\r\n\tBut hydrogenolysis has been also successfully applied in biorefineries to convert glycerol and other hydroxylated compounds to fine chemicals. Nowadays, a plenty of systems have been developed to accomplish each kind of processes and, actually, an immense ensemble of different routes are available on demand. Thus, this book will focus on research trends of the hydrogenation and hydrogenolysis covering several area (homogenous to heterogeneous catalysis, kinetic and computational studies, catalysts synthesis and characterization) offering a comprehensive \r\n\tview on this topics. \r\n\t
",isbn:null,printIsbn:null,pdfIsbn:null,doi:null,price:0,priceEur:null,priceUsd:null,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"30a3f86af67d090ca828bad17f314f19",bookSignature:"Dr. Marco Frediani and Dr. Mattia Bartoli",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/8276.jpg",keywords:"Hydrogenation, Hydrogenolysis, Catalysis, Homogeneous, Heterogeneous, Supported, Biphasic, Enzyme Based Catalysts, Electrolitical, Computational, Metal-Free, Batch",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"July 23rd 2018",dateEndSecondStepPublish:"August 13th 2018",dateEndThirdStepPublish:"October 12th 2018",dateEndFourthStepPublish:"December 31st 2018",dateEndFifthStepPublish:"March 1st 2019",remainingDaysToSecondStep:"2 years",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"53209",title:"Dr.",name:"Marco",middleName:null,surname:"Frediani",slug:"marco-frediani",fullName:"Marco Frediani",profilePictureURL:"https://mts.intechopen.com/storage/users/53209/images/system/53209.jpg",biography:"Marco Frediani is Professor of Industrial Chemistry at the University of Florence, Department of Chemistry “Ugo Schiff,” Italy. 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\n
1. Introduction
\n
By a topological dynamical system, we mean a pair \n\n\nX\nT\n\n\n, where \n\nX\n\n is a compact metric space with a metric d and T is a continuous surjective map from \n\nX\n\n 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.
\n
When a considered mapping T is invertible, it is well-known that \n\nT\n\n and the inverse mapping \n\n\nT\n\n−\n1\n\n\n\n have the same topological entropy. However, when the map \n\nT\n\n is not invertible, the “inverse” is set-valued, yielding the iterated preimage set \n\n\nT\n\n−\nn\n\n\n\nx\n\n=\n\n\nz\n∈\nX\n:\n\nT\nn\n\nz\n=\nx\n\n\n\n of a point \n\nx\n∈\nX\n\n 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.
\n
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].
\n
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.
\n
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.
\n
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).
\n
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 \n\n\nf\nn\n\n:\n\nX\nn\n\n→\n\nX\n\nn\n+\n1\n\n\n\n, which defined on a sequence on compact metric spaces (Xn\n, dn\n). 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).
\n
\n
\n
2. Topological entropy and pseudo-orbits
\n
\n
2.1 Topological entropy via open covers
\n
Topological entropy was defined originally by Adler et al. [2] for continuous maps on compact topological spaces. Let \n\n\nX\nT\n\n\n be a topological dynamical system. A finite open cover of \n\nX\n\n is a finite family of open sets whose union is \n\nX\n\n. Denoted by \n\n\nC\nX\no\n\n\n is the set of finite open covers of \n\nX\n\n. Given two open covers \n\nU\n,\nV\n∈\n\nC\nX\no\n\n\n, \n\nU\n\n is said to be finer than\n\n\nV\n\n (\n\nU\n≽\nV\n\n) if each element of \n\nU\n\n is contained in some element of \n\nV\n\n. Let \n\nU\n∨\nV\n=\n\n\nU\n∩\nV\n:\nU\n∈\nU\n\n\nV\n∈\nV\n\n\n\n. It is clear that \n\nU\n∨\nV\n≽\nU\n\n and \n\nU\n∨\nV\n≽\nV\n\n.
\n
Let \n\nU\n∈\n\nC\nX\no\n\n\n. For two nonnegative integers \n\nM\n≤\nN\n\n, denoted by \n\n\nU\nM\nN\n\n=\n\n∨\n\nn\n=\nM\n\nN\n\n\nT\n\n−\nn\n\n\nU\n\n, where \n\n\nT\n\n−\nn\n\n\nU\n=\n\n\n\nT\n\n−\nn\n\n\n\nU\n\n:\nU\n∈\nU\n\n\n\n for all positive integers \n\nn\n\n. For any \n\nK\n⊂\nX\n\n, define \n\nN\n\nU\nK\n\n\n as the minimal cardinality of any subcovers of \n\nU\n\n that covers \n\nK\n\n. We write \n\nN\n\nU\nX\n\n\n simply by \n\nN\n\nU\n\n\n. The topological entropy of\n\n\nU\n\n\nwith respect to\n\n\nT\n\n is defined by
2.2 Separated sets, spanning sets, and topological entropy
\n
In this subsection, we recall two equivalent definitions, which are given by Dinaburg [3] and Bowen [4]. Let \n\n\nX\nT\n\n\n be a topological dynamical system. Given a nonempty subset K of X, for any \n\nϵ\n>\n0\n\n and \n\nn\n∈\nℕ\n\n, a subset E of K is called an \n\n\nn\nϵ\n\n\n\n-separated set of K if any \n\nx\n≠\ny\n∈\nE\n\n implies \n\n\nd\nn\n\n\nx\ny\n\n≥\nϵ\n\n, where
Denote the maximal cardinality of any \n\n\nn\nϵ\n\n\n-separated subset of \n\nK\n\n by \n\ns\n\nn\nϵ\nK\n\n\n. A subset \n\nF\n\n of \n\nK\n\n is called an \n\n\nn\nϵ\n\n\n\n-spanning set of \n\nK\n\n, if for any \n\nx\n∈\nK\n\n, there exists \n\ny\n∈\nF\n\n with \n\n\nd\nn\n\n\nx\ny\n\n<\nϵ\n\n. Denote the minimal cardinality of any \n\n\nn\nϵ\n\n\n-spanning set for \n\nK\n\n by \n\nr\n\nn\nϵ\nK\n\n\n.
\n
The following lemma is well-known, and its proof is not difficult, so we omit its detail proof.
