5.3 Some relationships between htopð Þ F and hp,Pð Þ F

Theorem 5.2. Let ð Þ X, F be a nonautonomous discrete dynamical system, and P ¼ P<sup>i</sup> ð Þ : i≥1 be a cover of X. Then we have

$$h\_{p\_\bullet, \mathcal{P}}(\mathbf{F}) \le h\_{\text{top}}(\mathbf{F}) \le h\_{b\_\bullet, \mathcal{P}}(\mathbf{F}) + h\_{p\_\bullet, \mathcal{P}}(\mathbf{F}).$$

Proof. Note that s max ,P<sup>n</sup> ð Þ n, ϵ ≤s n, ð Þ ϵ, On for any cover P of X and any ϵ>0. Then the former inequality is obtained. Now we show the later one. If hb,Pð Þ¼ F ∞, then there is nothing to prove. Now assuming hb,Pð Þ F < ∞.

Fixing a sufficiently small ϵ>0 and an integer n≥ 1, let Y be a dH-ð Þ n, ϵ -separated subset of On,<sup>P</sup><sup>n</sup> with cardinality s n, ϵ, On,<sup>P</sup><sup>n</sup> . For each Wnð Þ <sup>D</sup> <sup>∈</sup>Y, let M Dð Þ be a ~ dn-ð Þ n, ϵ -separated subset of Wnð Þ D with cardinality s n, ð Þ ϵ, Wnð Þ D . Put <sup>M</sup> <sup>¼</sup> <sup>∪</sup>Wnð Þ <sup>D</sup> <sup>∈</sup>YM Dð Þ. We claim that <sup>M</sup> is a <sup>~</sup> dn-ð Þ n, 3ϵ -spanning subset of On.

In fact, for any x ¼ ð Þ x1, x2, ⋯, xn ∈ On, since Y is a dH-ð Þ n, ϵ -separated subset of On,<sup>P</sup><sup>n</sup> with maximum cardinality and P<sup>n</sup> covers Xn, there is an E∈P<sup>n</sup> with xn ∈ E and a Wnð Þ D ∈Y such that dHð Þ Wnð Þ D , Wnð Þ E ≤ϵ. Then it follows that there is a y ¼ y1, y2, ⋯, yn <sup>∈</sup> Wnð Þ <sup>D</sup> such that <sup>~</sup> dnð Þ x, y <sup>≤</sup> <sup>ϵ</sup>. Also note that M Dð Þ is a <sup>~</sup> dn-ð Þ n, ϵ separated subset of Wnð Þ D with maximum cardinality; there is a z∈ M Dð Þ such that ~ dnð Þ y, z ≤ϵ. Hence we have

$$
\tilde{d}\_n(\mathfrak{x}, z) \le \tilde{d}\_n(\mathfrak{x}, \mathfrak{y}) + \tilde{d}\_n(\mathfrak{y}, z) < 3\epsilon.
$$

This yields the claim that M is a ~ dn-ð Þ n, ϵ -spanning subset of On. So we have r n, ð Þ 3ϵ, On ≤ ∣M∣, where M denotes the cardinality of M. Using the claim we have

$$\begin{split} r(n, \mathfrak{z}, O\_{\mathfrak{n}}) \leq & |M| \leq |Y| \cdot \max\left\{ |M(D)| \colon W\_{\mathfrak{n}}(D) \in Y \right\} \\ \leq & s\left(n, \mathfrak{e}, O\_{\mathfrak{n}\_{\mathfrak{n}}, \mathcal{P}\_{\mathfrak{n}}}\right) \cdot s\_{\max, \mathcal{P}\_{\mathfrak{n}}}(n, \mathfrak{e}). \end{split}$$

Taking limits as the requirements of the related definitions of entropies establishes the desired inequality. This completes the proof. □

Let Pð Þδ be a finite cover of a compact metric space X consisting of open balls with radius less than some <sup>δ</sup>>0. Write <sup>F</sup><sup>X</sup> <sup>¼</sup> fi : fi : <sup>X</sup> ! <sup>X</sup> is continous, i <sup>≥</sup><sup>1</sup> and PXð Þ¼ P δ ð Þ ð Þδ ,Pð Þδ , ⋯ .

Theorem 5.3.

