1. Introduction

By a topological dynamical system, we mean a pairð Þ X,T , whereX is a compactmetric space with a metric d and T is a continuous surjective map from X to itself [1]. An important notion for understanding the complexity of dynamical systems is topological entropy, which was first introduced by Adler et al. [2] in 1965, and later Dinaburg [3] and Bowen [4] gave two equivalent definitions on a metric space by using separated sets and spanning sets. Roughly speaking, topological entropy measures the maximal exponential growth rate of orbits for an arbitrary topological dynamical system.

When a considered mapping T is invertible, it is well-known that T and the inverse mapping T�<sup>1</sup> have the same topological entropy. However, when the map T is not invertible, the "inverse" is set-valued, yielding the iterated preimage set <sup>T</sup>�<sup>n</sup>ð Þ¼ <sup>x</sup> <sup>z</sup>∈<sup>X</sup> : <sup>T</sup><sup>n</sup> f g <sup>z</sup> <sup>¼</sup> <sup>x</sup> of a point <sup>x</sup>∈<sup>X</sup> which is in general a set rather than a point, so different ways of "extending the procedure into the past" lead to several new entropy-like invariants for non-invertible maps.

In 1991, Langevin and Walczak [5] regard the "inverse" as a relation and formulate a notion of entropy for this relation (analogous to the entropy of a foliation [6]), based on distinguishing points by means of the structure of their "preimage trees," which is called preimage relation entropy. The interested reader can see [7] or [8] for more details on this invariant. Later, several important entropy-like invariants based on the preimage structure for non-invertible maps, such as pointwise preimage

entropies, preimage branch entropy [1, 8–10], partial preimage entropy, conditional preimage entropy [11], etc., have been introduced, and their relationships with topological entropy have been established. To learn more about the results related to the preimage entropy for noninvertible maps, one can see [12–23].

The local entropy theory for topological dynamical systems started in the early 1990s with the work of Blanchard (see [24, 25]). Nowadays this theory has become a very interesting topic in the field of dynamical systems and has also proven to be fundamental to many other related fields. For example, Blanchard defined the notion of entropy pairs and used it to obtain a disjointness theorem [26]. The notion of entropy pairs can also be used to show the existence of the maximal zero-entropy factor, called the topological Pinsker factor, for any topological dynamical system [25]. In order to gain a better understanding of the topological version of a Ksystem, the theory of entropy tuples [27–29] was developed. To learn more about the theory related to the local entropy, we refer the interested reader to see the survey paper [30] and references therein.

We remark that in reality, it is difficult to find a real orbit in the system, but a pseudo-orbit can be used to approximate the real orbit, and so there have been a lot of applications in many fields. Since the works of Bowen [31] and Conley [32], pseudoorbits have proved to be a powerful tool in dynamical systems. For instance, Hammel et al. [33, 34] have investigated the role of pseudo-orbits in computer simulations of certain dynamical systems; Barge and Swanson [35] made use of pseudo-orbits to study rotation sets of circle and annulus maps. Also, a remarkable result by Misiurewicz [36] showed that the topological entropy can be computed by measuring the exponential growth rate of the numbers of pseudo-orbits (related results can see [37]). In [1], Hurley showed that the point entropy with pseudo-orbits that is defined by replacing inverse orbit segments by inverse pseudo-orbit segments in the definition of pointwise preimage entropy is in fact equal to the topological entropy.

In this chapter, following Hurley [1] we further study the preimage entropy for topological dynamical system from the view of localization. In Section 2, we consider the calculation of topological entropy for open covers from pseudo-orbits (Theorem 2.3). In Section 3, we investigate the relationship among the topological entropy for open covers and several preimage entropy invariants, which is viewed as the local version of the Hurley inequality (Theorem 3.1). In Section 4, we show that the topological entropy for open covers can be computed by measuring the exponential growth rate of the number of pseudo-orbits that end at a particular point (Theorems 4.2 and 4.3).

A nonautonomous discrete dynamical system is a natural generalization of a classical dynamical system; its dynamics is determined by a sequence of continuous self-maps f <sup>n</sup> : Xn ! Xnþ1, which defined on a sequence on compact metric spaces (Xn, dn). The topological entropy of nonautonomous discrete dynamical systems was introduced by Kolyada and Snoha [38]. In Section 5, following the idea of [1, 39], we introduce two entropy-like invariants, which are called the partial entropy and bundle-like entropy, for nonautonomous discrete dynamical systems, and study the relationship among them and the topological entropy (Theorems 5.2, 5.3, and 5.5).