\n
\nLemma 2.1. Let\n\n\n\nX\nT\n\n\n\nbe a topological dynamical system. For any subset\n\n\nK\n\n\nof\n\n\nX\n\n\nand any integer\n\n\nn\n≥\n1\n\n\n, we have the following properties:
Let \n\n\nX\nd\n\n\n be a compact metric space. Denote \n\n\nX\nn\n\n\n as the \n\nn\n\n-fold Cartesian product of \n\nX\n\n (\n\nn\n≥\n1\n)\n\n. Fixing a positive number \n\nϵ\n\n, a subset \n\nE\n⊂\n\nX\nn\n\n\n is said to be \n\n\nn\nϵ\n\n\n\n-separated if for any two distinct points \n\n\nx\n˜\n\n=\n\n\nx\n0\n\n\nx\n1\n\n⋯\n\nx\n\nn\n−\n1\n\n\n\n,\n\ny\n˜\n\n=\n\n\ny\n0\n\n\ny\n1\n\n⋯\n\ny\n\nn\n−\n1\n\n\n\n∈\nE\n\n, there is a \n\n0\n≤\ni\n≤\nn\n−\n1\n\n such that \n\nd\n\n\nx\ni\n\n\ny\ni\n\n\n>\nϵ\n\n. By the compactness of \n\nX\n\n, any \n\n\nn\nϵ\n\n\n-separated set is finite. If \n\nZ\n⊂\n\nX\nn\n\n\n is a nonempty subset, then we denote the maximal cardinality of any \n\n\nn\nϵ\n\n\n-separated subset of \n\nZ\n\n by \n\ns\n\nn\nϵ\nZ\n\n\n.
\n
Let \n\nZ\n⊂\n\nX\nn\n\n\n be a nonempty subset. A subset \n\nF\n⊂\nZ\n\n is called \n\n\nn\nϵ\n\n\n\n-panning for \n\nZ\n\n if for each \n\n\nz\n˜\n\n=\n\n\nz\n0\n\n\nz\n1\n\n⋯\n\nz\n\nn\n−\n1\n\n\n\n∈\nZ\n\n, there is a \n\n\ny\n˜\n\n=\n\n\ny\n0\n\n\ny\n1\n\n⋯\n\ny\n\nn\n−\n1\n\n\n\n∈\nF\n\n with \n\nd\n\n\nz\ni\n\n\ny\ni\n\n\n<\nϵ\n\n for every \n\n0\n≤\ni\n≤\nn\n−\n1\n\n. We denote the minimal cardinality of any \n\n\nn\nϵ\n\n\n-spanning subset of \n\nZ\n\n by \n\nr\n\nn\nϵ\nZ\n\n\n.
\n
For each positive integer \n\nn\n≥\n1\n\n, we let \n\n\nO\nn\n\n\n denote the set of all orbit segments of length \n\nn\n\n, that is,
Note that a point \n\n\nw\n˜\n\n=\n\nx\nTx\n⋯\n\n\nT\n\nn\n−\n1\n\n\nx\n\n\n∈\n\nO\nn\n\n\n is uniquely determined by its initial point \n\nx\n∈\nX\n\n. Thus, we have
Topological entropy has been characterized by Misiurewicz [36] and Barge and Swanson [37] in terms of growth rates of pseudo-orbits. Let \n\n\nX\nT\n\n\n be a topological dynamical system. For \n\nα\n>\n0\n\n, an \n\nα\n\n\n-pseudo-orbit for \n\nT\n\n of length \n\nn\n\n is a point \n\n\nx\n˜\n\n=\n\n\nx\n0\n\n\nx\n1\n\n⋯\n\nx\n\nn\n−\n1\n\n\n\n∈\n\nX\nn\n\n\n with the property that \n\nd\n\n\nT\n\n\nx\n\nj\n−\n1\n\n\n\n\n\nx\nj\n\n\n<\nα\n\n for all \n\n1\n≤\nj\n≤\nn\n−\n1\n\n. Let \n\n\nΨ\nn\n\n\nα\n\n⊂\n\nX\nn\n\n\n denote all \n\nα\n\n-pseudo-orbits of length \n\nn\n\n. 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 \n\n\nX\nT\n\n\n be a topological dynamical system. For each integer \n\nn\n≥\n1\n\n and \n\nU\n∈\n\nC\nX\no\n\n\n, we define an open cover \n\n\nU\nn\n\n\n of the product space \n\n\nX\nn\n\n\n by
Given \n\nα\n>\n0\n\n, it is not hard to see that \n\n\na\nn\n\n=\nN\n\n\nU\nn\n\n\n\nΨ\nn\n\n\nα\n\n\n\n\n is a nonnegative sub-additive sequence, that is, \n\n\na\n\nn\n+\nm\n\n\n≤\n\na\nn\n\n+\n\na\nm\n\n\n for all positive integers \n\nn\n\n and \n\nm\n\n. The \n\nα\n\n\n-pseudo-orbit entropy of \n\nU\n\n is then defined by
\nProof. To prove (5), it suffices to note that \n\n\nh\nΨ\n\n\nT\nU\n\nα\n1\n\n\n≤\n\nh\nΨ\n\n\nT\nU\n\nα\n2\n\n\n\n whenever \n\n\nα\n1\n\n<\n\nα\n2\n\n\n and \n\n\ninf\n\n0\n<\nα\n≤\n1\n\n\n\nN\n\n\nU\nn\n\n\n\nΨ\nn\n\n\nα\n\n\n\n=\nN\n\n\nU\nn\n\n\nO\nn\n\n\n=\nN\n\n\nU\n0\n\nn\n−\n1\n\n\n\n\n. Thus, we have
On the other hand, let us define \n\n\nh\nΨ\n\n\nT\n\n=\n\nsup\n\nU\n∈\n\nC\nX\no\n\n\n\n\nh\nΨ\n\n\nT\nU\n\n\n, which is called the pseudo-orbit entropy of \n\nT\n\n. 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 [37] by Theorem 2.3.
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3. Pointwise preimage entropies for open covers and local Hurley’s inequality
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When T is not invertible, one can ask about growth rates of inverse images \n\n\nf\n\n−\nn\n\n\n\nx\n\n\n. In this section we describe two ways of doing this, which were introduced by Hurley in [1].