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

Proof. Note that lim <sup>n</sup>!<sup>∞</sup> <sup>1</sup> <sup>n</sup> log ∣Pð Þδ ∣ ¼ 0. Then, by Theorem 5.2, we have the former equality. Now we show the later equality.

Clearly, s n, ð Þ ϵ, On ≥s max ,<sup>P</sup><sup>X</sup> ð Þδ ð Þ n, ϵ for any δ> 0, so we have

$$\limsup\_{n \to \infty} \log s(n, \epsilon, O\_n) \ge \lim\_{\delta \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log s\_{\max, \mathcal{P}\_\lambda(\delta)}(n, \epsilon).$$

This implies

$$h\_{\text{top}}(\mathbf{F}\_X) \ge \lim\_{\varepsilon \to 0} \lim\_{\delta \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log s\_{\text{max}, \mathcal{P}\_\mathbf{X}(\delta)}(n, \varepsilon). \tag{22}$$

On the other hand, from the proof of Theorem 5.2, it follows that

$$r(n, \mathfrak{z}\epsilon, \mathcal{O}\_n) \le \mathfrak{s}\left(n, \mathfrak{e}, \mathcal{O}\_{n, \mathcal{P}(\delta)}\right) \cdot s\_{\max, \mathcal{P}\_\mathcal{X}(\delta)}(n, \mathfrak{e})$$

for any integer n ≥1, any sufficiently small ϵ> 0 and any δ>0. Noting that s n, ϵ, On,Pð Þ<sup>δ</sup> ≤ ∣Pð Þ<sup>δ</sup> <sup>∣</sup> for any integer <sup>n</sup>≥1, then we have

$$\lim\_{n \to \infty} \sup\_n \frac{1}{n} \log r(n, \mathfrak{A}; O\_n) \le \lim\_{\delta \to 0} \lim\_{n \to \infty} \sup\_n \frac{1}{n} \log \mathfrak{s}\_{\max, \mathcal{P}\_X(\delta)}(n, \mathfrak{e}).$$

This implies

$$h\_{\text{top}}(\mathbf{F}\_X) \le \lim\_{\epsilon \to 0} \lim\_{\delta \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log \varepsilon\_{\text{max}, \mathcal{P}\_\mathbf{X}(\delta)}(n, \epsilon). \tag{23}$$

Thus, combining (22) and (23) gets the later equality. This completes the proof. □

Remark 5.4. The first equality of Theorem 5.3 is in fact a simpler version of Theorem 7.6 of [40] (a useful result for calculating the classical topological entropy) when restricting to the autonomous discrete dynamical systems.

Given a nonautonomous discrete dynamical system ð Þ X, ℱ , when does htopð Þ¼ F hp,Pð Þ F for any cover P of X? The following theorem gives an answer to this question.

Theorem 5.5. Let ð Þ X, F be a nonautonomous discrete dynamical system. Then htopð Þ¼ F hp,Pð Þ F for any cover P of X if the following conditions hold:

(1) For each integer i ≥1, there exists δ<sup>i</sup> >0 such that diþ<sup>1</sup> fi ð Þ x , fi ð Þ<sup>y</sup> <sup>≥</sup>dið Þ x, y whenever dið Þ x, y ≤δ<sup>i</sup> for x, y∈Xi.

(2) For each integer i ≥1, every x∈Xiþ<sup>1</sup> has an open neighborhood Ux whose preimage f �<sup>1</sup> <sup>i</sup> ð Þ Ux is an union of disjoint open sets on each of which fi is a homeomorphism.

(3) lim sup<sup>n</sup>!<sup>∞</sup> <sup>1</sup> <sup>n</sup> log Nð Þ¼ ϵn, Xn 0 for every monotonic decreasing sequence f g ϵ<sup>n</sup> with lim <sup>n</sup>!<sup>∞</sup>ϵ ¼ 0, where each Nð Þ ϵn, Xn denotes the minimal cardinality of the open cover of X consisting of open ϵn-ball for the compact metric space Xn.

Proof. It suffices to show that hb,Pð Þ¼ F 0 for any cover P of X by Theorem 5.2. Let <sup>P</sup>max ¼ Pi, max : <sup>i</sup><sup>≥</sup> <sup>1</sup> be the cover of <sup>X</sup> in which each <sup>P</sup>i, max cover Xi consisting to singletons of Xi, i.e., Pi, max ¼ f g f gz : z∈Xi . It is easy to see that hb,Pð Þ F ≤hb,<sup>P</sup>max ð Þ F for any cover P of X. So from Theorem 5.2, it follows that what we want to prove is hb,<sup>P</sup>max ð Þ¼ F 0.

