Open access peer-reviewed chapter - ONLINE FIRST

Partial Entropy and Bundle-Like Entropy for Topological Dynamical Systems

By Kesong Yan and Fanping Zeng

Submitted: May 14th 2019Reviewed: August 5th 2019Published: September 23rd 2019

DOI: 10.5772/intechopen.89021

Downloaded: 45

Abstract

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.

Keywords

  • topological entropy
  • point entropy
  • pseudo-orbit
  • partial entropy
  • bundle-like entropy
  • 2000 Mathematics Subject Classification: Primary: 37B40
  • 37A35
  • 37B10
  • 37A05

1. Introduction

By a topological dynamical system, we mean a pair XT, where Xis a compact metric space with a metric d and T is a continuous surjective map from Xto 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 Tand the inverse mapping T1have the same topological entropy. However, when the map Tis not invertible, the “inverse” is set-valued, yielding the iterated preimage set Tnx=zX:Tnz=xof a point xXwhich 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, 9, 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, 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 [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 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 [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], 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 [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 fn:XnXn+1, 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 XTbe a topological dynamical system. A finite open cover of Xis a finite family of open sets whose union is X. Denoted by CXois the set of finite open covers of X. Given two open covers U,VCXo, Uis said to be finer thanV(UV) if each element of Uis contained in some element of V. Let UV=UV:UUVV. It is clear that UVUand UVV.

Let UCXo. For two nonnegative integers MN, denoted by UMN=n=MNTnU, where TnU=TnU:UUfor all positive integers n. For any KX, define NUKas the minimal cardinality of any subcovers of Uthat covers K. We write NUXsimply by NU. The topological entropy ofUwith respect toTis defined by

htopTU=limn1nlogNU0n1=infn11nlogNU0n1.E1

The topological entropy of T is

htopT=supUCXohtopTU.E2

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 XTbe a topological dynamical system. Given a nonempty subset K of X, for any ϵ>0and n, a subset E of K is called an nϵ-separated set of K if any xyEimplies dnxyϵ, where

dnxymax0in1dTixTiy.

Denote the maximal cardinality of any nϵ-separated subset of Kby snϵK. A subset Fof Kis called an nϵ-spanning set of K, if for any xK, there exists yFwith dnxy<ϵ. Denote the minimal cardinality of any nϵ-spanning set for Kby rnϵK.

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

Lemma 2.1. LetXTbe a topological dynamical system. For any subsetKofXand any integern1, we have the following properties:

  1. rnϵKsnϵKrnϵ/2Kfor allϵ>0.

  2. NU0n1KrnδKfor anynand anyUCXowith the Lebesgue number2δ.

  3. snϵKNU0n1Kfor anyUCXowithdiamU<ϵ.

By Lemma 2.1, we obtain directly the following result.

Theorem 2.2. (see [3, 4, 40]). LetXTbe a topological dynamical system. Then

htopT=limϵ0limsupn1nlogsnϵX=limϵ0limsupn1nlogrnϵX.

2.3 Topological entropy via pseudo-orbits

Let Xdbe a compact metric space. Denote Xnas the n-fold Cartesian product of X(n1). Fixing a positive number ϵ, a subset EXnis said to be nϵ-separated if for any two distinct points x˜=x0x1xn1,y˜=y0y1yn1E, there is a 0in1such that dxiyi>ϵ. By the compactness of X, any nϵ-separated set is finite. If ZXnis a nonempty subset, then we denote the maximal cardinality of any nϵ-separated subset of Zby snϵZ.

Let ZXnbe a nonempty subset. A subset FZis called nϵ-panning for Zif for each z˜=z0z1zn1Z, there is a y˜=y0y1yn1Fwith dziyi<ϵfor every 0in1. We denote the minimal cardinality of any nϵ-spanning subset of Zby rnϵZ.

For each positive integer n1, we let Ondenote the set of all orbit segments of length n, that is,

On=xTxTn1xXn:xX.

Note that a point w˜=xTxTn1xOnis uniquely determined by its initial point xX. Thus, we have

htopT=limϵ0limα0limsupn1nlogsnϵOn=limϵ0limα0limsupn1nlogrnϵOn.

