4. Point entropy for open covers with pseudo-orbits

In [1], Hurley considered pseudo-orbits for inverse images and showed that the topological entropy can be characterized in terms of growth rates of pseudo-orbits that end at a particular point. Let ð Þ X, T be a topological dynamical system. Recall that if <sup>α</sup><sup>&</sup>gt; 0, then an <sup>α</sup>-pseudo-orbit ð Þ <sup>x</sup>0, x1, …, xn�<sup>1</sup> <sup>∈</sup>X<sup>n</sup> is an approximate orbits segment for T in the sense that dTxj , xjþ<sup>1</sup> <sup>&</sup>lt; <sup>α</sup> for all 0<sup>≤</sup> <sup>j</sup>≤<sup>n</sup> � 1.

For each <sup>x</sup>∈X, let <sup>Ψ</sup>nð Þ <sup>α</sup>, x <sup>⊂</sup>X<sup>n</sup> denote the set of all <sup>α</sup>-pseudo-orbits of length <sup>n</sup> that end at <sup>x</sup>, i.e., an element of <sup>Ψ</sup>nð Þ <sup>α</sup>, x is an <sup>α</sup>-pseudo-orbit <sup>y</sup>0, y1, <sup>⋯</sup>, yn�<sup>1</sup> with yn�<sup>1</sup> <sup>¼</sup> <sup>x</sup>. It was shown in [1], (Propositions 4.2 and 4.3) that

$$\begin{split} h\_{\text{top}} &= \lim\_{\epsilon \to 0} \lim\_{a \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log \left( \max\_{\boldsymbol{x} \in X} s(n, \epsilon, \boldsymbol{\Psi}\_{n}(\boldsymbol{a}, \boldsymbol{x})) \right) \\ &= \sup\_{\boldsymbol{x} \in X} \lim\_{\epsilon \to 0} \lim\_{a \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log s(n, \epsilon, \boldsymbol{\Psi}\_{n}(\boldsymbol{a}, \boldsymbol{x})). \end{split} \tag{8}$$

In either formula s n, ð Þ ϵ, Ψnð Þ α, x can be replaced by r n, ð Þ ϵ, Ψnð Þ α, x .

In the following, we will show that the topological entropy for an open cover can be characterized by pseudo-orbits for inverse images. Before proceeding, let us

consider the following definitions, which use the notation introduced in Section 2.3. Let ð Þ X,T be a topological dynamical system. For each integer <sup>n</sup> <sup>≥</sup>1, <sup>U</sup> <sup>∈</sup>C<sup>o</sup> <sup>X</sup>, and α> 0, we define

$$N\_{\max}(\mathfrak{n}, \mathcal{U}, a) = \max\_{x \in X} N(\mathcal{U}^{\mathfrak{n}} | \Psi\_{\mathfrak{n}}(a, \mathfrak{x})).\tag{9}$$

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

Clearly,

$$N(\mathcal{U}''|\Psi\_n(a,\varkappa)) \le N\_{\text{max}}(n,\mathcal{U},a) \le N(\mathcal{U}''|\Psi\_n(a))\tag{10}$$

for every x∈X. In addition, by the compactness of X, there is some point y∈X such that

$$N(\mathcal{U}^\eta|\Psi\_\mathfrak{n}(a,\mathfrak{y})) = N\_{\max}(\mathfrak{n}, \mathcal{U}, a).$$

Lemma 4.1. Let X, T ð Þ be a topological dynamical system and <sup>U</sup> <sup>∈</sup> <sup>C</sup><sup>o</sup> X. Suppose that ε>0 is a Lebesgue number of U and 0< α<ε=4. Then there is a constant K ¼ Kð Þ α such that for every n ≥1,

$$N(\mathcal{U}''|\Psi\_n(a)) \le K \cdot N\_{\text{max}}(\mathfrak{n}, \mathcal{U}, a). \tag{11}$$