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3.1 Preimage branch entropy
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Let \n\n\nX\nT\n\n\n be a topological dynamical system. Given \n\nx\n∈\nX\n\n let \n\n\nT\nn\n\n\nx\n\n\n denote the tree of inverse images of\n\n\nx\n\n\nup to order\n\n\nn\n\n, which is defined by
Each \n\n\n\nz\n0\n\n\nz\n1\n\n⋯\n\nz\nn\n\n\n∈\n\nT\nn\n\n\nx\n\n\n is called a branch of \n\n\nT\nn\n\n\nx\n\n\n, and its length is \n\nn\n\n. Note that every branch of \n\n\nT\nn\n\n\nx\n\n\n ends with \n\nx\n\n. Let \n\n\nT\nn\n\n=\n\n∪\n\nx\n∈\nX\n\n\n\nT\nn\n\n\nx\n\n\n; we define a metric on \n\n\nT\nn\n\n\n as follows: suppose that \n\n\nz\n˜\n\n=\n\n\nz\n0\n\n\nz\n1\n\n⋯\n\nz\nn\n\n\n\n and \n\n\nw\n˜\n\n=\n\n\nw\n0\n\n\nw\n1\n\n⋯\n\nw\nn\n\n\n\n are two branches of the length \n\nn\n\n, the branch distance between them is defined as
Let \n\n\nO\nn\n\n=\n\n\n\nT\nn\n\n\nx\n\n:\nx\n∈\nX\n\n\n\n. Given two trees \n\n\nT\nn\n\n\nx\n\n\n and \n\n\nT\nn\n\n\ny\n\n\n in \n\n\nO\nn\n\n\n, the branch Hausdorff distance between them, \n\n\nd\nbH\n\n\n\n\nT\nn\n\n\nx\n\n\n\n\nT\nn\n\n\ny\n\n\n\n\n is the usual Hausdorff metric based upon \n\n\nd\n\nB\n,\nn\n\n\n\n; that is,
Note that \n\n\nd\nbH\n\n\n\n\nT\nn\n\n\nx\n\n\n\n\nT\nn\n\n\ny\n\n\n\n<\nϵ\n\n if and only if each branch of either tree is \n\n\nd\n\nB\n,\nn\n\n\n\n within \n\nϵ\n\n of at least one branch of the other tree. Two trees \n\n\nT\nn\n\n\nx\n\n\n and \n\n\nT\nn\n\n\ny\n\n\n in \n\n\nO\nn\n\n\n are said to be \n\n\nd\nbH\n\n\n\n-\n\n\n\nn\nϵ\n\n\n\n-separated if \n\n\nd\nbH\n\n\n\n\nT\nn\n\n\nx\n\n\n\n\nT\nn\n\n\ny\n\n\n\n<\nϵ\n\n, that is, there is a branch \n\n\nz\n˜\n\n\n of one of the trees with the property that \n\n\nd\n\nB\n,\nn\n\n\n\n\nz\n˜\n\n\nw\n˜\n\n\n>\nϵ\n\n for all branches \n\n\nw\n˜\n\n\n of the other tree. Let \n\nt\n\nn\nϵ\n\n\n denote the maximum cardinality of any \n\n\nd\nbH\n\n\n-\n\n\nn\nϵ\n\n\n-separated sets of \n\n\nO\nn\n\n\n. Define the entropy by
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:
It is clear that \n\n\nh\np\n\n\nT\n\n≤\n\nh\nm\n\n\nT\n\n\n, and in [18] the authors constructed an example for which \n\n\nh\np\n\n\nT\n\n<\n\nh\nm\n\n\nT\n\n\n. In addition, Hurley established the following relationships among preimage branch entropy, pointwise preimage entropy, and topological entropy (see [1], Theorem 3.1):
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.
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Let \n\n\nX\nT\n\n\n be a topological dynamical system. Given \n\nU\n∈\n\nC\nX\no\n\n\n, define two pointwise preimage entropies of\n\n\nU\n\n with respect to \n\nT\n\n by
\nProof. It is obvious that \n\nN\n\n\nU\n0\n\nn\n−\n1\n\n\n\n\nT\n\n−\nn\n\n\n\nx\n\n\n\n≤\nN\n\n\nU\n0\n\nn\n−\n1\n\n\n\n\n for every \n\nx\n∈\nX\n\n and every integer \n\nn\n≥\n1\n\n. So that \n\n\nh\np\n\n\nT\nU\n\n≤\n\nh\nm\n\n\nT\nU\n\n≤\n\nh\ntop\n\n\nT\nU\n\n\n. Now we show the last inequality \n\n\nh\ntop\n\n\nT\nU\n\n≤\n\nh\nm\n\n\nT\nU\n\n+\n\nh\nb\n\n\nT\n\n\n.
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Let \n\nϵ\n>\n0\n\n be a Lebesgue number of \n\nU\n\n. Fix \n\nn\n≥\n1\n\n, and let \n\nY\n\n denote a \n\n\nd\nbH\n\n\n-\n\n\nn\n\nϵ\n/\n3\n\n\n\n-separated set of \n\n\nO\nn\n\n\n with cardinality \n\nt\n\nn\n\nϵ\n/\n3\n\n\n\n. Let \n\nZ\n\n denote the set of all root points of trees in \n\nY\n\n, where the root point of the tree \n\n\nT\nn\n\n\nx\n\n\n is \n\nx\n\n. For each \n\nz\n∈\nZ\n\n, let \n\nV\n\nz\nU\n\n\n be a subcover of \n\n\nU\n0\n\nn\n−\n1\n\n\n\n with cardinality \n\nN\n\n\nU\n0\n\nn\n−\n1\n\n\n\n\nT\n\n−\nn\n\n\n\nz\n\n\n\n\n that covers \n\n\nT\n\n−\nn\n\n\n\nz\n\n\n, and let
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\n\nV\n=\n\n⋃\n\nz\n∈\nZ\n\n\nV\n\nz\nU\n\n.\n\n
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We claim that \n\nV\n\n is an open cover of \n\nX\n\n.