Topological entropy has been characterized by Misiurewicz [36] and Barge and Swanson [37] in terms of growth rates of pseudo-orbits. Let XTbe a topological dynamical system. For α>0, an α-pseudo-orbit for Tof length nis a point x˜=x0x1xn1Xnwith the property that dTxj1xj<αfor all 1jn1. Let ΨnαXndenote all α-pseudo-orbits of length n. It was shown in [36, 37] that

htopT=limϵ0limα0limsupn1nlogsnϵΨnα=limϵ0limα0limsupn1nlogrnϵΨnα.

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 XTbe a topological dynamical system. For each integer n1and UCXo, we define an open cover Unof the product space Xnby

Un=U0×U1××Un1:UjUforeachj=01n1,

where

U0×U1××Un1=u0u1un1:ujUjforeachj=01n1.

Given α>0, it is not hard to see that an=NUnΨnαis a nonnegative sub-additive sequence, that is, an+man+amfor all positive integers nand m. The α-pseudo-orbit entropy of Uis then defined by

hΨTUα=limn1nlogNUnΨnα=infn11nlogNUnΨnα,E3

and the pseudo-orbit entropy of Uis defined by

hΨTU=limα0hΨTUα.E4

Theorem 2.3. LetXTbe a topological dynamical system. IfUCXo, then we have

htopTU=hΨTU.E5

Proof. To prove (5), it suffices to note that hΨTUα1hΨTUα2whenever α1<α2and inf0<α1NUnΨnα=NUnOn=NU0n1. Thus, we have

hΨTU=limα0hΨTUα=inf0<α0infn11nlogNUnΨnα=infn1inf0<α11nlogNUnΨnα=infn11nlogNU0n1=htopTU.

This completes the proof of the theorem. □

Remark 2.4. Combining (2) and (5), we have

htopT=supUCXohΨTU.

On the other hand, let us define hΨT=supUCXohΨTU, 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

hΨT=limϵ0limα0limsupn1nlogsnϵΨnα=limϵ0limα0limsupn1nlogrnϵΨnα.

So, it is in fact to give a simpler proof of Theorem 1 of [37] 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 fnx. In this section we describe two ways of doing this, which were introduced by Hurley in [1].

3.1 Preimage branch entropy

Let XTbe a topological dynamical system. Given xXlet Tnxdenote the tree of inverse images ofxup to ordern, which is defined by

Tnx=z0z1zn:zn=xandzj=Tzj1forall1jn.

Each z0z1znTnxis called a branch of Tnx, and its length is n. Note that every branch of Tnxends with x. Let Tn=xXTnx; we define a metric on Tnas follows: suppose that z˜=z0z1znand w˜=w0w1wnare two branches of the length n, the branch distance between them is defined as

dB,nz˜w˜=max0jndzjwj.

Let On=Tnx:xX. Given two trees Tnxand Tnyin On, the branch Hausdorff distance between them, dbHTnxTnyis the usual Hausdorff metric based upon dB,n; that is,

dbHTnxTny=maxmaxz˜Tnxminw˜TnydB,nz˜w˜maxw˜Tnyminz˜TnxdB,nz˜w˜.

Note that dbHTnxTny<ϵif and only if each branch of either tree is dB,nwithin ϵof at least one branch of the other tree. Two trees Tnxand Tnyin Onare said to be dbH-nϵ-separated if dbHTnxTny<ϵ, that is, there is a branch z˜of one of the trees with the property that dB,nz˜w˜>ϵfor all branches w˜of the other tree. Let tnϵdenote the maximum cardinality of any dbH-nϵ-separated sets of On. Define the entropy by

hbT=limϵ0limsupn1nlogtnϵ,

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:

hpT=supxXlimϵ0limsupn1nlogsnϵTnx=supxXlimϵ0limsupn1nlogrnϵTnx,hmT=limϵ0limsupn1nlogsupxXsnϵTnx=limϵ0limsupn1nlogsupxXrnϵTnx.

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

hmThtopThmT+hbT.

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 XTbe a topological dynamical system. Given UCXo, define two pointwise preimage entropies ofUwith respect to Tby

hpTU=supxXlimsupn1nlogNU0n1Tnx

and

hmTU=limsupn1nlogsupxXN(U0n1Tnx).

