Entropy is an important notion for understanding the complexity of dynamical systems. Several important entropy-like invariants based on the preimage structure for noninvertible maps have been defined and studied by some authors. In this chapter, following the idea of Hurley, we first further study the relationship among the topological entropy, pseudo-orbit, and preimage entropies for topological dynamical systems from the view of localization. Secondly, two entropy-like invariants, which are called the partial entropy and bundle-like entropy, for nonautonomous discrete dynamical systems are introduced. A relationship between the topological entropy and such two entropies is established.
- topological entropy
- point entropy
- partial entropy
- bundle-like entropy
- 2000 Mathematics Subject Classification: Primary: 37B40
By a topological dynamical system, we mean a pair , where is a compact metric space with a metric d and T is a continuous surjective map from to itself . An important notion for understanding the complexity of dynamical systems is topological entropy, which was first introduced by Adler et al.  in 1965, and later Dinaburg  and Bowen  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 and the inverse mapping have the same topological entropy. However, when the map is not invertible, the “inverse” is set-valued, yielding the iterated preimage set of a point 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  regard the “inverse” as a relation and formulate a notion of entropy for this relation (analogous to the entropy of a foliation ), based on distinguishing points by means of the structure of their “preimage trees,” which is called preimage relation entropy. The interested reader can see  or  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, 9, 10], partial preimage entropy, conditional preimage entropy , 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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 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 . 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 . In order to gain a better understanding of the topological version of a K-system, the theory of entropy tuples [27, 28, 29] was developed. To learn more about the theory related to the local entropy, we refer the interested reader to see the survey paper  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  and Conley , pseudo-orbits 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  made use of pseudo-orbits to study rotation sets of circle and annulus maps. Also, a remarkable result by Misiurewicz  showed that the topological entropy can be computed by measuring the exponential growth rate of the numbers of pseudo-orbits (related results can see ). In , 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  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 , 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 . 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.  for continuous maps on compact topological spaces. Let be a topological dynamical system. A finite open cover of is a finite family of open sets whose union is . Denoted by is the set of finite open covers of . Given two open covers , is said to be finer than () if each element of is contained in some element of . Let . It is clear that and .
Let . For two nonnegative integers , denoted by , where for all positive integers . For any , define as the minimal cardinality of any subcovers of that covers . We write simply by . The topological entropy of with respect to is defined by
The topological entropy of T is
2.2 Separated sets, spanning sets, and topological entropy
In this subsection, we recall two equivalent definitions, which are given by Dinaburg  and Bowen . Let be a topological dynamical system. Given a nonempty subset K of X, for any and , a subset E of K is called an -separated set of K if any implies , where
Denote the maximal cardinality of any -separated subset of by . A subset of is called an -spanning set of , if for any , there exists with . Denote the minimal cardinality of any -spanning set for by .
The following lemma is well-known, and its proof is not difficult, so we omit its detail proof.
Lemma 2.1. Let be a topological dynamical system. For any subset of and any integer , we have the following properties:
for all .
for any and any with the Lebesgue number .
for any with .
By Lemma 2.1, we obtain directly the following result.
2.3 Topological entropy via pseudo-orbits
Let be a compact metric space. Denote as the -fold Cartesian product of (. Fixing a positive number , a subset is said to be -separated if for any two distinct points , there is a such that . By the compactness of , any -separated set is finite. If is a nonempty subset, then we denote the maximal cardinality of any -separated subset of by .
Let be a nonempty subset. A subset is called -panning for if for each , there is a with for every . We denote the minimal cardinality of any -spanning subset of by .
For each positive integer , we let denote the set of all orbit segments of length , that is,
Note that a point is uniquely determined by its initial point . Thus, we have
Topological entropy has been characterized by Misiurewicz  and Barge and Swanson  in terms of growth rates of pseudo-orbits. Let be a topological dynamical system. For , an -pseudo-orbit for of length is a point with the property that for all . Let denote all -pseudo-orbits of length . It was shown in [36, 37] that
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 be a topological dynamical system. For each integer and , we define an open cover of the product space by
Given , it is not hard to see that is a nonnegative sub-additive sequence, that is, for all positive integers and . The -pseudo-orbit entropy of is then defined by
and the pseudo-orbit entropy of is defined by
Theorem 2.3. Let be a topological dynamical system. If , then we have
Proof. To prove (5), it suffices to note that whenever and . Thus, we have
This completes the proof of the theorem. □
On the other hand, let us define , which is called the pseudo-orbit entropy of . Using the same techniques of topological entropy (see Lemma 2.1), we can easily show that
So, it is in fact to give a simpler proof of Theorem 1 of  by Theorem 2.3.
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 . In this section we describe two ways of doing this, which were introduced by Hurley in .
