*4.1. Co-Compact Entropy Less Than or Equal to Bowen's Entropy,* hc(f) ≤ hd(f)

First let us recall the definition of Bowen's entropy [13,38]. Let (X, d) be a metric space and f : X → X a continuous mapping. A compact subset, E of X, is called a (n, )-separated set with respect to f if for any different x, y ∈ E, there exists an integer, j, with 0 ≤ j<n, such that d(f<sup>j</sup> (x), f<sup>j</sup> (y)) > . A subset, F, of X is called a (n, )-spanning set of a compact set, K, relative to <sup>f</sup> if for any <sup>x</sup> <sup>∈</sup> <sup>K</sup>, there exists <sup>y</sup> <sup>∈</sup> <sup>F</sup>, such that for all <sup>j</sup> satisfying <sup>0</sup> <sup>≤</sup> j<n, <sup>d</sup>(f<sup>j</sup> (x), f<sup>j</sup> (y)) <sup>≤</sup> .

Let K be a compact subset of X. Put

$$r\_n(\epsilon, K, f) = \min\{ card(F) : F \text{ is a } (n, \epsilon)\text{-spanning set for } K \text{ with respect to } f\}$$

$$s\_n(\epsilon, K, f) = \max\{card(F) : F \subseteq K \text{ and } F \text{ is a } (n, \epsilon)\text{-separated set with respect to } f\}$$

$$r(\epsilon, K, f) = \lim\_{n \to \infty} \frac{1}{n} \log r\_n(\epsilon, K, f), \qquad s(\epsilon, K, f) = \lim\_{n \to \infty} \frac{1}{n} \log s\_n(\epsilon, K, f)$$

$$r(K, f) = \lim\_{\epsilon \to 0} r(\epsilon, K, f), \qquad s(K, f) = \lim\_{\epsilon \to 0} s(\epsilon, K, f)$$

Then, sup K r(K, f) = sup K s(K, f), and this non-negative number, denoted by hd(f), is the Bowen entropy of f.

It should be pointed out that Bowen's entropy, hd(f), is metric-dependent, see e.g., [13,39]. For the topology of the metrizable space, X, the selection of different metrics may result in different entropies.

Next, recall the Lebesgue Covering Theorem and Lebesgue Number [36]. Let (X, d) be a metric space and U an open cover of X. diam(U) = sup{d(A) | A ∈ U} is called the diameter of U, where d(A) = sup{d(x, y) | x, y ∈ A}. A real number, δ, is said to be a Lebesgue number of U if every open subset, U, of X, satisfying diam(U) < δ, is completely contained in an element of the cover, U.

The Lebesgue Covering Theorem (see [36]): Every open cover of a compact metric space has a Lebesgue number.

Our next theorem generalizes the Lebesgue Covering Theorem to all co-compact open covers of non-compact metric spaces.

Theorem 4.1. *Let* (X, d) *be a metric space, regardless of compactness. Then, every co-compact open cover of* X *has a Lebesgue number.*

Proof. Let U be any co-compact open cover of X. By Theorem 2.2, U has a finite subcover V = {V1, V2, ..., Vm}. Put Y = (X \ V1) ∪ (X \ V2) ∪ ... ∪ (X \ Vm). Then, Y is compact as Vi's are co-compact.

We will prove that V has a Lebesgue number, so does U. As it is obvious that the theorem holds when Y = ∅, thus in the following proof, we assume Y = ∅.

Assume that <sup>V</sup> does not have a Lebesgue number. Then, for any positive integer, <sup>n</sup>, <sup>1</sup> <sup>n</sup> is not a Lebesgue number of V. Consequently, for each positive integer, n, there exists an open subset, On, of X, satisfying diam(On) < <sup>1</sup> <sup>n</sup>, but O<sup>n</sup> is not completely contained in any element of V, *i.e*., O<sup>n</sup> ∩ (X \ V<sup>j</sup> ) = ∅, j = 1, 2, ..., m. Hence, O<sup>n</sup> ∩ Y = ∅. Take an x<sup>n</sup> ∈ O<sup>n</sup> ∩ Y . By the compactness of Y , the sequence x<sup>n</sup> has a subsequence, x<sup>n</sup><sup>i</sup> , that is convergent to some point, y ∈ Y , *i.e*., lim <sup>i</sup>→∞ <sup>x</sup><sup>n</sup><sup>i</sup> <sup>=</sup> <sup>y</sup> <sup>∈</sup> <sup>Y</sup> <sup>⊆</sup> <sup>X</sup>.