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In fact, let \n\nx\n∈\nX\n\n be given and let \n\nw\n=\n\nf\nn\n\n\nx\n\n\n. Since \n\nY\n\n is a \n\n\nd\nbH\n\n\n-\n\n\nn\n\nϵ\n/\n3\n\n\n\n-separated set of \n\n\nO\nn\n\n\n with maximal cardinality, there is a tree \n\n\nT\nn\n\n\ny\n\n∈\nY\n\n such that \n\n\nd\nbH\n\n\n\n\nT\nn\n\n\nw\n\n\n\n\nT\nn\n\n\ny\n\n\n\n<\nϵ\n/\n3\n\n. Now we consider the branch \n\n\nw\n˜\n\n\n of \n\n\nT\nn\n\n\nw\n\n\n begins with \n\nx\n\n, i.e., \n\n\nw\n˜\n\n=\n\nx\n\nf\n\nx\n\n\n⋯\n\n\nf\n\nn\n−\n1\n\n\n\nx\n\n\n\n\nf\nn\n\n\nx\n\n=\nw\n\n\n∈\n\nT\nn\n\n\nw\n\n\n. Then there exists a branch \n\n\ny\n˜\n\n=\n\n\ny\n0\n\n\ny\n1\n\n⋯\n\n\ny\nn\n\n=\ny\n\n\n∈\n\nT\nn\n\n\ny\n\n\n such that \n\n\nd\n\nB\n,\nn\n\n\n\n\nw\n˜\n\n\ny\n˜\n\n\n<\nϵ\n/\n3\n\n. This means that \n\nd\n\n\n\nT\nj\n\n\n\ny\n0\n\n\n\n\n\nT\nj\n\n\nx\n\n\n\n<\nϵ\n/\n3\n\n for each \n\n0\n≤\nj\n≤\nn\n\n. Thus, there exists \n\nV\n∈\nV\n\ny\nU\n\n\n such that \n\nx\n∈\nV\n\n. This yields the claim that \n\nV\n\n is an open cover of \n\nX\n\n. So that \n\nN\n\n\nU\n0\n\nn\n−\n1\n\n\n\n≤\n∣\nV\n∣\n\n, where \n\n∣\nV\n∣\n\n denotes the cardinality of \n\nV\n\n. Using the claim, we have
Thus, combining (2), (7), and Theorem 3.1 gives (6).□
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4. Point entropy for open covers with pseudo-orbits
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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 \n\n\nX\nT\n\n\n be a topological dynamical system. Recall that if \n\nα\n>\n0\n\n, then an \n\nα\n\n-pseudo-orbit \n\n\n\nx\n0\n\n\nx\n1\n\n…\n\nx\n\nn\n−\n1\n\n\n\n∈\n\nX\nn\n\n\n is an approximate orbits segment for \n\nT\n\n in the sense that \n\nd\n\n\nT\n\n\nx\nj\n\n\n\n\nx\n\nj\n+\n1\n\n\n\n<\nα\n\n for all \n\n0\n≤\nj\n≤\nn\n−\n1\n\n.
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For each \n\nx\n∈\nX\n\n, let \n\n\nΨ\nn\n\n\nα\nx\n\n⊂\n\nX\nn\n\n\n denote the set of all \n\nα\n\n-pseudo-orbits of length \n\nn\n\n that end at \n\nx\n\n, i.e., an element of \n\n\nΨ\nn\n\n\nα\nx\n\n\n is an \n\nα\n\n-pseudo-orbit \n\n\n\ny\n0\n\n\ny\n1\n\n⋯\n\ny\n\nn\n−\n1\n\n\n\n\n with \n\n\ny\n\nn\n−\n1\n\n\n=\nx\n\n. It was shown in [1], (Propositions 4.2 and 4.3) that
In either formula \n\ns\n\nn\nϵ\n\n\nΨ\nn\n\n\nα\nx\n\n\n\n\n can be replaced by \n\nr\n\nn\nϵ\n\n\nΨ\nn\n\n\nα\nx\n\n\n\n\n.
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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.
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Let \n\n\nX\nT\n\n\n be a topological dynamical system. For each integer \n\nn\n≥\n1\n\n, \n\nU\n∈\n\nC\nX\no\n\n\n, and \n\nα\n>\n0\n\n, we define
\nLemma 4.1. Let\n\n\n\nX\nT\n\n\n\nbe a topological dynamical system and\n\n\nU\n∈\n\nC\nX\no\n\n\n\n. Suppose that\n\n\nε\n>\n0\n\n\nis a Lebesgue number of\n\n\nU\n\n\nand\n\n\n0\n<\nα\n<\nε\n/\n4\n\n\n. Then there is a constant\n\n\nK\n=\nK\n\nα\n\n\n\nsuch that for every\n\n\nn\n≥\n1\n\n,
\nProof. Let \n\n\n\nx\n1\n\n\nx\n2\n\n⋯\n\nx\nK\n\n\n\n be a finite \n\nα\n\n-dense subset of \n\nX\n\n, i.e., \n\n\n⋃\n\ni\n=\n1\n\nn\n\nB\n\n\nx\ni\n\nα\n\n=\nX\n\n, where \n\nB\n\n\nx\ni\n\nα\n\n=\n\n\nz\n∈\nX\n:\nd\n\n\nx\ni\n\nz\n\n<\nα\n\n\n\n. For each \n\n1\n≤\ni\n≤\nK\n\n, let \n\n\nV\ni\n\n\n be a subcover of \n\n\nU\nn\n\n\n that covers \n\n\nΨ\nn\n\n\nα\n\nx\ni\n\n\n\n with cardinality \n\nN\n\n\nU\nn\n\n\n\nΨ\nn\n\n\nα\n\nx\ni\n\n\n\n\n\n. Define \n\nV\n=\n\n⋃\n\ni\n=\n1\n\nK\n\n\nV\ni\n\n\n. Clearly, \n\n∣\nV\n∣\n≤\n\n∑\n\ni\n=\n1\n\nK\n\n∣\n\nV\ni\n\n∣\n≤\nK\n⋅\n\nN\nmax\n\n\nn\nU\nα\n\n\n. So, to complete the proof of the lemma, it suffices to show \n\nV\n\n is a subcover of \n\n\nU\nn\n\n\n that covers \n\n\nΨ\nn\n\n\nα\n\n\n.