3.1 Preimage branch entropy
Let be a topological dynamical system. Given let denote the tree of inverse images of up to order , which is defined by
Each is called a branch of , and its length is . Note that every branch of ends with . Let ; we define a metric on as follows: suppose that and are two branches of the length , the branch distance between them is defined as
Let . Given two trees and in , the branch Hausdorff distance between them, is the usual Hausdorff metric based upon ; that is,
Note that if and only if each branch of either tree is within of at least one branch of the other tree. Two trees and in are said to be - -separated if , that is, there is a branch of one of the trees with the property that for all branches of the other tree. Let denote the maximum cardinality of any --separated sets of . Define the entropy by
which is called the preimage branch entropy of T.
3.2 Pointwise preimage entropies
Let us recall two non-invertible invariants defined by Hurley  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 . The difference between these two invariants is when the maximization takes place:
It is clear that , and in  the authors constructed an example for which . In addition, Hurley established the following relationships among preimage branch entropy, pointwise preimage entropy, and topological entropy (see , Theorem 3.1):
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 as the local version of the Hurley inequality. To do it, we first introduced a definition of preimage entropy via open covers.
Let be a topological dynamical system. Given , define two pointwise preimage entropies of with respect to by
Theorem 3.1. (Local Hurley’s inequality). Let be a topological dynamical system. If , then we have
Proof. It is obvious that for every and every integer . So that . Now we show the last inequality .
Let be a Lebesgue number of . Fix , and let denote a --separated set of with cardinality . Let denote the set of all root points of trees in , where the root point of the tree is . For each , let be a subcover of with cardinality that covers , and let
We claim that is an open cover of .
In fact, let be given and let . Since is a --separated set of with maximal cardinality, there is a tree such that . Now we consider the branch of begins with , i.e., . Then there exists a branch such that . This means that for each . Thus, there exists such that . This yields the claim that is an open cover of . So that , where denotes the cardinality of . Using the claim, we have
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 be a topological dynamical system. Then we have
Proof. It follows directly from Lemma 2.1 that
4. Point entropy for open covers with pseudo-orbits
In , 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 be a topological dynamical system. Recall that if , then an -pseudo-orbit is an approximate orbits segment for in the sense that for all .
For each , let denote the set of all -pseudo-orbits of length that end at , i.e., an element of is an -pseudo-orbit with . It was shown in , (Propositions 4.2 and 4.3) that
In either formula can be replaced by .
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 be a topological dynamical system. For each integer , , and , we define
for every . In addition, by the compactness of , there is some point such that
Lemma 4.1. Let be a topological dynamical system and . Suppose that is a Lebesgue number of and . Then there is a constant such that for every ,
Proof. Let be a finite -dense subset of , i.e., , where . For each , let be a subcover of that covers with cardinality . Define . Clearly, . So, to complete the proof of the lemma, it suffices to show is a subcover of that covers .
In fact, let be an -pseudo-orbit. Since is an -dense subset of , there is some satisfying . This implies is an -pseudo-orbit ending at . Since is a subcover of that covers , there is some such that . Since for all and is the Lebesgue number of , in order to show that , we need only to show that ; this is obviously, as .□
Theorem 4.2. Let be a topological dynamical system. If , then we have
for each fixed and all , where is a Lebesgue number of and in Lemma 4.1 is independent of . This implies that
for all positive number . Thus, we have
This completes the proof.□
Theorem 4.3. Let be a topological dynamical system. If , then we have
Now we start to prove the converse inequality.
Note that for the given and , there is a point such that
Taking a sequence of integers such that
By restricting to a subsequence, we can assume without loss of generality that the sequence converses to a limit .
Let be a Lebesgue number of . If and , then is a subcover of that covers whenever is a subcover of that covers . This implies that
Now we choose a sequence such that converges to some point . Similar to the proof as above we have
whenever and . If is a fixed integer with , then (17) holds for all sufficiently large integers . Thus,
Now let and use the fact that both sides (18) are nonincreasing as decreases to conclude that
This completes the proof.□
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 be a collection of countable infinitely many compact metric space and be a collection of countable infinite many continuous maps , . Then the pair is called a nonautonomous discrete dynamical system.
For any integer , we define a metric on as follows: for any two points ,
Fixing an integer and a positive number . A subset of is called - -separated if for any two distinct points we have . Denote the maximal cardinality of any -separated subset of by . A subset is called - -spanning for if for each , there is a such that . Denote the minimal cardinality of any -spanning subset of by .
The following result is trivial, so we omit its detail proof.
Lemma 5.1. Suppose that is a positive integer and is a nonempty subset of . Then for each , we have
For each let be a nonempty subset of . Then it follows immediately from Lemma 5.1 that
Given a nonautonomous discrete dynamical system , denoted by or for short the set of all orbit segments of length for each , i.e.,
Then the common limit in (21) by taking is defined to be the topological entropy of , written or for short if there is no confusion.