On the other hand, V is an open cover of X, thus there exists some V ∈ V, such that y ∈ V . As V is open, there exists an open neighborhood, S(y, ), of y, such that y ∈ S(y, ) ⊆ V . Since <sup>x</sup><sup>n</sup><sup>i</sup> converges to <sup>y</sup>, there exists a positive integer, <sup>M</sup>, such that <sup>x</sup><sup>n</sup><sup>i</sup> <sup>∈</sup> <sup>S</sup>(y,  <sup>2</sup> ) for i>M. Let k be any integer larger than M + <sup>2</sup>  . Then, for any z ∈ O<sup>n</sup><sup>k</sup> , we have d(z, y) ≤ d(z, x<sup>n</sup><sup>k</sup> ) + d(x<sup>n</sup><sup>k</sup> , y) < <sup>2</sup> +  <sup>2</sup> = , thus O<sup>n</sup><sup>k</sup> ⊆ S(y, ) ⊆ V ∈ V, which contradicts the selection of open sets, On's.

Therefore, V has a Lebesgue number.

Theorem 4.2. *Let* (X, d) *be a metric space,* U *be any co-compact open cover of* X*, and* f : X → X *be a perfect mapping. Then, there exists* δ > 0 *and a compact subset* K *of* X*, such that for all positive integers,* n*,*

$$N(\bigvee\_{i=0}^{n-1} f^{-i}(\mathcal{U})) \le n \cdot r\_n(\frac{\delta}{3}, K, f) + 1$$

Proof. Let U be any co-compact open cover of X. By Theorem 2.2, U has a finite subcover, V = {V1, V2, ..., Vm}. By Theorem 4.1, U has a Lebesgue number, δ. Put K = (X \ V1)∪(X \ V2)∪ ... ∪ (X \ Vm). If K = ∅, then X = V<sup>j</sup> for all j = 1, 2, ..., m, and in this case, the theorem clearly holds. Hence, we assume <sup>K</sup> <sup>=</sup> <sup>∅</sup>; thus, the compact set, <sup>K</sup>, has a (n, <sup>δ</sup> <sup>3</sup> )-spanning set, F, relative to f and satisfying card(F) = rn( <sup>δ</sup> <sup>3</sup> , K, f).

(a) For any <sup>x</sup> <sup>∈</sup> <sup>K</sup>, there exists, <sup>y</sup> <sup>∈</sup> <sup>F</sup>, such that <sup>d</sup>(f<sup>i</sup> (x), f<sup>i</sup> (y)) <sup>≤</sup> <sup>δ</sup> <sup>3</sup> , i = 0, 1, ..., n − 1; equivalently, <sup>x</sup> <sup>∈</sup> <sup>f</sup> <sup>−</sup><sup>i</sup> (S(f<sup>i</sup> (y), <sup>δ</sup> <sup>3</sup> )), i = 0, 1, ..., n − 1. Hence, K ⊆ 7 y∈F n6−1 i=0 f <sup>−</sup><sup>i</sup> (S(f<sup>i</sup> (y), <sup>δ</sup> <sup>3</sup> )). By the definition of the Lebesgue number, every S(f<sup>i</sup> (y), <sup>δ</sup> <sup>3</sup> ) is a subset of an element of <sup>V</sup>. Hence, <sup>n</sup>6<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (S(f<sup>i</sup> (y), <sup>δ</sup> <sup>3</sup> )) is a subset of an element of <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (V). Consequently, K can be covered by rn( <sup>δ</sup> <sup>3</sup> , K, f) elements of <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (V).

(b) For any x, ∈ X \ K, *i.e*., x ∈ V<sup>1</sup> ∩ V<sup>2</sup> ∩ ... ∩ Vm. In the following, we will consider points of X \ K, according to two further types of points.

