3. Pointwise preimage entropies for open covers and local Hurley's inequality

When T is not invertible, one can ask about growth rates of inverse images f �n ð Þ x . In this section we describe two ways of doing this, which were introduced by Hurley in [1].

#### 3.1 Preimage branch entropy

Let ð Þ X, T be a topological dynamical system. Given x∈X let Tnð Þ x denote the tree of inverse images of x up to order n, which is defined by

$$T\_n(\mathbf{x}) = \left\{ (\mathbf{z}\_0, \mathbf{z}\_1, \dots, \mathbf{z}\_n) : \mathbf{z}\_n = \mathbf{x} \text{ and } \mathbf{z}\_j = T(\mathbf{z}\_{j-1}) \text{ for all } 1 \le j \le n \right\}.$$

Each ð Þ z0, z1, ⋯, zn ∈Tnð Þ x is called a branch of Tnð Þ x , and its length is n. Note that every branch of Tnð Þ x ends with x. Let T <sup>n</sup> ¼ ∪<sup>x</sup>∈XTnð Þ x ; we define a metric on T <sup>n</sup> as follows: suppose that z~ ¼ ð Þ z0, z1, ⋯, zn and w~ ¼ ð Þ w0, w1, ⋯, wn are two branches of the length n, the branch distance between them is defined as


$$d\_{B\_{\mathfrak{b}},n}(\tilde{z},\tilde{w}) = \max\_{0 \le j \le n} d\left(z\_j, w\_j\right).$$

Let On ¼ f g Tnð Þ x : x∈X . Given two trees Tnð Þ x and Tnð Þy in On, the branch Hausdorff distance between them, dbHð Þ Tnð Þ x , Tnð Þy is the usual Hausdorff metric based upon dB, <sup>n</sup>; that is,

$$d\_{bH}(T\_n(\mathbf{x}), T\_n(\boldsymbol{y})) = \max\left\{ \max\_{\tilde{\mathbf{z}} \in T\_n(\mathbf{x})} \min\_{\tilde{w} \in T\_n(\boldsymbol{y})} d\_{\mathcal{B}, \mathbf{z}}(\tilde{\mathbf{z}}, \tilde{w}), \max\_{\tilde{w} \in T\_n(\boldsymbol{y})} \min\_{\tilde{\mathbf{z}} \in T\_n(\mathbf{x})} d\_{\mathcal{B}, \mathbf{z}}(\tilde{\mathbf{z}}, \tilde{w}) \right\}.$$

Note that dbHð Þ Tnð Þ x , Tnð Þy <ϵ if and only if each branch of either tree is dB, <sup>n</sup> within ϵ of at least one branch of the other tree. Two trees Tnð Þ x and Tnð Þy in On are said to be dbH- n, ð Þϵ -separated if dbHð Þ Tnð Þ x , Tnð Þy <ϵ, that is, there is a branch z~ of one of the trees with the property that dB, <sup>n</sup>ð Þ z,~ w~ > ϵ for all branches w~ of the other tree. Let t n, ð Þϵ denote the maximum cardinality of any dbH-ð Þ n, ϵ -separated sets of On. Define the entropy by

$$h\_b(T) = \lim\_{\epsilon \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log t(n, \epsilon),$$

which is called the preimage branch entropy of T.

#### 3.2 Pointwise preimage entropies

Let us recall two non-invertible invariants defined by Hurley [1] in 1995. Hurley's invariants are about the maximum rate of dispersal of the preimage sets of individual points, which are called pointwise preimage entropies in [8]. The difference between these two invariants is when the maximization takes place:

$$h\_p(T) = \sup\_{\boldsymbol{x} \in X} \lim\_{\epsilon \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log s(n, \epsilon, T^{-n}(\boldsymbol{x}))$$

$$= \sup\_{\boldsymbol{x} \in X} \lim\_{\epsilon \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log r(n, \epsilon, T^{-n}(\boldsymbol{x})),$$

$$h\_m(T) = \lim\_{\epsilon \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log \sup\_{\boldsymbol{x} \in X} s(n, \epsilon, T^{-n}(\boldsymbol{x}))$$

$$= \lim\_{\epsilon \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log \sup\_{\boldsymbol{x} \in X} r(n, \epsilon, T^{-n}(\boldsymbol{x})).$$

It is clear that hpð Þ T ≤hmð Þ T , and in [18] the authors constructed an example for which hpð Þ T <hmð Þ T . In addition, Hurley established the following relationships among preimage branch entropy, pointwise preimage entropy, and topological entropy (see [1], Theorem 3.1):

$$h\_m(T) \le h\_{\text{top}}(T) \le h\_m(T) + h\_b(T).$$

We call it the Hurley inequality.