5.2 Partial entropy and bundle-like entropy
Let be a nonautonomous discrete dynamical system. A collection is said to be a cover of if each covers , respectively. We now define two entropies, partial entropy and bundle-like entropy, for relative to .
For any integer and , let denote the set of all orbit segments of length that end at some point , i.e.,
Put . Define the entropy by
which is called the partial entropy of relative to and written shortly by if there is no confusion.
Let . For any two elements, and of , denoted by , the usual Hausdorff metric between them is based upon metric of defined as before and by the maximum cardinality of any --separated subset of . Define the entropy by
which is called the bundle-like entropy of relative to and written shortly by if there is no confusion.
Also, we have the spanning set versions of definitions of and , respectively.
5.3 Some relationships between and
Theorem 5.2. Let be a nonautonomous discrete dynamical system, and be a cover of . Then we have
Proof. Note that for any cover of and any . Then the former inequality is obtained. Now we show the later one. If , then there is nothing to prove. Now assuming .
Fixing a sufficiently small and an integer , let be a --separated subset of with cardinality . For each , let be a --separated subset of with cardinality . Put . We claim that is a --spanning subset of .
In fact, for any , since is a --separated subset of with maximum cardinality and covers , there is an with and a such that . Then it follows that there is a such that . Also note that is a --separated subset of with maximum cardinality; there is a such that . Hence we have
This yields the claim that is a --spanning subset of . So we have , where denotes the cardinality of . Using the claim we have
Taking limits as the requirements of the related definitions of entropies establishes the desired inequality. This completes the proof.□
Let be a finite cover of a compact metric space consisting of open balls with radius less than some . Write and .
Proof. Note that . Then, by Theorem 5.2, we have the former equality. Now we show the later equality.
Clearly, for any , so we have
On the other hand, from the proof of Theorem 5.2, it follows that
for any integer , any sufficiently small and any . Noting that for any integer , then we have
Remark 5.4. The first equality of Theorem 5.3 is in fact a simpler version of Theorem 7.6 of  (a useful result for calculating the classical topological entropy) when restricting to the autonomous discrete dynamical systems.
Given a nonautonomous discrete dynamical system , when does for any cover of ? The following theorem gives an answer to this question.
Theorem 5.5. Let be a nonautonomous discrete dynamical system. Then for any cover of if the following conditions hold:
(1) For each integer , there exists such that whenever for .
(2) For each integer , every has an open neighborhood whose preimage is an union of disjoint open sets on each of which is a homeomorphism.
(3) for every monotonic decreasing sequence with , where each denotes the minimal cardinality of the open cover of consisting of open -ball for the compact metric space .
Proof. It suffices to show that for any cover of by Theorem 5.2. Let be the cover of in which each cover consisting to singletons of , i.e., . It is easy to see that for any cover of . So from Theorem 5.2, it follows that what we want to prove is .
For each , by condition (1), there exists a such that
for any whenever . Also, by condition (2) and the compactness of , there exists an such that the -ball about any point has preimage equals the union of disjoint open sets of diameter less than . Then we get a sequence . Furthermore, we can take such that is monotonic decreasing sequence and .
Now, given and , we want to find a point with and then . In fact, for , we can easily find a point with and , for . Let be the piece of with . Since , there is a unique point such that . Then we have
This argument shows that . Thus, by condition (3), we get
For any sufficiently small , there exists such that for any . Then we have and hence . This completes the proof.□
Several important entropy-like invariants based on the preimage structure for non-invertible maps have been defined and studied by some authors. In this chapter, we first further study the preimage entropy for topological dynamical system from the view of localization. We show that the topological entropy for an open cover can be characterized by pseudo-orbits (Theorems 2.3, 4.2, and 4.3). We also establish an inequality relating the topological entropy for open covers and several preimage entropy invariants, which is viewed as the local version of the Hurley’s inequality (Theorem 3.1). Finally, we discuss the topological entropy for nonautonomous discrete dynamical systems by introducing two entropy-like invariants called the partial entropy and bundle-like entropy. We establish some relationships among such two invariants and the topological entropy (Theorem 5.2, 5.3, and 5.5).
This work was carried out when Kesong Yan visited the Michigan State University. Kesong Yan sincerely appreciates the warm hospitality of Professor Huyi Hu. We thank the anonymous referees for their useful comments and helpful suggestions that improved the manuscript. The authors are supported by NNSF of China (11861010,11761012) and NSF for Distinguished Young Scholar of Guangxi Province (2018GXNSFFA281008). The first author is supported by the Cultivation Plan of Thousands of Young Backbone Teachers in Higher Education Institutions of Guangxi Province, Program for Innovative Team of Guangxi University of Finance and Economics, and Project of Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce Intelligent Information Processing (201801ZZ03).