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In fact, let \n\n\ny\n˜\n\n=\n\n\ny\n0\n\n\ny\n1\n\n⋯\n\ny\n\nn\n−\n1\n\n\n\n\n be an \n\nα\n\n-pseudo-orbit. Since \n\n\n\nx\n1\n\n\nx\n2\n\n⋯\n\nx\nK\n\n\n\n is an \n\nα\n\n-dense subset of \n\nX\n\n, there is some \n\n\nx\ni\n\n\n satisfying \n\nd\n\n\nT\n\n\ny\n\nn\n−\n2\n\n\n\n\n\nx\ni\n\n\n<\nα\n\n. This implies \n\n\nz\n˜\n\n=\n\n\nz\n0\n\n\nz\n1\n\n⋯\n\nz\n\nn\n−\n2\n\n\n\nz\n\nn\n−\n1\n\n\n\n=\n\n\ny\n0\n\n\ny\n1\n\n⋯\n\ny\n\nn\n−\n2\n\n\n\nx\ni\n\n\n\n is an \n\nα\n\n-pseudo-orbit ending at \n\n\nx\ni\n\n\n. Since \n\n\nV\ni\n\n\n is a subcover of \n\n\nU\nn\n\n\n that covers \n\n\nΨ\nn\n\n\nα\n\nx\ni\n\n\n\n, there is some \n\nV\n∈\n\nV\ni\n\n\n such that \n\n\nz\n˜\n\n∈\nV\n\n. Since \n\n\nz\nj\n\n=\n\ny\nj\n\n\n for all \n\n0\n≤\nj\n≤\nn\n−\n2\n\n and \n\nϵ\n\n is the Lebesgue number of \n\nU\n\n, in order to show that \n\n\ny\n˜\n\n∈\nV\n\n, we need only to show that \n\nd\n\n\ny\n\nn\n−\n1\n\n\n\nx\ni\n\n\n<\nϵ\n/\n2\n\n; this is obviously, as \n\nd\n\n\ny\n\nn\n−\n1\n\n\n\nx\ni\n\n\n≤\nd\n\n\ny\n\nn\n−\n1\n\n\n\nT\n\n\ny\n\nn\n−\n2\n\n\n\n\n\n+\nd\n\n\nT\n\n\ny\n\nn\n−\n2\n\n\n\n\n\nx\ni\n\n\n<\n2\nα\n<\nϵ\n/\n2\n\n.□
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\nTheorem 4.2. Let\n\n\n\nX\nT\n\n\n\nbe a topological dynamical system. If\n\n\nU\n∈\n\nC\nX\no\n\n\n\n, then we have\n
for each fixed \n\n0\n<\nα\n<\nϵ\n/\n4\n\n and all \n\nn\n≥\n1\n\n, where \n\nϵ\n\n is a Lebesgue number of \n\nU\n\n and \n\nK\n=\nK\n\nα\n\n\n in Lemma 4.1 is independent of \n\nn\n\n. This implies that
By restricting to a subsequence, we can assume without loss of generality that the sequence \n\n\ny\ni\n\n\nα\n\n=\ny\n\n\nn\ni\n\nU\nα\n\n\n converses to a limit \n\nq\n\nα\n\n\n.
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Let \n\nϵ\n\n be a Lebesgue number of \n\nU\n\n. If \n\n0\n<\nβ\n<\nϵ\n/\n4\n\n and \n\nd\n\n\n\ny\ni\n\n\nα\n\n\n\nq\n\nα\n\n\n\n<\nβ\n\n, then \n\nV\n\n is a subcover of \n\n\nU\nn\n\n\n that covers \n\n\nΨ\nn\n\n\nα\n\n\ny\ni\n\n\nα\n\n\n\n\n whenever \n\nV\n\n is a subcover of \n\n\nU\nn\n\n\n that covers \n\n\nΨ\nn\n\n\n\nα\n+\nβ\n\n\nq\n\nα\n\n\n\n\n. This implies that
Now we choose a sequence \n\n\nα\nj\n\n→\n0\n\n such that \n\nq\n\n\nα\nj\n\n\n\n converges to some point \n\nq\n∈\nX\n\n. Similar to the proof as above we have
whenever \n\nd\n\n\n\ny\ni\n\n\n\nα\nj\n\n\n\n\nq\n\n\nα\nj\n\n\n\n\n<\nβ\n\n and \n\nd\n\n\nq\n\n\nα\nj\n\n\n\nq\n\n<\nβ\n\n. If \n\nj\n\n is a fixed integer with \n\nd\n\n\nq\n\n\nα\nj\n\n\n\nq\n\n<\nβ\n\n, then (17) holds for all sufficiently large integers \n\ni\n\n. Thus,
5. Partial entropy and bundle-like entropy for nonautonomous discrete dynamical systems
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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].
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5.1 Topological entropy for nonautonomous discrete dynamical systems
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Let \n\nX\n\n be a collection of countable infinitely many compact metric space \n\n\n\nX\ni\n\n\nd\ni\n\n\n\n and \n\nF\n\n be a collection of countable infinite many continuous maps \n\n\nf\ni\n\n:\n\nX\ni\n\n→\n\nX\n\ni\n+\n1\n\n\n\n, \n\ni\n=\n1\n,\n2\n,\n⋯\n\n. Then the pair \n\n\nX\nF\n\n\n is called a nonautonomous discrete dynamical system.
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For any integer \n\nn\n≥\n1\n\n, we define a metric \n\n\n\nd\n˜\n\nn\n\n\n on \n\n\n∏\n\ni\n=\n1\n\nn\n\n\n\nX\ni\n\n\n as follows: for any two points \n\n\n\nx\n˜\n\nn\n\n=\n\n\nx\n1\n\n\nx\n2\n\n⋯\n\nx\nn\n\n\n,\n\n\ny\n˜\n\nn\n\n=\n\n\ny\n1\n\n\ny\n2\n\n⋯\n\ny\nn\n\n\n∈\n\n∏\n\ni\n=\n1\n\nn\n\n\n\nX\ni\n\n\n,
Fixing an integer \n\nn\n≥\n1\n\n and a positive number \n\nϵ\n\n. A subset \n\nZ\n\n of \n\n\n∏\n\ni\n=\n1\n\nn\n\n\nX\ni\n\n\n is called \n\n\n\nd\n˜\n\nn\n\n\n\n-\n\n\n\nn\nϵ\n\n\n\n-separated if for any two distinct points \n\n\n\nx\n˜\n\nn\n\n,\n\n\ny\n˜\n\nn\n\n∈\nZ\n\n we have \n\n\n\nd\n˜\n\nn\n\n\n\n\nx\n˜\n\nn\n\n\n\ny\n˜\n\nn\n\n\n>\nϵ\n\n. Denote the maximal cardinality of any \n\n\n\nd\n˜\n\nn\n\n\n-separated subset of \n\nZ\n\n by \n\ns\n\nn\nϵ\nZ\n\n\n. A subset \n\nW\n⊂\nZ\n\n is called \n\n\n\nd\n˜\n\nn\n\n\n\n-\n\n\n\nn\nϵ\n\n\n\n-spanning for\n\n\nZ\n\n if for each \n\n\n\nz\n˜\n\nn\n\n∈\nZ\n\n, there is a \n\n\n\nw\n˜\n\nn\n\n∈\nW\n\n such that \n\n\n\nd\n˜\n\nn\n\n\n\n\nz\n˜\n\nn\n\n\n\nw\n˜\n\nn\n\n\n<\nϵ\n\n. Denote the minimal cardinality of any \n\n\n\nd\n˜\n\nn\n\n\n-spanning subset of \n\nZ\n\n by \n\nr\n\nn\nϵ\nZ\n\n\n.