First, consider those <sup>x</sup> for which there exists <sup>l</sup> with <sup>1</sup> <sup>≤</sup> <sup>l</sup> <sup>≤</sup> <sup>n</sup> <sup>−</sup> <sup>1</sup>, such that <sup>f</sup><sup>l</sup> (x) ∈ K and x, f(x), f <sup>2</sup>(x), ..., f<sup>l</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>X</sup> \ <sup>K</sup> (<sup>l</sup> depends on <sup>x</sup>, but for convenience, we use <sup>l</sup> instead of <sup>l</sup>x). Namely, we consider the set, {<sup>x</sup> <sup>∈</sup> <sup>X</sup> \ <sup>K</sup> : <sup>x</sup> <sup>∈</sup> <sup>X</sup> \ K, x, f(x), f <sup>2</sup>(x), ..., f<sup>l</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>X</sup> \ K, f<sup>l</sup> (x) <sup>∈</sup> <sup>K</sup>}. For every such <sup>x</sup>, there exists <sup>y</sup> <sup>∈</sup> <sup>F</sup>, such that <sup>d</sup>(f<sup>l</sup>+<sup>i</sup> (x), f<sup>i</sup> (y)) <sup>≤</sup> <sup>δ</sup> <sup>3</sup> , i = <sup>0</sup>, <sup>1</sup>, ..., n <sup>−</sup> <sup>l</sup> <sup>−</sup> <sup>1</sup>; equivalently, <sup>x</sup> <sup>∈</sup> <sup>f</sup> <sup>−</sup>(l+i) (S(f<sup>i</sup> (y), <sup>δ</sup> <sup>3</sup> )), i = 0, 1, ..., n − l − 1. By the definition of the Lebesgue number, every S(f<sup>i</sup> (y), <sup>δ</sup> <sup>3</sup> ) is a subset of an element of <sup>V</sup>. Hence, <sup>V</sup><sup>1</sup> <sup>∩</sup> <sup>f</sup> <sup>−</sup><sup>1</sup>(V1) <sup>∩</sup> ... <sup>∩</sup> <sup>f</sup> <sup>−</sup>(l−1)(V1) <sup>∩</sup> ( n−6 l−1 i=0 f <sup>−</sup>(l+i) (S(f<sup>i</sup> (y), <sup>δ</sup> <sup>3</sup> ))) is a subset of an element of <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (V) and x ∈ <sup>V</sup><sup>1</sup> <sup>∩</sup><sup>f</sup> <sup>−</sup><sup>1</sup>(V1)∩...∩<sup>f</sup> <sup>−</sup>(l−1)(V1)∩( n−6 l−1 i=0 f <sup>−</sup>(l+i) (S(f<sup>i</sup> (y), <sup>δ</sup> <sup>3</sup> ))). There are rn( <sup>δ</sup> <sup>3</sup> , K, f) such open sets, implying that <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (V) has <sup>r</sup>n( <sup>δ</sup> <sup>3</sup> , K, f) elements that cover this type of point, x. As 1 ≤ l ≤ n − 1, n8−1 f <sup>−</sup><sup>i</sup> (V) has (<sup>n</sup> <sup>−</sup> 1) · <sup>r</sup>n( <sup>δ</sup> <sup>3</sup> , K, f) elements that actually cover this type of points, x.

i=0 Next, consider those x for which f<sup>i</sup> (x) ∈ X \ K for every i = 0, 1, ..., n − 1. One (any) element of <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (V) covers all such points, x. Hence, X \ K can be covered by no more than (<sup>n</sup> <sup>−</sup> 1) · <sup>r</sup>n( <sup>δ</sup> <sup>3</sup> , K, f)+1 elements of <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (V).

By (a) and (b), for any n > 0, it holds N( n8−1 i=0 f <sup>−</sup><sup>i</sup> (V)) <sup>≤</sup> <sup>n</sup> · <sup>r</sup>n( <sup>δ</sup> <sup>3</sup> , K, f)+1. Now, it follows from U≺V and Fact 4, N( n8−1 i=0 f <sup>−</sup><sup>i</sup> (U)) ≤ N( n8−1 i=0 f <sup>−</sup><sup>i</sup> (V)) <sup>≤</sup> <sup>n</sup> · <sup>r</sup>n( <sup>δ</sup> <sup>3</sup> , K, f)+1.

Theorem 4.3. *Let* (X, d) *be a metric space and* f : X → X *be a perfect mapping. Then* hc(f) ≤ hd(f)*.*