## 2. Topological entropy and pseudo-orbits

#### 2.1 Topological entropy via open covers

Topological entropy was defined originally by Adler et al. [2] for continuous maps on compact topological spaces. Let ð Þ X,T be a topological dynamical system. A finite open cover of X is a finite family of open sets whose union is X. Denoted by Partial Entropy and Bundle-Like Entropy for Topological Dynamical Systems DOI: http://dx.doi.org/10.5772/intechopen.89021

Co <sup>X</sup> is the set of finite open covers of <sup>X</sup>. Given two open covers <sup>U</sup>, <sup>V</sup> <sup>∈</sup>C<sup>o</sup> <sup>X</sup>, U is said to be finer than V (U ≽V) if each element of U is contained in some element of V. Let U∨V ¼ f g U∩V : U ∈U, V ∈V . It is clear that U∨V ≽U and U∨V ≽V.

Let <sup>U</sup> <sup>∈</sup>C<sup>o</sup> <sup>X</sup>. For two nonnegative integers <sup>M</sup> <sup>≤</sup> <sup>N</sup>, denoted by <sup>U</sup><sup>N</sup> <sup>M</sup> <sup>¼</sup> <sup>∨</sup><sup>N</sup> <sup>n</sup>¼MT�nU, where <sup>T</sup>�nU ¼ <sup>T</sup>�<sup>n</sup> f g ð Þ <sup>U</sup> : <sup>U</sup> <sup>∈</sup><sup>U</sup> for all positive integers <sup>n</sup>. For any <sup>K</sup> <sup>⊂</sup>X, define Nð Þ UjK as the minimal cardinality of any subcovers of U that covers K. We write Nð Þ UjX simply by Nð Þ U . The topological entropy of U with respect to T is defined by

$$h\_{\text{top}}(T, \mathcal{U}) = \lim\_{n \to \infty} \frac{1}{n} \log N(\mathcal{U}\_0^{n-1}) = \inf\_{n \ge 1} \frac{1}{n} \log N(\mathcal{U}\_0^{n-1}).\tag{1}$$

The topological entropy of T is

$$h\_{\text{top}}(T) = \sup\_{\mathcal{U} \in \mathcal{E}\_X^{\text{v}}} h\_{\text{top}}(T, \mathcal{U}). \tag{2}$$

#### 2.2 Separated sets, spanning sets, and topological entropy

In this subsection, we recall two equivalent definitions, which are given by Dinaburg [3] and Bowen [4]. Let ð Þ X,T be a topological dynamical system. Given a nonempty subset K of X, for any ϵ>0 and n ∈ℕ, a subset E of K is called an ð Þ n, ϵ -separated set of K if any x 6¼ y∈E implies dnð Þ x, y ≥ ϵ, where

$$d\_n(\mathfrak{x}, \mathfrak{y}) \coloneqq \max\_{0 \le i \le n-1} d\left(T^i \mathfrak{x}, T^i \mathfrak{y}\right).$$

Denote the maximal cardinality of any ð Þ n, ϵ -separated subset of K by s n, ð Þ ϵ, K . A subset F of K is called an ð Þ n, ϵ -spanning set of K, if for any x∈ K, there exists y∈F with dnð Þ x, y <ϵ. Denote the minimal cardinality of any ð Þ n, ϵ -spanning set for K by r n, ð Þ ϵ, K .

The following lemma is well-known, and its proof is not difficult, so we omit its detail proof.

Lemma 2.1. Let X, T ð Þ be a topological dynamical system. For any subset K of X and any integer n≥1, we have the following properties:

$$\mathbf{1}. r(n, \epsilon, K) \le s(n, \epsilon, K) \le r(n, \epsilon/2, K) \text{ for all } \epsilon > 0.$$

$$2. N(\mathcal{U}\_0^{n-1}|K) \le r(n, \delta, K) \text{ for any } n \in \mathbb{N} \text{ and any } \mathcal{U} \in \mathcal{C}\_X^\diamondsuit \text{ with the Lebesgue number } 2\delta.$$

$$\mathfrak{Z}.s(n,\epsilon,K) \le N\left(\mathcal{U}\_0^{n-1}|K\right) \text{ for any } \mathcal{U} \in \mathcal{C}\_X^\circ \text{ with } \text{diam}(\mathcal{U}) < \epsilon.$$

By Lemma 2.1, we obtain directly the following result.