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The following result is trivial, so we omit its detail proof.
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\nLemma 5.1. Suppose that\n\n\nn\n\n\nis a positive integer and\n\n\nZ\n\n\nis a nonempty subset of\n\n\n\n∏\n\ni\n=\n1\n\nn\n\n\nX\ni\n\n\n\n. Then for each\n\n\nϵ\n>\n0\n\n\n, we have\n
For each \n\nn\n≥\n1\n\n let \n\n\nZ\nn\n\n\n be a nonempty subset of \n\n\n∏\n\ni\n=\n1\n\nn\n\n\nX\ni\n\n\n. Then it follows immediately from Lemma 5.1 that
Given a nonautonomous discrete dynamical system \n\n\nX\nF\n\n\n, denoted by \n\n\nO\n\nn\n,\nF\n\n\n\n or \n\n\nO\nn\n\n\n for short the set of all orbit segments of length \n\nn\n\n for each \n\nn\n≥\n1\n\n, i.e.,
Then the common limit in (21) by taking \n\n\nZ\nn\n\n=\n\nO\nn\n\n\n is defined to be the topological entropy of\n\n\n\nX\nF\n\n\n, written \n\n\nh\ntop\n\n\nX\nF\n\n\n or \n\n\nh\ntop\n\n\nF\n\n\n for short if there is no confusion.
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5.2 Partial entropy and bundle-like entropy
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Let \n\n\nX\nF\n\n\n be a nonautonomous discrete dynamical system. A collection \n\nP\n=\n\n\n\nP\ni\n\n:\ni\n≥\n1\n\n\n\n is said to be a cover of \n\nX\n\n if each \n\n\nP\ni\n\n\n covers \n\n\nX\ni\n\n\n, respectively. We now define two entropies, partial entropy and bundle-like entropy, for \n\n\nX\nF\n\n\n relative to \n\nP\n\n.
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For any integer \n\nn\n≥\n1\n\n and \n\nD\n∈\n\nP\nn\n\n\n, let \n\n\nW\nn\n\n\nD\n\n⊂\n\n∏\n\ni\n=\n1\n\nn\n\n\nX\ni\n\n\n denote the set of all orbit segments of length that end at some point \n\n\nx\nn\n\n∈\nD\n\n, i.e.,
which is called the partial entropy of\n\n\n\nX\nF\n\n\n\nrelative to\n\n\nP\n\n and written shortly by \n\n\nh\n\np\n,\nP\n\n\n\nF\n\n\n if there is no confusion.
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Let \n\n\nO\n\nn\n,\n\nP\nn\n\n\n\n=\n\n\n\nW\nn\n\n\nD\n\n:\nD\n∈\n\nP\nn\n\n\n\n\n. For any two elements, \n\n\nW\nn\n\n\nD\n\n\n and \n\n\nW\nn\n\n\nE\n\n\n of \n\n\nO\n\nn\n,\n\nP\nn\n\n\n\n\n, denoted by \n\n\nd\nH\n\n\n\n\nW\nn\n\n\nD\n\n\n\n\nW\nn\n\n\nE\n\n\n\n\n, the usual Hausdorff metric between them is based upon metric \n\n\n\nd\n˜\n\nn\n\n\n of \n\n\n∏\n\ni\n=\n1\n\nn\n\n\n\nX\ni\n\n\n defined as before and by \n\ns\n\nn\nϵ\n\nO\n\nn\n,\n\nP\nn\n\n\n\n\n\n the maximum cardinality of any \n\n\nd\nH\n\n\n-\n\n\nn\nϵ\n\n\n-separated subset of \n\n\nO\n\nn\n,\n\nP\nn\n\n\n\n\n. Define the entropy by
which is called the bundle-like entropy of\n\n\n\nX\nF\n\n\n\nrelative to\n\n\nP\n\n and written shortly by \n\n\nh\n\nb\n,\nP\n\n\n\nF\n\n\n if there is no confusion.
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Also, we have the spanning set versions of definitions of \n\n\nh\n\np\n,\nP\n\n\n\nF\n\n\n and \n\n\nh\n\nb\n,\nP\n\n\n\nF\n\n\n, respectively.
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5.3 Some relationships between \n\n\nh\ntop\n\n\nF\n\n\n and \n\n\nh\n\np\n,\nP\n\n\n\nF\n\n\n\n
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\nTheorem 5.2. Let\n\n\n\nX\nF\n\n\n\nbe a nonautonomous discrete dynamical system, and\n\n\nP\n=\n\n\n\nP\ni\n\n:\ni\n≥\n1\n\n\n\n\nbe a cover of\n\n\nX\n\n\n. Then we have\n
\nProof. Note that \n\n\ns\n\nmax\n,\n\nP\nn\n\n\n\n\nn\nϵ\n\n≤\ns\n\nn\nϵ\n\nO\nn\n\n\n\n for any cover \n\nP\n\n of \n\nX\n\n and any \n\nϵ\n>\n0\n\n. Then the former inequality is obtained. Now we show the later one. If \n\n\nh\n\nb\n,\nP\n\n\n\nF\n\n=\n∞\n\n, then there is nothing to prove. Now assuming \n\n\nh\n\nb\n,\nP\n\n\n\nF\n\n<\n∞\n\n.
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Fixing a sufficiently small \n\nϵ\n>\n0\n\n and an integer \n\nn\n≥\n1\n\n, let \n\nY\n\n be a \n\n\nd\nH\n\n\n-\n\n\nn\nϵ\n\n\n-separated subset of \n\n\nO\n\nn\n,\n\nP\nn\n\n\n\n\n with cardinality \n\ns\n\nn\nϵ\n\nO\n\nn\n,\n\nP\nn\n\n\n\n\n\n. For each \n\n\nW\nn\n\n\nD\n\n∈\nY\n\n, let \n\nM\n\nD\n\n\n be a \n\n\n\nd\n˜\n\nn\n\n\n-\n\n\nn\nϵ\n\n\n-separated subset of \n\n\nW\nn\n\n\nD\n\n\n with cardinality \n\ns\n\nn\nϵ\n\n\nW\nn\n\n\nD\n\n\n\n\n. Put \n\nM\n=\n\n∪\n\n\nW\nn\n\n\nD\n\n∈\nY\n\n\nM\n\nD\n\n\n. We claim that \n\nM\n\n is a \n\n\n\nd\n˜\n\nn\n\n\n-\n\n\nn\n\n3\nϵ\n\n\n\n-spanning subset of \n\n\nO\nn\n\n\n.