Proof. For any co-compact open cover, U of X, if X ∈ U, then hc(f, U)=0. Hence, we can assume X ∈ U. By Theorem 4.2, there exists δ > 0 and a non-empty compact subset, K, of X, such that for any n > 0, it holds N( n8−1 i=0 f <sup>−</sup><sup>i</sup> (U)) <sup>≤</sup> <sup>n</sup> · <sup>r</sup>n( <sup>δ</sup> <sup>3</sup> , K, f)+1. Hence, <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)) <sup>≤</sup> lim<sup>n</sup>→∞ 1 <sup>n</sup> log(<sup>n</sup> · <sup>r</sup>n( <sup>δ</sup> <sup>3</sup> , K, f) + 1) = r( <sup>δ</sup> <sup>3</sup> , K, f). Let δ → 0. It follows from the definition of Bowen's entropy (Walters' book [13], P.168, Definition 7.8 and Remark (2)) that r( <sup>δ</sup> <sup>3</sup> , K, f) is decreasing on <sup>δ</sup> and <sup>r</sup>(K, f) = lim<sup>δ</sup>→<sup>0</sup> r( <sup>δ</sup> <sup>3</sup> , K, f). Therefore, <sup>h</sup>c(f, <sup>U</sup>) <sup>≤</sup> <sup>r</sup>( <sup>δ</sup> <sup>3</sup> , K, f) ≤ r(K, f). Moreover, r(K, f) ≤ hd(f). Finally, because U is arbitrarily selected, hc(f) ≤ hd(f).

Bowen's entropy, hd(f), is metric-dependent. Theorem 4.3 indicates that the co-compact entropy, which is metric-independent, is always bounded by Bowen's entropy, *i.e*., hc(f) ≤ hd(f), regardless of the choice of a metric for the calculation of Bowen's entropy. In the next section, we will give an example where co-compact entropy is strictly less than Bowen's entropy.

#### *4.2. An Example*

In this section, R denotes the one-dimensional Euclidean space equipped with the usual metric d(x, y) = |x − y|, x, y ∈ R. The mapping, f : R → R, is defined by f(x)=2x, x ∈ R. f is clearly a perfect mapping. It is known that hd(f) ≥ log 2 [13]. We will show hc(f)=0.

Let V be any co-compact open cover of R. By Theorem 2.2, V has a finite co-compact subcover, U. Let m = card(U). As compact subsets of R are closed and bounded sets, there exist Ur, U<sup>l</sup> ∈ U, such that for any U ∈ U, sup {R \ U} ≤ sup {R \ Ur} and inf {R \ U} ≥ inf {R \ Ul}. Let <sup>a</sup><sup>r</sup> = sup {<sup>R</sup> \ <sup>U</sup>r} and <sup>b</sup><sup>l</sup> = inf {<sup>R</sup> \ <sup>U</sup>l}. Observe that for any n > <sup>0</sup>, <sup>x</sup> <sup>∈</sup> <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (Ui) ⇐⇒ x ∈ <sup>U</sup>0, f(x) <sup>∈</sup> <sup>U</sup>1, ..., f <sup>n</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>U</sup><sup>n</sup>−<sup>1</sup>, where <sup>U</sup><sup>i</sup> ∈ U, i = 0, <sup>1</sup>, ..., n <sup>−</sup> <sup>1</sup>.

Case 1: <sup>0</sup> < b<sup>l</sup> < ar. For any n > <sup>0</sup> and <sup>x</sup> <sup>∈</sup> (ar, <sup>+</sup>∞), <sup>x</sup> <sup>∈</sup> <sup>U</sup>r, f(x) <sup>∈</sup> <sup>U</sup>r, ..., f <sup>n</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>U</sup>r. So (ar, <sup>+</sup>∞) <sup>⊆</sup> <sup>n</sup>6<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (Ur). For any <sup>x</sup> <sup>∈</sup> (−∞, 0], <sup>x</sup> <sup>∈</sup> <sup>U</sup>l, f(x) <sup>∈</sup> <sup>U</sup>l, ..., f <sup>n</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>U</sup>l, thus (−∞, 0] <sup>⊆</sup> <sup>n</sup>6<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (Ul).

As f is a monotone increasing mapping, there exists k > 0, such that f <sup>k</sup>(bl) > ar. We can assume n>k> 0. Consider the following two possibilities (1.1 and 1.2).

(1.1) x ∈ [bl, ar].

This requires at most <sup>k</sup> iterations, so that <sup>f</sup> <sup>k</sup>(x) <sup>∈</sup> <sup>U</sup>r. Hence, <sup>x</sup> <sup>∈</sup> <sup>U</sup><sup>j</sup><sup>0</sup> , f(x) <sup>∈</sup> <sup>U</sup><sup>j</sup><sup>1</sup> , ..., f <sup>k</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>U</sup><sup>j</sup>k−<sup>1</sup> , <sup>f</sup> <sup>k</sup>(x) <sup>∈</sup> <sup>U</sup>r, ..., f <sup>n</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>U</sup>r, where <sup>U</sup><sup>j</sup><sup>0</sup> , U<sup>j</sup><sup>1</sup> , ., U<sup>j</sup>k−<sup>1</sup> ∈ U. Since card(U) = <sup>m</sup>, [bl, ar] can be covered by <sup>m</sup><sup>k</sup> elements of <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (U).