Proof. Let f g <sup>x</sup>1, x2, <sup>⋯</sup>, xK be a finite <sup>α</sup>-dense subset of <sup>X</sup>, i.e., <sup>⋃</sup><sup>n</sup> <sup>i</sup>¼<sup>1</sup>B xi ð Þ¼ , <sup>α</sup> <sup>X</sup>, where B xi ð Þ¼ , <sup>α</sup> <sup>z</sup><sup>∈</sup> <sup>X</sup> : d xi f g ð Þ , z <sup>&</sup>lt;<sup>α</sup> . For each 1<sup>≤</sup> <sup>i</sup><sup>≤</sup> <sup>K</sup>, let <sup>V</sup><sup>i</sup> be a subcover of <sup>U</sup><sup>n</sup> that covers <sup>Ψ</sup>nð Þ <sup>α</sup>, xi with cardinality <sup>N</sup> <sup>U</sup><sup>n</sup> ð Þ <sup>j</sup>Ψnð Þ <sup>α</sup>, xi . Define V ¼ <sup>⋃</sup><sup>K</sup> <sup>i</sup>¼<sup>1</sup>Vi. Clearly, <sup>∣</sup>V∣ ≤P<sup>K</sup> <sup>i</sup>¼<sup>1</sup>∣Vi∣ ≤<sup>K</sup> � <sup>N</sup>maxð Þ n, <sup>U</sup>, <sup>α</sup> . So, to complete the proof of the lemma, it suffices to show <sup>V</sup> is a subcover of <sup>U</sup><sup>n</sup> that covers <sup>Ψ</sup>nð Þ <sup>α</sup> .

In fact, let <sup>y</sup><sup>~</sup> <sup>¼</sup> <sup>y</sup>0, y1, <sup>⋯</sup>, yn�<sup>1</sup> � � be an <sup>α</sup>-pseudo-orbit. Since f g <sup>x</sup>1, x2, <sup>⋯</sup>, xK is an <sup>α</sup>-dense subset of <sup>X</sup>, there is some xi satisfying dTyn�<sup>2</sup> � �, xi � �<sup>&</sup>lt; <sup>α</sup>. This implies <sup>z</sup><sup>~</sup> <sup>¼</sup> <sup>ð</sup>z0, z1, <sup>⋯</sup>, zn�<sup>2</sup>, zn�<sup>1</sup>Þ ¼ <sup>y</sup>0, y1, <sup>⋯</sup>, yn�<sup>2</sup>, xi � � is an <sup>α</sup>-pseudo-orbit ending at xi. Since <sup>V</sup><sup>i</sup> is a subcover of <sup>U</sup><sup>n</sup> that covers <sup>Ψ</sup>nð Þ <sup>α</sup>, xi , there is some <sup>V</sup> <sup>∈</sup>V<sup>i</sup> such that <sup>z</sup>~∈V. Since zj ¼ yj for all 0 ≤j≤n � 2 and ϵ is the Lebesgue number of U, in order to show that <sup>y</sup>~∈V, we need only to show that d yn�<sup>1</sup>, xi � �< ϵ=2; this is obviously, as d yn�<sup>1</sup>, xi � �≤d yn�<sup>1</sup>,T yn�<sup>2</sup> � � � � <sup>þ</sup> dTyn�<sup>2</sup> � �, xi � �<2<sup>α</sup> <sup>&</sup>lt;ϵ=2. □

Theorem 4.2. Let X, T ð Þ be a topological dynamical system. If <sup>U</sup> <sup>∈</sup>C<sup>o</sup> X, then we have

$$h\_{\text{top}}(T, \mathcal{U}) = \lim\_{a \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log \left( \sup\_{\varkappa \in X} \mathcal{N}(\mathcal{U}^n | \Psi\_n(a, \varkappa)) \right). \tag{12}$$

Proof. Combining (10) and (11), we have

$$N\_{\max}(n, \mathcal{U}) \le \mathcal{N}(\mathcal{U}^n | \Psi\_n(a)) \le \mathcal{K} \cdot N\_{\max}(n, \mathcal{U})$$

for each fixed 0 <α <ϵ=4 and all n ≥1, where ϵ is a Lebesgue number of U and K ¼ Kð Þ α in Lemma 4.1 is independent of n. This implies that

$$\limsup\_{n \to \infty} \frac{1}{n} \log N\_{\text{max}}(n, \mathcal{U}, a) = \limsup\_{n \to \infty} \frac{1}{n} \log N(\mathcal{U}^{\boldsymbol{\eta}} | \Psi(a)) \tag{13}$$

for all positive number 0< α<ϵ=2. Thus, we have

$$h\_{\text{top}}(T,\mathcal{U}) = \lim\_{a \to 0} \lim\_{n \to \infty} \frac{1}{n} \log N(\mathcal{U}^{\boldsymbol{\mu}} | \Psi\_n(a)) \quad \text{(by Theorem 2.3)}$$