#### 3.3 Local Hurley's inequality

In this subsection, we mainly study the relationship among the topological entropy for open covers and several preimage entropy invariants, which is viewed Partial Entropy and Bundle-Like Entropy for Topological Dynamical Systems DOI: http://dx.doi.org/10.5772/intechopen.89021

as the local version of the Hurley inequality. To do it, we first introduced a definition of preimage entropy via open covers.

Let ð Þ X,T be a topological dynamical system. Given <sup>U</sup> <sup>∈</sup>C<sup>o</sup> <sup>X</sup>, define two pointwise preimage entropies of U with respect to T by

$$h\_p(T, \mathcal{U}) = \sup\_{\boldsymbol{x} \in \mathcal{X}} \limsup\_{n \to \infty} \frac{1}{n} \log N\left(\mathcal{U}\_0^{n-1} | T^{-n}(\boldsymbol{x})\right),$$

and

$$h\_m(T, \mathcal{U}) = \limsup\_{n \to \infty} \frac{1}{n} \log \left( \sup\_{\varkappa \in X} N(\mathcal{U}\_0^{n-1} | T^{-n}(\varkappa)) \right).$$

Theorem 3.1. (Local Hurley's inequality). Let X, T ð Þ be a topological dynamical system. If <sup>U</sup> <sup>∈</sup>C<sup>o</sup> X, then we have

$$h\_p(T, \mathcal{U}) \le h\_m(T, \mathcal{U}) \le h\_{\text{top}}(T, \mathcal{U}) \le h\_m(T, \mathcal{U}) + h\_b(T).$$

Proof. It is obvious that <sup>N</sup> <sup>U</sup><sup>n</sup>�<sup>1</sup> <sup>0</sup> <sup>j</sup>T�<sup>n</sup>ð Þ <sup>x</sup> � �<sup>≤</sup> <sup>N</sup> <sup>U</sup><sup>n</sup>�<sup>1</sup> 0 � � for every x∈X and every integer n ≥1. So that hpð Þ T, U ≤hmð Þ T, U ≤htopð Þ T, U . Now we show the last inequality htopð Þ T, U ≤hmð Þþ T, U hbð Þ T .

Let ϵ>0 be a Lebesgue number of U. Fix n≥1, and let Y denote a dbH-ð Þ n, ϵ=3 separated set of On with cardinality t n, ð Þ ϵ=3 . Let Z denote the set of all root points of trees in Y, where the root point of the tree Tnð Þ x is x. For each z∈Z, let Vð Þ z, <sup>U</sup> be a subcover of <sup>U</sup><sup>n</sup>�<sup>1</sup> <sup>0</sup> with cardinality <sup>N</sup> <sup>U</sup><sup>n</sup>�<sup>1</sup> <sup>0</sup> <sup>j</sup>T�<sup>n</sup>ð Þ<sup>z</sup> � � that covers <sup>T</sup>�<sup>n</sup>ð Þ<sup>z</sup> , and let

$$\mathcal{V} = \bigcup\_{\mathfrak{z} \in Z} \mathcal{V}(\mathfrak{z}, \mathcal{U}).$$

We claim that V is an open cover of X.