(1.2) x ∈ (0, bl).

This is divided into three further possibilities as follows.

(1.2.1) f <sup>n</sup>−<sup>1</sup>(x) > ar.

Choose <sup>j</sup> with <sup>0</sup> <j<n, such that <sup>f</sup><sup>j</sup>−<sup>1</sup>(x) < bl, but <sup>f</sup><sup>j</sup> (x) <sup>≥</sup> <sup>b</sup>l. Then, <sup>x</sup> <sup>∈</sup> <sup>U</sup>l, f(x) <sup>∈</sup> <sup>U</sup>l, ..., f<sup>j</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>U</sup>l, f<sup>j</sup> (x) <sup>∈</sup> <sup>U</sup><sup>j</sup><sup>0</sup> , ..., f<sup>j</sup>+k−<sup>1</sup>(x) <sup>∈</sup> <sup>U</sup><sup>j</sup>k−<sup>1</sup> , f<sup>j</sup>+<sup>k</sup>(k) <sup>∈</sup> <sup>U</sup>r, ..., f <sup>n</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>U</sup>r, where U<sup>j</sup><sup>0</sup> , U<sup>j</sup><sup>1</sup> , ..., U<sup>j</sup>k−<sup>1</sup> ∈ U. Since card(U) = m, n8−1 i=0 f <sup>−</sup><sup>i</sup> (U) has <sup>m</sup><sup>k</sup> elements that cover this kind of point, x.

(1.2.2) <sup>b</sup><sup>l</sup> <sup>≤</sup> <sup>f</sup><sup>n</sup>−<sup>1</sup>(x) <sup>≤</sup> <sup>a</sup>r.

If <sup>f</sup> <sup>n</sup>−<sup>2</sup>(x) < bl, *i.e*., for the last jump getting into [bl, ar], it holds <sup>x</sup> <sup>∈</sup> <sup>U</sup>l, ..., f <sup>n</sup>−<sup>2</sup>(x) <sup>∈</sup> <sup>U</sup>l, f<sup>n</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>U</sup><sup>j</sup><sup>0</sup> , where <sup>U</sup><sup>j</sup><sup>0</sup> ∈ U, while card(U) = <sup>m</sup>; there are <sup>m</sup> elements of <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (U) that cover these kind of points, x.

If <sup>f</sup> <sup>n</sup>−<sup>3</sup>(x) < b<sup>l</sup> and <sup>f</sup> <sup>n</sup>−<sup>2</sup>(x) <sup>≥</sup> <sup>b</sup>l, *i.e*., for the second jump to the last before getting into [bl, ar], it holds <sup>x</sup> <sup>∈</sup> <sup>U</sup>l, ..., f <sup>n</sup>−<sup>3</sup>(x) <sup>∈</sup> <sup>U</sup>l, f<sup>n</sup>−<sup>2</sup>(x) <sup>∈</sup> <sup>U</sup><sup>j</sup><sup>2</sup> , f<sup>n</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>U</sup><sup>j</sup><sup>1</sup> , where <sup>U</sup><sup>j</sup><sup>2</sup> , U<sup>j</sup><sup>1</sup> ∈ U, while card(U) = m, n8−1 i=0 f <sup>−</sup><sup>i</sup> (U) has <sup>m</sup><sup>2</sup> elements that cover this kind of point, <sup>x</sup>.

Continue in this fashion: if <sup>f</sup><sup>n</sup>−<sup>k</sup>(x) < b<sup>l</sup> and <sup>f</sup><sup>n</sup>−(k−1)(x) <sup>≥</sup> <sup>b</sup>l, *i.e*., for the (<sup>k</sup> <sup>−</sup> 1)th jump from the last before getting into [bl, ar], it holds <sup>x</sup> <sup>∈</sup> <sup>U</sup>l, ..., f <sup>n</sup>−<sup>k</sup>(x) <sup>∈</sup> <sup>U</sup>l, f<sup>n</sup>−(k−1)(x) <sup>∈</sup> <sup>U</sup><sup>j</sup>k−<sup>1</sup> , ., f <sup>n</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>U</sup><sup>j</sup><sup>1</sup> , where <sup>U</sup><sup>j</sup><sup>1</sup> , ..., U<sup>j</sup>k−<sup>1</sup> ∈ U, while card(U) = <sup>m</sup>, n8−1 i=0 f <sup>−</sup><sup>i</sup> (U) has <sup>m</sup><sup>k</sup>−<sup>1</sup> elements that cover this kind of point, x.