$$= \lim\_{a \to 0} \lim\_{n \to \infty} \sup\_n \frac{1}{n} \log N(\mathcal{U}^{\boldsymbol{\mu}} | \Psi\_n(a))$$

$$= \lim\_{a \to 0} \lim\_{n \to \infty} \sup\_n \frac{1}{n} \log N\_{\text{max}}(n, \mathcal{U}, a) \quad \text{(by (4.6))}$$

$$= \lim\_{a \to 0} \lim\_{n \to \infty} \sup\_n \frac{1}{n} \log \left( \sup\_{x \in X} N(\mathcal{U}^{\boldsymbol{\mu}} | \Psi\_n(a, x)) \right) \quad \text{(by (4.1))}$$

This completes the proof. □ Theorem 4.3. Let X, T ð Þ be a topological dynamical system. If <sup>U</sup> <sup>∈</sup>C<sup>o</sup> X, then we have

$$h\_{\text{top}}(T, \mathcal{U}) = \sup\_{x \in X} \lim\_{a \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log N(\mathcal{U}^{\mathbb{R}} | \Psi\_n(a, \infty)). \tag{14}$$

Proof. It follows directly from (10) and (12) that

$$h\_{\text{top}}(T, \mathcal{U}) = \lim\_{a \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log \left( \sup\_{\mathbf{x} \in X} N(\mathcal{U}^n | \Psi\_n(a, \mathbf{x})) \right).$$

$$\geq \sup\_{\mathbf{x} \in X} \limsup\_{n \to 0} \frac{1}{n} \log N(\mathcal{U}^n | \Psi\_n(a, \mathbf{x})).$$

Now we start to prove the converse inequality.

Note that for the given n ≥1 and α> 0, there is a point y n, ð Þ U, α ∈X such that

$$N(\mathcal{U}^\mathfrak{n}|\Psi\_n(a,\mathcal{y}(n,\mathcal{U},a)) = \max\_{\mathcal{X}\in\mathcal{X}} N(\mathcal{U}^\mathfrak{n}|\Psi\_n(a,\mathfrak{x})).$$

Taking a sequence of integers ni ¼ nið Þ! α ∞ such that

$$\limsup\_{n \to \infty} \frac{1}{n} \log N(\mathcal{U}^{\boldsymbol{a}} | \Psi\_n(\boldsymbol{a}, \boldsymbol{\jmath}(\boldsymbol{n}, \boldsymbol{\mathcal{U}}, \boldsymbol{a}))) = \lim\_{i \to \infty} \frac{1}{\eta\_i} \log N(\mathcal{U}^{\boldsymbol{a}\_i} | \Psi\_{\boldsymbol{n}\_i}(\boldsymbol{a}, \boldsymbol{\jmath}(\boldsymbol{n}\_i, \boldsymbol{\mathcal{U}}, \boldsymbol{a}))).$$

By restricting to a subsequence, we can assume without loss of generality that the sequence yi ð Þ¼ α y ni ð Þ , U, α converses to a limit qð Þ α .

Let ϵ be a Lebesgue number of U. If 0<β <ϵ=4 and d yi ð Þ <sup>α</sup> , qð Þ <sup>α</sup> <sup>&</sup>lt;β, then <sup>V</sup> is a subcover of <sup>U</sup><sup>n</sup> that covers <sup>Ψ</sup><sup>n</sup> <sup>α</sup>, yi ð Þ <sup>α</sup> whenever <sup>V</sup> is a subcover of <sup>U</sup><sup>n</sup> that covers Ψnð Þ α þ β, qð Þ α . This implies that

$$N(\mathcal{U}^{\boldsymbol{n}}|\Psi\_{\boldsymbol{n}}(\boldsymbol{a}+\boldsymbol{\beta},\boldsymbol{q}(\boldsymbol{a}))) \geq N(\mathcal{U}^{\boldsymbol{n}}|\Psi\_{\boldsymbol{n}}(\boldsymbol{a},\boldsymbol{\gamma}\_{\boldsymbol{i}}(\boldsymbol{a}))) \tag{15}$$

whenever d yi ð Þ <sup>α</sup> , qð Þ <sup>α</sup> <sup>&</sup>lt;β.

