lim<sup>n</sup>→∞ 1 <sup>n</sup>H( n8−1 i=0 f <sup>−</sup><sup>i</sup> (U)) *exists.*

Proof. Let a<sup>n</sup> = H( n8−1 i=0 f <sup>−</sup><sup>i</sup> (U)). By Lemma 3.1, it suffices to show an+<sup>k</sup> ≤ a<sup>n</sup> + ak. Now, Fact 6 gives <sup>H</sup>(<sup>f</sup> <sup>−</sup><sup>1</sup>(U)) <sup>≤</sup> <sup>H</sup>(U), and more generally, <sup>H</sup>(<sup>f</sup> <sup>−</sup><sup>j</sup> (U)) <sup>≤</sup> <sup>H</sup>(U), j = 0, <sup>1</sup>, <sup>2</sup>, .... Hence, by applying Fact 5, we have an+<sup>k</sup> = H( n+ 8 k−1 i=0 f <sup>−</sup><sup>i</sup> (U)) = H(( n8−1 i=0 f <sup>−</sup><sup>i</sup> (U)) 8( n+ 8 k−1 j=n <sup>f</sup> <sup>−</sup><sup>j</sup> (U))) = H( n8−1 i=0 f <sup>−</sup><sup>i</sup> (U) 8( k 8−1 j=0 <sup>f</sup> <sup>−</sup><sup>n</sup>(<sup>f</sup> <sup>−</sup><sup>j</sup> (U)))) <sup>≤</sup> <sup>H</sup>( n8−1 i=0 f <sup>−</sup><sup>i</sup> (U)) + H( k 8−1 j=0 <sup>f</sup> <sup>−</sup><sup>j</sup> (U)) = <sup>a</sup><sup>n</sup> <sup>+</sup> <sup>a</sup>k.

Next, we introduce the concept of entropy for co-compact open covers.

Definition 3.1. Let X be a Hausdorff space, f : X → X be a perfect mapping, and U be a co-compact open cover of <sup>X</sup>. The non-negative number, <sup>h</sup>c(f, <sup>U</sup>) = lim<sup>n</sup>→∞ 1 <sup>n</sup>H( n8−1 i=0 f <sup>−</sup><sup>i</sup> (U)), is said to be the co-compact entropy of f relative to U, and the non-negative number, hc(f) = sup U {hc(f, U)}, is said to be the co-compact entropy of f.

In particular, when X is compact Hausdorff, any open set of X is co-compact, and any continuous mapping f : X → X is perfect. Hence, Adler, Konheim and McAndrew's topological entropy is a special case of our co-compact entropy. It should be made aware that the new entropy is well defined for perfect mappings on non-compact spaces, e.g., on R<sup>n</sup>, but Adler, Konheim and McAndrew's topological entropy requires that the space be compact.

Co-compact entropy generalizes Adler, Konheim and McAndrew's topological entropy, and yet, it holds various similar properties, as well, as demonstrated by the fact that co-compact entropy is an invariant of topological conjugation (next theorem) and more explored in the next section.

Recall that ent denotes Adler, Konheim and McAndrew's topological entropy, and h<sup>c</sup> denotes the co-compact entropy.

Theorem 3.2. *Let* (X, f) *and* (Y,g) *be two topological dynamical systems, where* X *and* Y *are Hausdorff,* f : X → X *and* g : Y → Y *are perfect mappings. If there exists a semi-topological conjugation,* h : X → Y *, where* h *is also perfect, then* hc(f) ≥ hc(g)*. Consequently, when* h *is a topological conjugation, we have* hc(f) = hc(g)*.*

Proof. Let U be any co-compact open cover of Y . As h is perfect and U is a co-compact open cover of <sup>Y</sup> , <sup>h</sup>−<sup>1</sup>(U) is co-compact open cover of <sup>X</sup> by applying Theorem 2.3. Hence, we have:

$$\begin{aligned} h\_c(g, \mathcal{U}) &= \lim\_{n \to \infty} \frac{1}{n} H(\bigvee\_{i=0}^{n-1} g^{-i}(\mathcal{U})) = \lim\_{n \to \infty} \frac{1}{n} H(h^{-1}(\bigvee\_{i=0}^{n-1} g^{-i}(\mathcal{U}))) \\ &= \lim\_{n \to \infty} \frac{1}{n} H(\bigvee\_{i=0}^{n-1} h^{-1}(g^{-i}(\mathcal{U}))) = \lim\_{n \to \infty} \frac{1}{n} H(\bigvee\_{i=0}^{n-1} f^{-i}(h^{-1}(\mathcal{U}))) \\ &= h\_c(f, h^{-1}(\mathcal{U})) \le h\_c(f, \mathcal{U}) \end{aligned}$$

Therefore, hc(f) ≥ hc(g).

When h is a topological conjugation, it is, of course, perfect, too. Hence, we have both hc(f) ≥ hc(g) and hc(g) ≥ hc(f) from the above proof, implying hc(f) = hc(g).

Remark: The condition that the conjugation map is perfect is crucial in this result. In the general case, the inequality given by conjugacy need not hold. Cánovas and Rodríguez [14] defined an entropy for non-compact spaces that has this property (Theorem 2.1 (a)), which can be applied for non-perfect maps. Notice that Cánovas and Rodríguez's definition does not depend on the metric that generates the given topology of X. This is due to the fact that for compact metric spaces, the definition of Bowen's entropy does not depend on the metric. Since Cánovas and Rodríguez's definition is based on invariant compact sets, and they are the same for equivalent metrics, that is, metrics that generate the same topology of X, Cánovas and Rodríguez's definition does not depend on the metric when topology is fixed [37].

We sum up some properties of the new definition of topological entropy in the following results. A minor adaptation of the proof of standard techniques on topological entropy (e.g., [13]) gives the proof of these results. These properties are comparable to that of Adler, Konheim and McAndrew's topological entropy.

Theorem 3.3. *Let* X *be Hausdorff and* id : X → X *be the identity mapping. Then* hc(id)=0*.*

When <sup>X</sup> is Hausdorff and <sup>f</sup> : <sup>X</sup> <sup>→</sup> <sup>X</sup> is perfect, <sup>f</sup> <sup>m</sup> : <sup>X</sup> <sup>→</sup> <sup>X</sup> is also a perfect mapping [36].

Theorem 3.4. *. Let* <sup>X</sup> *be Hausdorff and* <sup>f</sup> : <sup>X</sup> <sup>→</sup> <sup>X</sup> *be perfect. Then,* <sup>h</sup>c(<sup>f</sup> <sup>m</sup>) = <sup>m</sup> · <sup>h</sup>c(f)*.*

Theorem 3.5. *Let* X *be Hausdorff and* f : X → X *be perfect. If* Λ *is a closed subset of* X *and invariant under* f*, i.e.,* f(Λ) ⊆ Λ*, then* hc(f|Λ) ≤ hc(f)*.*

#### 4. Relations between Co-Compact Entropy and Bowen's Entropy