If <sup>f</sup> <sup>n</sup>−(k+1)(x) < b<sup>l</sup> and <sup>f</sup><sup>n</sup>−<sup>k</sup>(x) <sup>≥</sup> <sup>b</sup>l, *i.e*., jump into [bl, ar] on the <sup>k</sup>th, <sup>f</sup><sup>n</sup>−<sup>1</sup>(x) > ar, and this is Case (1.2.1).

$$\begin{aligned} &(1.2.3) \ f^{n-1}(x) < b\_l. \\ &\text{Clearly, } x \in \bigcap\_{i=0}^{n-1} f^{-i}(U\_l) \in \bigvee\_{i=0}^{n-1} f^{-i}(\mathcal{U}). \end{aligned}$$

Hence, in Case 1, where 0 < b<sup>l</sup> < ar, for any n>k> 0, it holds N( n8−1 i=0 f <sup>−</sup><sup>i</sup> (U)) <sup>≤</sup> 2+m<sup>k</sup>+m<sup>k</sup><sup>+</sup> <sup>m</sup>+m<sup>2</sup>+...+m<sup>k</sup>−<sup>1</sup>, and by the definition of co-compact entropy, <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)) ≤ lim<sup>n</sup>→∞ 1 <sup>n</sup> log(2 + m<sup>k</sup> + m<sup>k</sup> + m + m<sup>2</sup> + ... + m<sup>k</sup>−<sup>1</sup>)=0.

Case 2: b<sup>l</sup> < a<sup>r</sup> < 0. This is similar to Case 1 above.

Case 3: <sup>b</sup><sup>l</sup> <sup>&</sup>lt; <sup>0</sup> < ar. For any n > <sup>0</sup> and <sup>x</sup> <sup>∈</sup> (ar, <sup>+</sup>∞), <sup>x</sup> <sup>∈</sup> <sup>U</sup>r, f(x) <sup>∈</sup> <sup>U</sup>r, ..., f <sup>n</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>U</sup>r, thus (ar, <sup>+</sup>∞) <sup>⊆</sup> <sup>n</sup>6<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (Ur).

Similarly, for <sup>x</sup> <sup>∈</sup> (−∞, bl), <sup>x</sup> <sup>∈</sup> <sup>U</sup>l, f(x) <sup>∈</sup> <sup>U</sup>l, ..., f <sup>n</sup>−<sup>1</sup>(x) <sup>∈</sup> <sup>U</sup>l, thus (−∞, bl) <sup>⊆</sup> <sup>n</sup>6<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (Ul). As <sup>U</sup> is an open cover of <sup>R</sup>, there exists <sup>U</sup><sup>0</sup> ∈ U, such that <sup>0</sup> <sup>∈</sup> <sup>U</sup>0, f(0) = 0 <sup>∈</sup> <sup>U</sup>0, ..., f <sup>n</sup>−<sup>1</sup>(0) = 0 ∈ U0, and hence, 0 ∈ n6−1 i=0 f <sup>−</sup><sup>i</sup> (U0).

For x ∈ [bl, ar], U0, as an open set of R, can be decomposed into a union of many countably open intervals. Noting that 0 ∈ U0, there are two further possibilities, as given in (3.1) and (3.2) below.