Theorem 3.1. (Local Hurley’s inequality). LetXTbe a topological dynamical system. IfUCXo, then we have

hpTUhmTUhtopTUhmTU+hbT.

Proof. It is obvious that NU0n1TnxNU0n1for every xXand every integer n1. So that hpTUhmTUhtopTU. Now we show the last inequality htopTUhmTU+hbT.

Let ϵ>0be a Lebesgue number of U. Fix n1, and let Ydenote a dbH-nϵ/3-separated set of Onwith cardinality tnϵ/3. Let Zdenote the set of all root points of trees in Y, where the root point of the tree Tnxis x. For each zZ, let VzUbe a subcover of U0n1with cardinality NU0n1Tnzthat covers Tnz, and let

V=zZVzU.

We claim that Vis an open cover of X.

In fact, let xXbe given and let w=fnx. Since Yis a dbH-nϵ/3-separated set of Onwith maximal cardinality, there is a tree TnyYsuch that dbHTnwTny<ϵ/3. Now we consider the branch w˜of Tnwbegins with x, i.e., w˜=xfxfn1xfnx=wTnw. Then there exists a branch y˜=y0y1yn=yTnysuch that dB,nw˜y˜<ϵ/3. This means that dTjy0Tjx<ϵ/3for each 0jn. Thus, there exists VVyUsuch that xV. This yields the claim that Vis an open cover of X. So that NU0n1V, where Vdenotes the cardinality of V. Using the claim, we have

NU0n1VzZVzU=zZNU0n1TnzZsupxXN(U0n1Tnx)=YsupxXN(U0n1Tnx)=tnϵ/3supxXN(U0n1Tnx).

So that,

htopTU=limn1nlogNU0n1limsupn1nlogtnϵ/3+logsupxXN(U0n1Tnx)limsupn1nlogtnϵ/3+limsupn1nlogsupxXN(U0n1Tnx)=limsupn1nlogtnϵ/3+hmTUhbT+hmTU.

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). LetXTbe a topological dynamical system. Then we have

hpThmThtopThmT+hbT.E6

Proof. It follows directly from Lemma 2.1 that

hpT=supUCXohpTUandhmT=supUCXohpTU.E7

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

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 XTbe a topological dynamical system. Recall that if α>0, then an α-pseudo-orbit x0x1xn1Xnis an approximate orbits segment for Tin the sense that dTxjxj+1<αfor all 0jn1.

For each xX, let ΨnαxXndenote the set of all α-pseudo-orbits of length nthat end at x, i.e., an element of Ψnαxis an α-pseudo-orbit y0y1yn1with yn1=x. It was shown in [1], (Propositions 4.2 and 4.3) that

htop=limϵ0limα0limsupn1nlogmaxxXs(nϵΨnαx)=supxXlimϵ0limα0limsupn1nlogsnϵΨnαx.E8

In either formula snϵΨnαxcan be replaced by rnϵΨnαx.

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 XTbe a topological dynamical system. For each integer n1, UCXo, and α>0, we define

NmaxnUα=maxxXNUnΨnαx.E9

Clearly,

NUnΨnαxNmaxnUαNUnΨnαE10

for every xX. In addition, by the compactness of X, there is some point yXsuch that

NUnΨnαy=NmaxnUα.

Lemma 4.1. LetXTbe a topological dynamical system andUCXo. Suppose thatε>0is a Lebesgue number ofUand0<α<ε/4. Then there is a constantK=Kαsuch that for everyn1,

NUnΨnαKNmaxnUα.E11

Proof. Let x1x2xKbe a finite α-dense subset of X, i.e., i=1nBxiα=X, where Bxiα=zX:dxiz<α. For each 1iK, let Vibe a subcover of Unthat covers Ψnαxiwith cardinality NUnΨnαxi. Define V=i=1KVi. Clearly, Vi=1KViKNmaxnUα. So, to complete the proof of the lemma, it suffices to show Vis a subcover of Unthat covers Ψnα.