In fact, let x∈X be given and let w ¼ f n ð Þ x . Since Y is a dbH-ð Þ n, ϵ=3 -separated set of On with maximal cardinality, there is a tree Tnð Þy ∈ Y such that dbHð Þ Tnð Þ w , Tnð Þy <ϵ=3. Now we consider the branch w~ of Tnð Þ w begins with x, i.e., <sup>w</sup><sup>~</sup> <sup>¼</sup> x, f xð Þ, <sup>⋯</sup>, f <sup>n</sup>�<sup>1</sup> ð Þ <sup>x</sup> , f <sup>n</sup> ð Þ¼ <sup>x</sup> <sup>w</sup> � �∈Tnð Þ <sup>w</sup> . Then there exists a branch <sup>y</sup><sup>~</sup> <sup>¼</sup> <sup>y</sup>0, y1, <sup>⋯</sup>, yn <sup>¼</sup> <sup>y</sup> � �<sup>∈</sup> Tnð Þ<sup>y</sup> such that dB, <sup>n</sup>ð Þ w, <sup>~</sup> <sup>y</sup><sup>~</sup> <sup>&</sup>lt;ϵ=3. This means that d T<sup>j</sup> y<sup>0</sup> � �, T<sup>j</sup> ð Þ <sup>x</sup> � �<ϵ=3 for each 0≤j≤n. Thus, there exists <sup>V</sup> <sup>∈</sup> Vð Þ y, <sup>U</sup> such that <sup>x</sup>∈V. This yields the claim that <sup>V</sup> is an open cover of <sup>X</sup>. So that <sup>N</sup> <sup>U</sup><sup>n</sup>�<sup>1</sup> 0 � �≤ ∣V∣, where ∣V∣ denotes the cardinality of V. Using the claim, we have

$$\begin{split} N(\mathcal{U}\_0^{n-1}) &\leq |\mathcal{V}| \leq \sum\_{x \in \mathcal{Z}} |\mathcal{V}(x, \mathcal{U})| = \sum\_{x \in \mathcal{Z}} N(\mathcal{U}\_0^{n-1} | T^{-n}(x)) \\\\ &\leq |\mathcal{Z}| \cdot \left( \sup\_{x \in \mathcal{X}} N(\mathcal{U}\_0^{n-1} | T^{-n}(x)) \right) \\\\ &= |Y| \cdot \left( \sup\_{x \in \mathcal{X}} N(\mathcal{U}\_0^{n-1} | T^{-n}(x)) \right) = t(n, \epsilon/3) \cdot \left( \sup\_{x \in \mathcal{X}} N(\mathcal{U}\_0^{n-1} | T^{-n}(x)) \right). \end{split}$$

So that,

$$\begin{split} h\_{\text{top}}(T, \mathcal{U}) &= \lim\_{n \to \infty} \frac{1}{n} \log N(\mathcal{U}\_0^{n-1}) \\ &\leq \limsup\_{n \to \infty} \frac{1}{n} \left[ \log t(n, \epsilon/3) + \log \left( \sup\_{x \in X} N(\mathcal{U}\_0^{n-1} | T^{-n}(x)) \right) \right] \\ &\leq \limsup\_{n \to \infty} \frac{1}{n} \log t(n, \epsilon/3) + \limsup\_{n \to \infty} \frac{1}{n} \log \left( \sup\_{x \in X} N(\mathcal{U}\_0^{n-1} | T^{-n}(x)) \right) \\ &= \limsup\_{n \to \infty} \frac{1}{n} \log t(n, \epsilon/3) + h\_m(T, \mathcal{U}) \leq h\_b(T) + h\_m(T, \mathcal{U}). \end{split}$$

This completes the proof of the theorem. □ We remark that Theorem 3.1 generalizes the classical Hurley's inequality given in [26, Theorem 3.1]. A direct consequence of Theorem 3.1 is.

Corollary 3.2. (Hurley's inequality). Let X, T ð Þ be a topological dynamical system. Then we have

$$h\_P(T) \le h\_m(T) \le h\_{\text{top}}(T) \le h\_m(T) + h\_b(T). \tag{6}$$

Proof. It follows directly from Lemma 2.1 that

$$h\_p(T) = \sup\_{\mathcal{U} \in \mathcal{C}\_\mathcal{X}^\circ} h\_p(T, \mathcal{U}) \text{ and } h\_m(T) = \sup\_{\mathcal{U} \in \mathcal{C}\_\mathcal{X}^\circ} h\_p(T, \mathcal{U}). \tag{7}$$

Thus, combining (2), (7), and Theorem 3.1 gives (6). □