(3.1) The stated decomposition of U<sup>0</sup> has an interval, (b0, a0), that contains zero, *i.e*., 0 ∈ (b0, a0). Since f is a monotone increasing mapping, there exists k > 0, such that f <sup>k</sup>(b0) < b<sup>l</sup> and f <sup>k</sup>(a0) > ar. Here, we can assume n>k> 0. Similar to Case 1, [bl, b0] can be covered by m<sup>k</sup> elements of n8−1 i=0 f <sup>−</sup><sup>i</sup> (U), (b0, 0) can be covered by <sup>m</sup><sup>k</sup> <sup>+</sup> <sup>m</sup> <sup>+</sup> <sup>m</sup><sup>2</sup> <sup>+</sup> ... <sup>+</sup> <sup>m</sup><sup>k</sup>−<sup>1</sup> elements of <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (U), (0, a0) can be covered by <sup>m</sup><sup>k</sup> <sup>+</sup> <sup>m</sup> <sup>+</sup> <sup>m</sup><sup>2</sup> <sup>+</sup> ... <sup>+</sup> <sup>m</sup><sup>k</sup>−<sup>1</sup> elements of <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (U) and [a0, ar] can be covered by <sup>m</sup><sup>k</sup> elements of <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (U). Hence, for any n>k> 0, N( n8−1 i=0 f <sup>−</sup><sup>i</sup> (U)) <sup>≤</sup> 3 + <sup>m</sup><sup>k</sup> <sup>+</sup> <sup>m</sup> <sup>+</sup> <sup>m</sup><sup>2</sup> <sup>+</sup> ... <sup>+</sup> <sup>m</sup><sup>k</sup>−<sup>1</sup> <sup>+</sup> <sup>m</sup><sup>k</sup> <sup>+</sup> <sup>m</sup> <sup>+</sup> <sup>m</sup><sup>2</sup> <sup>+</sup> ... <sup>+</sup> <sup>m</sup><sup>k</sup>−<sup>1</sup>. Therefore, by the definition of co-compact entropy, <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)) <sup>≤</sup> lim<sup>n</sup>→∞ 1 <sup>n</sup> log(3+m<sup>k</sup> +m+m<sup>2</sup> +...+m<sup>k</sup>−<sup>1</sup> +m<sup>k</sup> +m+m<sup>2</sup> +...+m<sup>k</sup>−<sup>1</sup>)=0.

(3.2) The only intervals covering zero are of the forms (−∞, a0) or (b0, +∞).

Consider the case, 0 ∈ (−∞, a0). As f is a monotone increasing mapping, there exists k > 0, such that f <sup>k</sup>(a0) > ar. We can assume n>k> 0. Similar to Case 1, (0, a0) can be covered by <sup>m</sup><sup>k</sup> <sup>+</sup> <sup>m</sup> <sup>+</sup> <sup>m</sup><sup>2</sup> <sup>+</sup> ... <sup>+</sup> <sup>m</sup><sup>k</sup>−<sup>1</sup> elements of <sup>n</sup>8<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (U) and [a0, ar] can be covered by <sup>m</sup><sup>k</sup> elements of n8−1 i=0 f <sup>−</sup><sup>i</sup> (U), and it also holds [bl, 0) <sup>⊆</sup> <sup>n</sup>6<sup>−</sup><sup>1</sup> i=0 f <sup>−</sup><sup>i</sup> (U0). Hence, for any n>k> 0, N( n8−1 i=0 f <sup>−</sup><sup>i</sup> (U)) ≤ 3 + <sup>m</sup><sup>k</sup>+m+m<sup>2</sup>+...+m<sup>k</sup>. By the definition of co-compact entropy, <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)) ≤ lim<sup>n</sup>→∞ 1 <sup>n</sup> log(3 + <sup>m</sup><sup>k</sup> <sup>+</sup> <sup>m</sup> <sup>+</sup> <sup>m</sup><sup>2</sup> <sup>+</sup> ... <sup>+</sup> <sup>m</sup><sup>k</sup>)=0. Therefore, when <sup>b</sup><sup>l</sup> <sup>&</sup>lt; <sup>0</sup> < ar, it holds <sup>h</sup>c(f, <sup>U</sup>)=0. The case, 0 ∈ (b0, +∞), is similar.

Now, by Cases 1, 2 and 3, it holds that hc(f, U)=0. Noting that V≺U, it holds that hc(f, V) ≤ hc(f, U)=0. Since V is arbitrary, hc(f)=0.

#### 5. Concluding Remarks

The investigation of dynamical systems could be tracked back to Isaac Newton's era, when calculus and his laws of motion and universal gravitation were invented or discovered. Then, differential equations with time as a parameter played a dominant role. However, it was not realized until the end of the 19th century that the hope of solving all kinds of problems in celestial mechanics by following Newton's frame and methodology, e.g., the two body problem, becomes unrealistic when Jules Henri Poincaré's New Methods of Celestial Mechanics was publicized (shortly after this, in the early 20th century, fundamental changes in electrodynamics occurred when Albert Einstein's historical papers appeared: reconciling Newtonian mechanics with Maxwell's electrodynamics, separating Newtonian mechanics from quantum mechanics and extending the principle of relativity to non-uniform motion), in which the space of all potential values of the parameters of the system is included in the analysis, and the attention to the system was changed from individual solutions to dynamical properties of all solutions, as well as the relation among all solutions. Although this approach may not provide much information on individual solutions, it can obtain important information on most of the solutions. For example, by taking an approach similar to that in ergodic theory, Poincaré concluded that for all Hamiltonian systems, most solutions are stable [40].