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In fact, for any \n\nx\n=\n\n\nx\n1\n\n\nx\n2\n\n⋯\n\nx\nn\n\n\n∈\n\nO\nn\n\n\n, since \n\nY\n\n is a \n\n\nd\nH\n\n\n-\n\n\nn\nϵ\n\n\n-separated subset of \n\n\nO\n\nn\n,\n\nP\nn\n\n\n\n\n with maximum cardinality and \n\n\nP\nn\n\n\n covers \n\n\nX\nn\n\n\n, there is an \n\nE\n∈\n\nP\nn\n\n\n with \n\n\nx\nn\n\n∈\nE\n\n and a \n\n\nW\nn\n\n\nD\n\n∈\nY\n\n such that \n\n\nd\nH\n\n\n\n\nW\nn\n\n\nD\n\n\n\n\nW\nn\n\n\nE\n\n\n\n≤\nϵ\n\n. Then it follows that there is a \n\ny\n=\n\n\ny\n1\n\n\ny\n2\n\n⋯\n\ny\nn\n\n\n∈\n\nW\nn\n\n\nD\n\n\n such that \n\n\n\nd\n˜\n\nn\n\n\nx\ny\n\n≤\nϵ\n\n. Also note that \n\nM\n\nD\n\n\n is a \n\n\n\nd\n˜\n\nn\n\n\n-\n\n\nn\nϵ\n\n\n-separated subset of \n\n\nW\nn\n\n\nD\n\n\n with maximum cardinality; there is a \n\nz\n∈\nM\n\nD\n\n\n such that \n\n\n\nd\n˜\n\nn\n\n\ny\nz\n\n≤\nϵ\n\n. Hence we have
This yields the claim that \n\nM\n\n is a \n\n\n\nd\n˜\n\nn\n\n\n-\n\n\nn\nϵ\n\n\n-spanning subset of \n\n\nO\nn\n\n\n. So we have \n\nr\n\nn\n\n3\nϵ\n\n\nO\nn\n\n\n≤\n∣\nM\n∣\n\n, where \n\nM\n\n denotes the cardinality of \n\nM\n\n. Using the claim we have
Taking limits as the requirements of the related definitions of entropies establishes the desired inequality. This completes the proof.□
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Let \n\nP\n\nδ\n\n\n be a finite cover of a compact metric space \n\nX\n\n consisting of open balls with radius less than some \n\nδ\n>\n0\n\n. Write \n\n\nF\nX\n\n=\n\n\n\nf\ni\n\n:\n\nf\ni\n\n:\nX\n→\nX\n\nis\n\ncontinous\n\n\ni\n≥\n1\n\n\n\n and \n\n\nP\nX\n\n\nδ\n\n=\n\n\nP\n\nδ\n\n\n\nP\n\nδ\n\n\n⋯\n\n\n.
\nProof. Note that \n\n\nlim\n\nn\n→\n∞\n\n\n\n1\nn\n\nlog\n∣\nP\n\nδ\n\n∣\n=\n0\n\n. Then, by Theorem 5.2, we have the former equality. Now we show the later equality.
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Clearly, \n\ns\n\nn\nϵ\n\nO\nn\n\n\n≥\n\ns\n\nmax\n,\n\nP\nX\n\n\n\n\nδ\n\n\nn\nϵ\n\n\n for any \n\nδ\n>\n0\n\n, so we have
for any integer \n\nn\n≥\n1\n\n, any sufficiently small \n\nϵ\n>\n0\n\n and any \n\nδ\n>\n0\n\n. Noting that \n\ns\n\nn\nϵ\n\nO\n\nn\n,\nP\n\nδ\n\n\n\n\n≤\n∣\nP\n\nδ\n\n∣\n\n for any integer \n\nn\n≥\n1\n\n, then we have
Thus, combining (22) and (23) gets the later equality. This completes the proof.□
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\nRemark 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.
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Given a nonautonomous discrete dynamical system \n\n\nX\nℱ\n\n\n, when does \n\n\nh\ntop\n\n\nF\n\n=\n\nh\n\np\n,\nP\n\n\n\nF\n\n\n for any cover \n\nP\n\n of \n\nX\n\n? The following theorem gives an answer to this question.
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\nTheorem 5.5. Let\n\n\n\nX\nF\n\n\n\nbe a nonautonomous discrete dynamical system. Then\n\n\n\nh\ntop\n\n\nF\n\n=\n\nh\n\np\n,\nP\n\n\n\nF\n\n\n\nfor any cover\n\n\nP\n\n\nof\n\n\nX\n\n\nif the following conditions hold:
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(1) For each integer\n\n\ni\n≥\n1\n\n\n, there exists\n\n\n\nδ\ni\n\n>\n0\n\n\nsuch that\n\n\n\nd\n\ni\n+\n1\n\n\n\n\n\nf\ni\n\n\nx\n\n\n\n\nf\ni\n\n\ny\n\n\n\n≥\n\nd\ni\n\n\nx\ny\n\n\n\nwhenever\n\n\n\nd\ni\n\n\nx\ny\n\n≤\n\nδ\ni\n\n\n\nfor\n\n\nx\n,\ny\n∈\n\nX\ni\n\n\n.
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(2) For each integer\n\n\ni\n≥\n1\n\n\n, every\n\n\nx\n∈\n\nX\n\ni\n+\n1\n\n\n\n\nhas an open neighborhood\n\n\n\nU\nx\n\n\n\nwhose preimage\n\n\n\nf\ni\n\n−\n1\n\n\n\n\nU\nx\n\n\n\n\nis an union of disjoint open sets on each of which\n\n\n\nf\ni\n\n\n\nis a homeomorphism.\n
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(3) \n\nlim\n\n\nsup\n\nn\n→\n∞\n\n\n\n1\nn\n\nlog\nN\n\n\nϵ\nn\n\n\nX\nn\n\n\n=\n0\n\n\nfor every monotonic decreasing sequence\n\n\n\n\nϵ\nn\n\n\n\n\nwith\n\n\n\nlim\n\nn\n→\n∞\n\n\nϵ\n=\n0\n\n\n, where each\n\n\nN\n\n\nϵ\nn\n\n\nX\nn\n\n\n\n\ndenotes the minimal cardinality of the open cover of\n\n\nX\n\n\nconsisting of open\n\n\n\nϵ\nn\n\n\n\n-ball for the compact metric space\n\n\n\nX\nn\n\n\n.