Theorem 2.2. (see [3, 4, 40]). Let X, T ð Þ be a topological dynamical system. Then

$$h\_{\text{top}}(T) = \lim\_{\epsilon \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log \kappa(n, \epsilon, X) = \lim\_{\epsilon \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log r(n, \epsilon, X).$$

#### 2.3 Topological entropy via pseudo-orbits

Let ð Þ X, d be a compact metric space. Denote <sup>X</sup><sup>n</sup> as the <sup>n</sup>-fold Cartesian product of <sup>X</sup> (<sup>n</sup> <sup>≥</sup>1Þ. Fixing a positive number <sup>ϵ</sup>, a subset <sup>E</sup> <sup>⊂</sup>X<sup>n</sup> is said to be ð Þ n, <sup>ϵ</sup> -separated if for any two distinct points <sup>x</sup><sup>~</sup> <sup>¼</sup> ð Þ <sup>x</sup>0, x1, <sup>⋯</sup>, xn�<sup>1</sup> , <sup>y</sup><sup>~</sup> <sup>¼</sup> <sup>y</sup>0, y1, <sup>⋯</sup>, yn�<sup>1</sup> ∈ E, there is a 0≤i≤ n � 1 such that d xi, yi <sup>&</sup>gt;ϵ. By the compactness of <sup>X</sup>, any ð Þ n, <sup>ϵ</sup> -separated set is finite. If Z ⊂X<sup>n</sup> is a nonempty subset, then we denote the maximal cardinality of any ð Þ n, ϵ -separated subset of Z by s n, ð Þ ϵ, Z .

Let <sup>Z</sup> <sup>⊂</sup>X<sup>n</sup> be a nonempty subset. A subset <sup>F</sup> <sup>⊂</sup><sup>Z</sup> is called ð Þ n, <sup>ϵ</sup> -panning for <sup>Z</sup> if for each <sup>z</sup><sup>~</sup> <sup>¼</sup> ð Þ <sup>z</sup>0, z1, <sup>⋯</sup>, zn�<sup>1</sup> <sup>∈</sup>Z, there is a <sup>y</sup><sup>~</sup> <sup>¼</sup> <sup>y</sup>0, y1, <sup>⋯</sup>, yn�<sup>1</sup> <sup>∈</sup><sup>F</sup> with d zi, yi < ϵ for every 0 ≤i ≤n � 1. We denote the minimal cardinality of any ð Þ n, ϵ -spanning subset of Z by r n, ð Þ ϵ, Z .

For each positive integer n≥ 1, we let On denote the set of all orbit segments of length n, that is,

$$O\_n = \left\{ \left( \mathfrak{x}, T\mathfrak{x}, \dots, T^{n-1}\mathfrak{x} \right) \in X^n : \mathfrak{x} \in X \right\}.$$

Note that a point <sup>w</sup><sup>~</sup> <sup>¼</sup> x, Tx, <sup>⋯</sup>, Tn�<sup>1</sup> x ∈ On is uniquely determined by its initial point x∈X. Thus, we have

$$h\_{\text{top}}(T) = \lim\_{\epsilon \to 0} \lim\_{a \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log s(n, \epsilon, O\_n)$$

$$= \lim\_{\epsilon \to 0} \lim\_{a \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log r(n, \epsilon, O\_n).$$

Topological entropy has been characterized by Misiurewicz [36] and Barge and Swanson [37] in terms of growth rates of pseudo-orbits. Let ð Þ X, T be a topological dynamical system. For α> 0, an α-pseudo-orbit for T of length n is a point x~ ¼ ð Þ <sup>x</sup>0, x1, <sup>⋯</sup>, xn�<sup>1</sup> <sup>∈</sup>X<sup>n</sup> with the property that dTxj�<sup>1</sup> , xj <sup>&</sup>lt;<sup>α</sup> for all 1≤j<sup>≤</sup> <sup>n</sup> � 1. Let <sup>Ψ</sup>nð Þ <sup>α</sup> <sup>⊂</sup> <sup>X</sup><sup>n</sup> denote all <sup>α</sup>-pseudo-orbits of length <sup>n</sup>. It was shown in [36, 37] that

$$h\_{\text{top}}(T) = \lim\_{\epsilon \to 0} \lim\_{a \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log s(n, \epsilon, \Psi\_n(a))$$

$$= \lim\_{\epsilon \to 0} \lim\_{a \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log r(n, \epsilon, \Psi\_n(a)).$$