For each n ≥2, by condition (1), there exists a δ<sup>n</sup>�<sup>1</sup> > 0 such that

$$d\_n(f\_{n-1}(\mathfrak{x}), f\_{n-1}(\mathfrak{y})) \ge d\_{n-1}(\mathfrak{x}, \mathfrak{y}),$$

for any x, y∈Xn�<sup>1</sup> whenever dn�<sup>1</sup>ð Þ x, y ≤δ<sup>n</sup>�1. Also, by condition (2) and the compactness of Xn, there exists an ϵ<sup>n</sup> >0 such that the ϵn-ball B xð Þ n, ϵ<sup>n</sup> about any point xn ∈Xn has preimage f �1 <sup>n</sup>�<sup>1</sup>ð Þ B xð Þ n, <sup>ϵ</sup> equals the union of disjoint open sets of

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

diameter less than δn�1. Then we get a sequence f g ϵ<sup>n</sup> . Furthermore, we can take ϵ<sup>n</sup> such that f g ϵ<sup>n</sup> is monotonic decreasing sequence and lim <sup>n</sup>!<sup>∞</sup> ϵ<sup>n</sup> ¼ 0.

Now, given yn <sup>∈</sup>Xn and <sup>x</sup><sup>~</sup> <sup>¼</sup> ð Þ <sup>x</sup>1, x2, <sup>⋯</sup>, xn <sup>∈</sup> Wn B yn, <sup>ϵ</sup>n, � � � � , we want to find a point y~ ¼ y1, y2, ⋯, yn � �∈ On with ~ dnð Þ¼ x, ~ y~ dn xn, yn � � and then ~ dnð Þ x, ~ y~ <ϵn. In fact, for 1<k<n, we can easily find a point yk, ⋯, yn � �∈ Q<sup>n</sup> <sup>i</sup>¼<sup>k</sup>Xi with dj xj, yj � �<sup>≤</sup> <sup>ϵ</sup><sup>j</sup> and djþ<sup>1</sup> xjþ<sup>1</sup>, yjþ<sup>1</sup> � �≥dj xj, yj � �, for <sup>j</sup> <sup>¼</sup> <sup>n</sup> � <sup>1</sup>, n � <sup>2</sup>, <sup>⋯</sup>, k. Let <sup>V</sup> be the piece of f �1 <sup>k</sup>�1ð Þ B xð Þ k, <sup>ϵ</sup><sup>k</sup> with xk�<sup>1</sup> <sup>∈</sup>V. Since yk <sup>∈</sup>B xð Þ k, <sup>ϵ</sup><sup>k</sup> , there is a unique point yk�<sup>1</sup> <sup>∈</sup>V∩<sup>f</sup> �1 <sup>k</sup>�<sup>1</sup> yk � � such that dk�<sup>1</sup> xk�<sup>1</sup>, yk�<sup>1</sup> � �<δk�1. Then we have

$$d\_{k-1}(\chi\_{k-1}, \mathcal{y}\_{k-1}) \le d\_k(\chi\_k, \mathcal{y}\_k) \le d\_n(\chi\_n, \mathcal{y}\_n) < \epsilon\_n < \epsilon\_{k-1}.$$

This argument shows that r n, <sup>ϵ</sup>n, On,comma max � �<sup>≤</sup> <sup>N</sup>ð Þ <sup>ϵ</sup>n, Xn . Thus, by condition (3), we get

$$\limsup\_{n \to \infty} \frac{1}{n} \log r(n, \epsilon\_n, O\_{n, \text{comm}} \max) \le \limsup\_{n \to \infty} \frac{1}{n} \log N(\epsilon\_n, X\_n) = 0.$$

For any sufficiently small ϵ >0, there exists N >0 such that ϵ<sup>n</sup> <ϵ for any n≥ N. Then we have r n, <sup>ϵ</sup>, On,Pn, max � �≤r n, <sup>ϵ</sup>n, On,Pn, max � � and hence hb,<sup>P</sup>max ð Þ¼ <sup>F</sup> 0. This completes the proof. □