In fact, let y˜=y0y1yn1be an α-pseudo-orbit. Since x1x2xKis an α-dense subset of X, there is some xisatisfying dTyn2xi<α. This implies z˜=z0z1zn2zn1=y0y1yn2xiis an α-pseudo-orbit ending at xi. Since Viis a subcover of Unthat covers Ψnαxi, there is some VVisuch that z˜V. Since zj=yjfor all 0jn2and ϵis the Lebesgue number of U, in order to show that y˜V, we need only to show that dyn1xi<ϵ/2; this is obviously, as dyn1xidyn1Tyn2+dTyn2xi<2α<ϵ/2.□

Theorem 4.2. LetXTbe a topological dynamical system. IfUCXo, then we have

htopTU=limα0limsupn1nlogsupxXN(UnΨnαx).E12

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

NmaxnUNUnΨnαKNmaxnU

for each fixed 0<α<ϵ/4and all n1, where ϵis a Lebesgue number of Uand K=Kαin Lemma 4.1 is independent of n. This implies that

limsupn1nlogNmaxnUα=limsupn1nlogNUnΨαE13

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

htopTU=limα0limn1nlogNUnΨnαbyTheorem2.3=limα0limsupn1nlogNUnΨnα=limα0limsupn1nlogNmaxnUαby4.6=limα0limsupn1nlogsupxXN(UnΨnαx)by4.1

This completes the proof.□

Theorem 4.3. LetXTbe a topological dynamical system. IfUCXo, then we have

htopTU=supxXlimα0limsupn1nlogNUnΨnαx.E14

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

htopTU=limα0limsupn1nlogsupxXN(UnΨnαx)supxXlimα0limsupn1nlogNUnΨnαx.

Now we start to prove the converse inequality.

Note that for the given n1and α>0, there is a point ynUαXsuch that

N(UnΨnαynUα=maxxXNUnΨnαx.

Taking a sequence of integers ni=niαsuch that

limsupn1nlogNUnΨnαynUα=limi1nilogNUniΨniαyniUα.

By restricting to a subsequence, we can assume without loss of generality that the sequence yiα=yniUαconverses to a limit qα.

Let ϵbe a Lebesgue number of U. If 0<β<ϵ/4and dyiαqα<β, then Vis a subcover of Unthat covers Ψnαyiαwhenever Vis a subcover of Unthat covers Ψnα+βqα. This implies that

NUnΨnα+βqαNUnΨnαyiαE15

whenever dyiαqα<β.

Now we choose a sequence αj0such that qαjconverges to some point qX. Similar to the proof as above we have

N(UnΨnαj+2βqNUnΨnαjqαjE16

whenever dqαjq<β. Combining inequalities (15) and (16), one has

NUnΨnαj+2βqNUnΨnαjyiαjE17

whenever dyiαjqαj<βand dqαjq<β. If jis a fixed integer with dqαjq<β, then (17) holds for all sufficiently large integers i. Thus,

limsupn1nlogNUnΨnαj+2βqlimsupi1nilogNUniΨniαj+2βqlimi1nilogNUniΨniαjyiαj=limsupn1nlogmaxxXN(UnΨnαjx)).E18

Now let jand use the fact that both sides (18) are nonincreasing as αdecreases to conclude that

limsupn1nlogNUnΨn3βqlimj0limsupn1nlogNUnΨnαj+2βqlimjlimsupn1nlogmaxxXN(UnΨnαjx))=infα>0limsupn1nlogmaxxXN(UnΨnαx)=limα0limsupn1nlogmaxxXN(UnΨnαx)).E19

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

htopTU=limα0limsupn1nlogmaxxXN(UnΨnαx))infβ>0limsupn1nlogNUnΨn3βq=infα>0limsupn1nlogNUnΨnαq=limα0limsupn1nlogNUnΨnαqsupxXlimα0limsupn1nlogNUnΨnαx.E20

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 Xbe a collection of countable infinitely many compact metric space Xidiand Fbe a collection of countable infinite many continuous maps fi:XiXi+1, i=1,2,. Then the pair XFis called a nonautonomous discrete dynamical system.

For any integer n1, we define a metric d˜non i=1nXias follows: for any two points x˜n=x1x2xn,y˜n=y1y2yni=1nXi,

d˜nx˜ny˜n=max1indixiyi.