In the following, we will show that the topological entropy for an open cover can be characterized by pseudo-orbits. Before proceeding, let us first introduce a definition of pseudo-orbit entropy via open covers. Let ð Þ X,T be a topological dynamical system. For each integer <sup>n</sup><sup>≥</sup> 1 and <sup>U</sup> <sup>∈</sup>C<sup>o</sup> <sup>X</sup>, we define an open cover <sup>U</sup><sup>n</sup> of the product space X<sup>n</sup> by

$$\mathcal{U}^{\mathfrak{n}} = \left\{ U\_0 \times U\_1 \times \dots \times U\_{n-1} : U\_j \in \mathcal{U} \text{ for each } j = 0, 1, \dots, n - 1 \right\},$$

where

$$\{U\_0 \times U\_1 \times \dots \times U\_{n-1} = \{ (u\_0, u\_1, \dots, u\_{n-1}) : u\_j \in U\_j \text{ for each } j = 0, 1, \dots, n - 1 \} \} $$

Given <sup>α</sup> <sup>&</sup>gt;0, it is not hard to see that an <sup>¼</sup> <sup>N</sup> <sup>U</sup><sup>n</sup> ð Þ <sup>j</sup>Ψnð Þ <sup>α</sup> is a nonnegative subadditive sequence, that is, anþ<sup>m</sup> ≤an þ am for all positive integers n and m. The α-pseudo-orbit entropy of U is then defined by

$$h\_{\Psi}(T, \mathcal{U}, a) = \lim\_{n \to \infty} \frac{1}{n} \log N(\mathcal{U}^{n} | \Psi\_{n}(a)) = \inf\_{n \ge 1} \frac{1}{n} \log N(\mathcal{U}^{n} | \Psi\_{n}(a)),\tag{3}$$

and the pseudo-orbit entropy of U is defined by

$$h\_{\Psi}(T, \mathcal{U}) = \lim\_{a \to 0} h\_{\Psi}(T, \mathcal{U}, a). \tag{4}$$

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

Theorem 2.3. Let X, T ð Þ be a topological dynamical system. If <sup>U</sup> <sup>∈</sup>C<sup>o</sup> X, then we have

$$h\_{\text{top}}(T, \mathcal{U}) = h\_{\Psi}(T, \mathcal{U}).\tag{5}$$

Proof. To prove (5), it suffices to note that hΨð Þ T, U, α<sup>1</sup> ≤hΨð Þ T, U, α<sup>2</sup> whenever <sup>α</sup><sup>1</sup> <sup>&</sup>lt;α<sup>2</sup> and inf0<sup>&</sup>lt;α≤<sup>1</sup> <sup>N</sup> <sup>U</sup><sup>n</sup> ð Þ¼ <sup>j</sup>Ψnð Þ <sup>α</sup> <sup>N</sup> <sup>U</sup><sup>n</sup> ð Þ¼ <sup>j</sup>On <sup>N</sup> <sup>U</sup>n�<sup>1</sup> 0 . Thus, we have

$$\begin{aligned} h\_{\Psi}(T,\mathcal{U}) &= \lim\_{a \to 0} h\_{\Psi}(T,\mathcal{U},a) \\ &= \inf\_{0 < a \le 0} \inf\_{n \ge 1} \frac{1}{n} \log N(\mathcal{U}^{a}|\Psi\_{n}(a)) \\ &= \inf\_{n \ge 1} \inf\_{0 < a \le 1} \frac{1}{n} \log N(\mathcal{U}^{a}|\Psi\_{n}(a)) \\ &= \inf\_{n \ge 1} \frac{1}{n} \log N(\mathcal{U}\_{0}^{a-1}) = h\_{\text{top}}(T,\mathcal{U}). \end{aligned}$$

This completes the proof of the theorem. □ Remark 2.4. Combining (2) and (5), we have

$$h\_{\text{top}}(T) = \sup\_{\mathcal{U} \in \mathcal{C}\_{\mathcal{X}}^{\text{w}}} h\_{\Psi}(T, \mathcal{U}).$$

On the other hand, let us define <sup>h</sup>Ψð Þ¼ <sup>T</sup> sup<sup>U</sup> <sup>∈</sup>C<sup>o</sup> X hΨð Þ T, U , which is called the pseudo-orbit entropy of T. Using the same techniques of topological entropy (see Lemma 2.1), we can easily show that

$$h\_{\Psi}(T) = \lim\_{\epsilon \to 0} \lim\_{a \to 0} \limsup\_{n \to \infty} \frac{1}{n} \log s(n, \epsilon, \Psi\_n(a)).$$

$$= \lim\_{\epsilon \to 0} \limsup\_{a \to 0} \frac{1}{n} \log r(n, \epsilon, \Psi\_n(a)).$$

So, it is in fact to give a simpler proof of Theorem 1 of [37] by Theorem 2.3.