The study of dynamical systems has become a central part of mathematics and its applications since the middle of the 20th century, when scientists from all related disciplines realized the power and beauty of the geometric and qualitative techniques developed during this period for nonlinear systems (see e.g., Robinson [31]).

Chaotic and random behavior of solutions of deterministic systems is now understood to be an inherent feature of many nonlinear systems (Devaney [41], 1989). Chaos and related concepts as main concerns in mathematics and physics were investigated through differentiable dynamical systems, differential equations, geometric structures, differential topology and ergodic theory, *etc*., by S. Smale, J. Moser, M. Peixoto, V.I. Arnol'd, Ya. Sinai, J.E. Littlewood, M.L. Cartwright, A.N. Kolmogorov and G.D. Birkhoff, among others, and even as early as H. Poincaré (global properties, nonperiodicity; 1900s) and J. Hadamard (stability of trajectories; 1890s).

Kolmogorov's metric entropy as an invariant of measure theoretical dynamical systems is a powerful concept, because it controls the top of the hierarchy of ergodic properties and plays a remarkable role in investigating the complexity and other properties of the systems. As an analogous invariant under conjugation of topological dynamical systems, topological entropy plays a prominent role for the study of dynamical systems and is often used as a measure in determining dynamical behavior (e.g., chaos) and the complexity of systems. In particular, topological entropy bounds measure-theoretic entropy (Goodwyn [10,11]). Other relations between various entropy characterizations were extensively studied, e.g., Dinaburg [42]. It is a common understanding that topological entropy, as a non-negative number and invariant of conjugation in describing dynamical systems, serves a unique and unsubstitutable role in dynamics. Consequently, an appropriate definition of topological entropy becomes important and difficult.

In the theory and applications of dynamical systems, locally compact systems appear commonly, e.g., R<sup>n</sup> or other manifolds. The introduced concept of co-compact open covers is fundamental for describing the dynamical behaviors of systems as, for example, for locally compact systems, co-compact open sets are the neighborhoods of the infinity point in the Alexandroff compactification and, hence, admit the investigation of the dynamical properties near infinity.

The co-compact entropy introduced in this paper is defined based on the co-compact open covers. In the special case of compact systems, this new entropy coincides with the topological entropy introduced by Adler, Konheim and McAndrew (Sections 3 and 4). For non-compact systems, this new entropy retains various fundamental properties of Adler, Konheim and McAndrew's entropy (e.g., invariant under conjugation, entropy of a subsystem does not exceed that of the whole system).

Another noticeable property of the co-compact entropy is that it is metric-independent for dynamical systems defined on metric spaces, thus different from the entropy defined by Bowen. In particular, for the linear mapping given in Section 4.2 (locally compact system), its co-compact entropy is zero, which would be at least log 2 according to Bowen's definition; as a positive entropy usually reflects certain dynamical complexity of a system, this new entropy is more appropriate.

For a dynamical system defined on a metric space, Bowen's definition may result in different entropies when different metrics are employed. As proven in Section 4, the co-compact entropy is a lower bound for Bowen's entropies, where the traditional Lebesgue Covering Theorem for open covers of compact metric spaces is generalized for co-compact open covers of non-compact metric spaces. As studied by Goodwyn in [10,11] and Goodman [43], when the space is compact, topological entropy bounds measure-theoretic entropy. The relation between co-compact entropy and measure-theoretic entropy (K-S entropy) remains an open question. Of course, when the space is compact, this relation degenerates to the variational principle [43]. Recently, M. Patrao (2010) [44] explored entropy and its variational principle for dynamical systems on locally compact metric spaces by utilizing one point compactification.

#### Acknowledgments

The authors are grateful to the referees for their detailed and informative reviews and to the editor for his guidance of revising the original manuscript. Their suggestions are now adopted in the paper.

#### Conflict of Interest

The authors declare no conflict of interest.

#### References


Reprinted from *Entropy*. Cite as: Chen, Y.-L.; Yau, H.-T.; Yang, G.-J. A Maximum Entropy-Based Chaotic Time-Variant Fragile Watermarking Scheme for Image Tampering Detection. *Entropy* **2013**, *15*, 3170-3185.
