5. Partial entropy and bundle-like entropy for nonautonomous discrete dynamical systems

In [38, 41], topological entropy for certain nonautonomous discrete dynamical system was defined and studied. In this section, we study the topological entropy for nonautonomous discrete dynamical systems by introducing two entropy-like invariants called the partial entropy and bundle-like entropy as being motivated by the idea of [1, 39].

#### 5.1 Topological entropy for nonautonomous discrete dynamical systems

Let X be a collection of countable infinitely many compact metric space Xi ð Þ , di and F be a collection of countable infinite many continuous maps fi : Xi ! Xiþ1, i ¼ 1, 2, ⋯. Then the pair ð Þ X, F is called a nonautonomous discrete dynamical system.

For any integer n≥1, we define a metric ~ dn on Q<sup>n</sup> <sup>i</sup>¼<sup>1</sup>Xi as follows: for any two points x~<sup>n</sup> ¼ ð Þ x1, x2, ⋯, xn , y~<sup>n</sup> ¼ y1, y2, ⋯, yn � �∈ Q<sup>n</sup> <sup>i</sup>¼<sup>1</sup> Xi,

$$
\tilde{d}\_n\left(\tilde{\mathfrak{x}}\_n, \tilde{\mathfrak{y}}\_n\right) = \max\_{1 \le i \le n} d\_i\left(\mathfrak{x}\_i, \mathfrak{y}\_i\right).
$$

Fixing an integer n ≥1 and a positive number ϵ. A subset Z of Q<sup>n</sup> <sup>i</sup>¼<sup>1</sup>Xi is called <sup>~</sup> dnð Þ n, <sup>ϵ</sup> -separated if for any two distinct points <sup>x</sup>~n, <sup>y</sup>~<sup>n</sup> <sup>∈</sup><sup>Z</sup> we have <sup>~</sup> dn x~n, y~<sup>n</sup> � �> ϵ. Denote the maximal cardinality of any ~ dn-separated subset of Z by s n, ð Þ ϵ, Z . A subset W ⊂Z is called ~ dn- n, ð Þϵ -spanning for Z if for each z~<sup>n</sup> ∈ Z, there is a w~<sup>n</sup> ∈W such that ~ dnð Þ <sup>z</sup>~n, <sup>w</sup>~<sup>n</sup> <sup>&</sup>lt;ϵ. Denote the minimal cardinality of any <sup>~</sup> dn-spanning subset of Z by r n, ð Þ ϵ, Z .

The following result is trivial, so we omit its detail proof.

Lemma 5.1. Suppose that n is a positive integer and Z is a nonempty subset of Q<sup>n</sup> <sup>i</sup>¼<sup>1</sup>Xi. Then for each ϵ>0, we have

$$r(n, \epsilon, Z) \le s(n, \epsilon, Z) \le r(n, \epsilon/2, Z).$$

For each n ≥1 let Zn be a nonempty subset of Q<sup>n</sup> <sup>i</sup>¼<sup>1</sup>Xi. Then it follows immediately from Lemma 5.1 that

$$\lim\_{\epsilon \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log r(n, \epsilon, Z\_n) = \lim\_{\epsilon \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log s(n, \epsilon, Z\_n). \tag{21}$$

Given a nonautonomous discrete dynamical system ð Þ X, F , denoted by On, <sup>F</sup> or On for short the set of all orbit segments of length n for each n ≥1, i.e.,

$$\mathcal{O}\_{\mathtt{n}} = \mathcal{O}\_{\mathtt{n}} \ \_{\mathtt{F}} \coloneqq \left\{ (\mathfrak{x}\_{1}, \mathfrak{x}\_{2}, \dots, \mathfrak{x}\_{n}) : \mathfrak{x}\_{1} \in X\_{1} \text{ and } \quad \mathfrak{x}\_{i+1} = f\_{i}(\mathfrak{x}\_{i}), i = 1, 2, \dots, n-1 \right\}.$$

Then the common limit in (21) by taking Zn ¼ On is defined to be the topological entropy of ð Þ X, F , written htopð Þ X, F or htopð Þ F for short if there is no confusion.

#### 5.2 Partial entropy and bundle-like entropy

Let ð Þ X, F be a nonautonomous discrete dynamical system. A collection P ¼ P<sup>i</sup> ð Þ : i≥1 is said to be a cover of X if each P<sup>i</sup> covers Xi, respectively. We now define two entropies, partial entropy and bundle-like entropy, for ð Þ X, F relative to P.

For any integer <sup>n</sup>≥1 and <sup>D</sup> <sup>∈</sup>Pn, let Wnð Þ <sup>D</sup> <sup>⊂</sup> <sup>Q</sup><sup>n</sup> <sup>i</sup>¼<sup>1</sup>Xi denote the set of all orbit segments of length that end at some point xn ∈ D, i.e.,

$$\mathcal{W}\_n(D) = \{ (\varkappa\_1, \varkappa\_2, \cdots, \varkappa\_n) \in O\_n : \varkappa\_n \in D \}.$$

Put <sup>s</sup> max ,<sup>P</sup><sup>n</sup> ð Þ¼ n, <sup>ϵ</sup> sup<sup>D</sup> <sup>∈</sup>P<sup>n</sup> s n, ð Þ ϵ, Wnð Þ D . Define the entropy by

$$h\_{p, \mathcal{P}}(\mathbf{X}, \mathbf{F}) = \lim\_{\epsilon \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log s\_{\max, \mathcal{P}\_n}(n, \epsilon),$$

which is called the partial entropy of ð Þ X, F relative to P and written shortly by hp,Pð Þ F if there is no confusion.

Let On,<sup>P</sup><sup>n</sup> ¼ f g Wnð Þ D : D ∈P<sup>n</sup> . For any two elements, Wnð Þ D and Wnð Þ E of On,<sup>P</sup><sup>n</sup> , denoted by dHð Þ Wnð Þ D , Wnð Þ E , the usual Hausdorff metric between them is based upon metric ~ dn of Q<sup>n</sup> <sup>i</sup>¼<sup>1</sup> Xi defined as before and by s n, <sup>ϵ</sup>, On,<sup>P</sup><sup>n</sup> � � the maximum cardinality of any dH-ð Þ n, ϵ -separated subset of On,<sup>P</sup><sup>n</sup> . Define the entropy by

Partial Entropy and Bundle-Like Entropy for Topological Dynamical Systems DOI: http://dx.doi.org/10.5772/intechopen.89021

$$h\_{b, \mathcal{P}}(\mathbf{X}, \mathbf{F}) = \lim\_{\epsilon \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log s(n, \epsilon, O\_{n, \mathcal{P}\_n}),$$

which is called the bundle-like entropy of ð Þ X, F relative to P and written shortly by hb,Pð Þ F if there is no confusion.

Also, we have the spanning set versions of definitions of hp,Pð Þ F and hb,Pð Þ F , respectively.