Now we choose a sequence α<sup>j</sup> ! 0 such that q α<sup>j</sup> converges to some point q∈X. Similar to the proof as above we have

$$N\left(\mathcal{U}^{\mathfrak{n}}|\Psi\_{n}\left(a\_{\mathfrak{j}}+2\beta,q\right)\geq N\left(\mathcal{U}^{\mathfrak{n}}|\Psi\_{n}\left(a\_{\mathfrak{j}},q\left(a\_{\mathfrak{j}}\right)\right)\right)\tag{16}$$

whenever d q α<sup>j</sup> , q <β. Combining inequalities (15) and (16), one has

$$N\left(\mathcal{U}^n|\Psi\_n\left(a\_\circ+2\beta,q\right)\right)\geq N\left(\mathcal{U}^n|\Psi\_n\left(a\_\circ,\mathcal{y}\_i\left(a\_\circ\right)\right)\right)\tag{17}$$

whenever d yi α<sup>j</sup> , q α<sup>j</sup> < β and d q α<sup>j</sup> , q <β. If j is a fixed integer with d q α<sup>j</sup> , q <β, then (17) holds for all sufficiently large integers i. Thus,

$$\begin{split} \limsup\_{n \to \infty} &\frac{1}{n} \log N\left(\mathcal{U}^{n} | \Psi\_{n}\left(a\_{j} + 2\beta, q\right)\right) \\ &\geq \limsup\_{i \to \infty} \frac{1}{n\_{i}} \log N\left(\mathcal{U}^{n\_{i}} | \Psi\_{n\_{i}}\left(a\_{j} + 2\beta, q\right)\right) \\ &\geq \lim\_{i \to \infty} \frac{1}{n\_{i}} \log N\left(\mathcal{U}^{n\_{i}} | \Psi\_{n\_{i}}\left(a\_{j}, y\_{i}\left(a\_{j}\right)\right)\right) \\ &= \limsup\_{n \to \infty} \frac{1}{n} \log \left(\max\_{x \in X} N(\mathcal{U}^{n} | \Psi\_{n}\left(a\_{j}\right), x)\right). \end{split} \tag{18}$$

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

Now let j ! ∞ and use the fact that both sides (18) are nonincreasing as α decreases to conclude that

$$\begin{split} \limsup\_{n \to \infty} \frac{1}{n} \log N(\mathcal{U}^{\boldsymbol{\mu}} | \Psi\_n(\boldsymbol{\beta}\boldsymbol{\beta}, q)) \\ \geq & \lim\_{j \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log N(\mathcal{U}^{\boldsymbol{\mu}} | \Psi\_n(\boldsymbol{a}\_j + 2\boldsymbol{\beta}, q)) \\ \geq & \lim\_{j \to \infty} \limsup\_{n \to \infty} \frac{1}{n} \log \left( \max\_{x \in X} N(\mathcal{U}^{\boldsymbol{\mu}} | \Psi\_n(\boldsymbol{a}\_j), \boldsymbol{x}) \right) \\ = & \inf\_{a > 0} \limsup\_{n \to \infty} \frac{1}{n} \log \left( \max\_{x \in X} N(\mathcal{U}^{\boldsymbol{\mu}} | \Psi\_n(\boldsymbol{a}, x)) \right) \\ = & \lim\_{a \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log \left( \max\_{x \in X} N(\mathcal{U}^{\boldsymbol{\mu}} | \Psi\_n(\boldsymbol{a}), \boldsymbol{x}) \right). \end{split} \tag{19}$$

Therefore, combining (12) and (19), we have

$$h\_{\text{top}}(T, \mathcal{U}) = \lim\_{a \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log \left( \max\_{x \in X} N(\mathcal{U}^a | \Psi\_n(a), x) \right)$$

$$\leq \inf\_{\beta > 0} \limsup\_{n \to \infty} \frac{1}{n} \log N(\mathcal{U}^a | \Psi\_n(3\beta, q))$$

$$= \inf\_{a > 0} \limsup\_{n \to \infty} \frac{1}{n} \log N(\mathcal{U}^a | \Psi\_n(a, q)) \tag{20}$$

$$= \limsup\_{a \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log N(\mathcal{U}^a | \Psi\_n(a, q))$$

$$\leq \sup\_{x \in X} \limsup\_{n \to 0} \frac{1}{n} \log N(\mathcal{U}^a | \Psi\_n(a, x)).$$

This completes the proof. □