Fixing an integer n1and a positive number ϵ. A subset Zof i=1nXiis called d˜n-nϵ-separated if for any two distinct points x˜n,y˜nZwe have d˜nx˜ny˜n>ϵ. Denote the maximal cardinality of any d˜n-separated subset of Zby snϵZ. A subset WZis called d˜n-nϵ-spanning forZif for each z˜nZ, there is a w˜nWsuch that d˜nz˜nw˜n<ϵ. Denote the minimal cardinality of any d˜n-spanning subset of Zby rnϵZ.

The following result is trivial, so we omit its detail proof.

Lemma 5.1. Suppose thatnis a positive integer andZis a nonempty subset ofi=1nXi. Then for eachϵ>0, we have

rnϵZsnϵZrnϵ/2Z.

For each n1let Znbe a nonempty subset of i=1nXi. Then it follows immediately from Lemma 5.1 that

limϵ0limsupn1nlogrnϵZn=limϵ0limsupn1nlogsnϵZn.E21

Given a nonautonomous discrete dynamical system XF, denoted by On,For Onfor short the set of all orbit segments of length nfor each n1, i.e.,

On=On,Fx1x2xn:x1X1andxi+1=fixii=12n1.

Then the common limit in (21) by taking Zn=Onis defined to be the topological entropy ofXF, written htopXFor htopFfor short if there is no confusion.

5.2 Partial entropy and bundle-like entropy

Let XFbe a nonautonomous discrete dynamical system. A collection P=Pi:i1is said to be a cover of Xif each Picovers Xi, respectively. We now define two entropies, partial entropy and bundle-like entropy, for XFrelative to P.

For any integer n1and DPn, let WnDi=1nXidenote the set of all orbit segments of length that end at some point xnD, i.e.,

WnD=x1x2xnOn:xnD.

Put smax,Pnnϵ=supDPnsnϵWnD. Define the entropy by

hp,PXF=limϵ0limsupn1nlogsmax,Pnnϵ,

which is called the partial entropy ofXFrelative toPand written shortly by hp,PFif there is no confusion.

Let On,Pn=WnD:DPn. For any two elements, WnDand WnEof On,Pn, denoted by dHWnDWnE, the usual Hausdorff metric between them is based upon metric d˜nof i=1nXidefined as before and by snϵOn,Pnthe maximum cardinality of any dH-nϵ-separated subset of On,Pn. Define the entropy by

hb,PXF=limϵ0limsupn1nlogsnϵOn,Pn,

which is called the bundle-like entropy ofXFrelative toPand written shortly by hb,PFif there is no confusion.

Also, we have the spanning set versions of definitions of hp,PFand hb,PF, respectively.

5.3 Some relationships between htopFand hp,PF

Theorem 5.2. LetXFbe a nonautonomous discrete dynamical system, andP=Pi:i1be a cover ofX. Then we have

hp,PFhtopFhb,PF+hp,PF.

Proof. Note that smax,PnnϵsnϵOnfor any cover Pof Xand any ϵ>0. Then the former inequality is obtained. Now we show the later one. If hb,PF=, then there is nothing to prove. Now assuming hb,PF<.

Fixing a sufficiently small ϵ>0and an integer n1, let Ybe a dH-nϵ-separated subset of On,Pnwith cardinality snϵOn,Pn. For each WnDY, let MDbe a d˜n-nϵ-separated subset of WnDwith cardinality snϵWnD. Put M=WnDYMD. We claim that Mis a d˜n-n3ϵ-spanning subset of On.

In fact, for any x=x1x2xnOn, since Yis a dH-nϵ-separated subset of On,Pnwith maximum cardinality and Pncovers Xn, there is an EPnwith xnEand a WnDYsuch that dHWnDWnEϵ. Then it follows that there is a y=y1y2ynWnDsuch that d˜nxyϵ. Also note that MDis a d˜n-nϵ-separated subset of WnDwith maximum cardinality; there is a zMDsuch that d˜nyzϵ. Hence we have

d˜nxzd˜nxy+d˜nyz<3ϵ.

This yields the claim that Mis a d˜n-nϵ-spanning subset of On. So we have rn3ϵOnM, where Mdenotes the cardinality of M. Using the claim we have

rn3ϵOnMYmaxMD:WnDYsnϵOn,Pnsmax,Pnnϵ.

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

Let Pδbe a finite cover of a compact metric space Xconsisting of open balls with radius less than some δ>0. Write FX=fi:fi:XXiscontinousi1and PXδ=PδPδ.

Theorem 5.3.

htopFX=hp,PXδFX=limϵ0limδ0limsupn1nlogsmax,Pδnϵ.

Proof. Note that limn1nlogPδ=0. Then, by Theorem 5.2, we have the former equality. Now we show the later equality.

Clearly, snϵOnsmax,PXδnϵfor any δ>0, so we have

limsupnlogsnϵOnlimδ0limsupn1nlogsmax,PXδnϵ.

This implies

htopFXlimε0limδ0limsupn1nlogsmax,PXδnε.E22

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

rn3ϵOnsnϵOn,Pδsmax,PXδnϵ

for any integer n1, any sufficiently small ϵ>0and any δ>0. Noting that snϵOn,PδPδfor any integer n1, then we have

limsupn1nlogrn3ϵOnlimδ0limsupn1nlogsmax,PXδnϵ.

This implies

htopFXlimϵ0limδ0limsupn1nlogsmax,PXδnϵ.E23

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 X, when does htopF=hp,PFfor any cover Pof X? The following theorem gives an answer to this question.

Theorem 5.5. LetXFbe a nonautonomous discrete dynamical system. ThenhtopF=hp,PFfor any coverPofXif the following conditions hold:

(1) For each integeri1, there existsδi>0such thatdi+1fixfiydixywheneverdixyδiforx,yXi.

(2) For each integeri1, everyxXi+1has an open neighborhoodUxwhose preimagefi1Uxis an union of disjoint open sets on each of whichfiis a homeomorphism.

(3) limsupn1nlogNϵnXn=0for every monotonic decreasing sequenceϵnwithlimnϵ=0, where eachNϵnXndenotes the minimal cardinality of the open cover ofXconsisting of openϵn-ball for the compact metric spaceXn.

Proof. It suffices to show that hb,PF=0for any cover Pof Xby Theorem 5.2. Let Pmax=Pi,max:i1be the cover of Xin which each Pi,maxcover Xiconsisting to singletons of Xi, i.e., Pi,max=z:zXi. It is easy to see that hb,PFhb,PmaxFfor any cover Pof X. So from Theorem 5.2, it follows that what we want to prove is hb,PmaxF=0.

For each n2, by condition (1), there exists a δn1>0such that

dnfn1xfn1ydn1xy

for any x,yXn1whenever dn1xyδn1. Also, by condition (2) and the compactness of Xn, there exists an ϵn>0such that the ϵn-ball Bxnϵnabout any point xnXnhas preimage fn11Bxnϵequals the union of disjoint open sets of diameter less than δn1. Then we get a sequence ϵn. Furthermore, we can take ϵnsuch that ϵnis monotonic decreasing sequence and limnϵn=0.

Now, given ynXnand x˜=x1x2xnWnBynϵn, we want to find a point y˜=y1y2ynOnwith d˜nx˜y˜=dnxnynand then d˜nx˜y˜<ϵn. In fact, for 1<k<n, we can easily find a point ykyni=knXiwith djxjyjϵjand dj+1xj+1yj+1djxjyj, for j=n1,n2,,k. Let Vbe the piece of fk11Bxkϵkwith xk1V. Since ykBxkϵk, there is a unique point yk1Vfk11yksuch that dk1xk1yk1<δk1. Then we have

dk1xk1yk1dkxkykdnxnyn<ϵn<ϵk1.

This argument shows that rnϵnOn,\,maxNϵnXn. Thus, by condition (3), we get

limsupn1nlogrnϵnOn,\,maxlimsupn1nlogNϵnXn=0.

For any sufficiently small ϵ>0, there exists N>0such that ϵn<ϵfor any nN. Then we have rnϵOn,Pn,maxrnϵnOn,Pn,maxand hence hb,PmaxF=0. This completes the proof.□

6. Conclusion

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).

Acknowledgments

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).

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Kesong Yan and Fanping Zeng (September 23rd 2019). Partial Entropy and Bundle-Like Entropy for Topological Dynamical Systems [Online First], IntechOpen, DOI: 10.5772/intechopen.89021. Available from:

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