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\nProof. It suffices to show that \n\n\nh\n\nb\n,\nP\n\n\n\nF\n\n=\n0\n\n for any cover \n\nP\n\n of \n\nX\n\n by Theorem 5.2. Let \n\n\nP\nmax\n\n=\n\n\n\nP\n\ni\n,\nmax\n\n\n:\ni\n≥\n1\n\n\n\n be the cover of \n\nX\n\n in which each \n\n\nP\n\ni\n,\nmax\n\n\n\n cover \n\n\nX\ni\n\n\n consisting to singletons of \n\n\nX\ni\n\n\n, i.e., \n\n\nP\n\ni\n,\nmax\n\n\n=\n\n\n\nz\n\n:\nz\n∈\n\nX\ni\n\n\n\n\n. It is easy to see that \n\n\nh\n\nb\n,\nP\n\n\n\nF\n\n≤\n\nh\n\nb\n,\n\nP\nmax\n\n\n\n\nF\n\n\n for any cover \n\nP\n\n of \n\nX\n\n. So from Theorem 5.2, it follows that what we want to prove is \n\n\nh\n\nb\n,\n\nP\nmax\n\n\n\n\nF\n\n=\n0\n\n.
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For each \n\nn\n≥\n2\n\n, by condition (1), there exists a \n\n\nδ\n\nn\n−\n1\n\n\n>\n0\n\n such that
for any \n\nx\n,\ny\n∈\n\nX\n\nn\n−\n1\n\n\n\n whenever \n\n\nd\n\nn\n−\n1\n\n\n\nx\ny\n\n≤\n\nδ\n\nn\n−\n1\n\n\n\n. Also, by condition (2) and the compactness of \n\n\nX\nn\n\n\n, there exists an \n\n\nϵ\nn\n\n>\n0\n\n such that the \n\n\nϵ\nn\n\n\n-ball \n\nB\n\n\nx\nn\n\n\nϵ\nn\n\n\n\n about any point \n\n\nx\nn\n\n∈\n\nX\nn\n\n\n has preimage \n\n\nf\n\nn\n−\n1\n\n\n−\n1\n\n\n\n\nB\n\n\nx\nn\n\nϵ\n\n\n\n\n equals the union of disjoint open sets of diameter less than \n\n\nδ\n\nn\n−\n1\n\n\n\n. Then we get a sequence \n\n\n\nϵ\nn\n\n\n\n. Furthermore, we can take \n\n\nϵ\nn\n\n\n such that \n\n\n\nϵ\nn\n\n\n\n is monotonic decreasing sequence and \n\n\nlim\n\nn\n→\n∞\n\n\n\n\nϵ\nn\n\n=\n0\n\n.
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Now, given \n\n\ny\nn\n\n∈\n\nX\nn\n\n\n and \n\n\nx\n˜\n\n=\n\n\nx\n1\n\n\nx\n2\n\n⋯\n\nx\nn\n\n\n∈\n\nW\nn\n\n\n\nB\n\n\n\ny\nn\n\n\nϵ\nn\n\n\n\n\n\n, we want to find a point \n\n\ny\n˜\n\n=\n\n\ny\n1\n\n\ny\n2\n\n⋯\n\ny\nn\n\n\n∈\n\nO\nn\n\n\n with \n\n\n\nd\n˜\n\nn\n\n\n\nx\n˜\n\n\ny\n˜\n\n\n=\n\nd\nn\n\n\n\nx\nn\n\n\ny\nn\n\n\n\n and then \n\n\n\nd\n˜\n\nn\n\n\n\nx\n˜\n\n\ny\n˜\n\n\n<\n\nϵ\nn\n\n\n. In fact, for \n\n1\n<\nk\n<\nn\n\n, we can easily find a point \n\n\n\ny\nk\n\n⋯\n\ny\nn\n\n\n∈\n\n∏\n\ni\n=\nk\n\nn\n\n\nX\ni\n\n\n with \n\n\nd\nj\n\n\n\nx\nj\n\n\ny\nj\n\n\n≤\n\nϵ\nj\n\n\n and \n\n\nd\n\nj\n+\n1\n\n\n\n\nx\n\nj\n+\n1\n\n\n\ny\n\nj\n+\n1\n\n\n\n≥\n\nd\nj\n\n\n\nx\nj\n\n\ny\nj\n\n\n\n, for \n\nj\n=\nn\n−\n1\n,\nn\n−\n2\n,\n⋯\n,\nk\n\n. Let \n\nV\n\n be the piece of \n\n\nf\n\nk\n−\n1\n\n\n−\n1\n\n\n\n\nB\n\n\nx\nk\n\n\nϵ\nk\n\n\n\n\n\n with \n\n\nx\n\nk\n−\n1\n\n\n∈\nV\n\n. Since \n\n\ny\nk\n\n∈\nB\n\n\nx\nk\n\n\nϵ\nk\n\n\n\n, there is a unique point \n\n\ny\n\nk\n−\n1\n\n\n∈\nV\n∩\n\nf\n\nk\n−\n1\n\n\n−\n1\n\n\n\n\ny\nk\n\n\n\n such that \n\n\nd\n\nk\n−\n1\n\n\n\n\nx\n\nk\n−\n1\n\n\n\ny\n\nk\n−\n1\n\n\n\n<\n\nδ\n\nk\n−\n1\n\n\n\n. Then we have
For any sufficiently small \n\nϵ\n>\n0\n\n, there exists \n\nN\n>\n0\n\n such that \n\n\nϵ\nn\n\n<\nϵ\n\n for any \n\nn\n≥\nN\n\n. Then we have \n\nr\n\nn\nϵ\n\nO\n\nn\n,\n\nP\n\nn\n,\nmax\n\n\n\n\n\n≤\nr\n\nn\n\nϵ\nn\n\n\nO\n\nn\n,\n\nP\n\nn\n,\nmax\n\n\n\n\n\n\n and hence \n\n\nh\n\nb\n,\n\nP\nmax\n\n\n\n\nF\n\n=\n0\n\n. This completes the proof.□
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6. Conclusion
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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).
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Acknowledgments
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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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\n',keywords:"topological entropy, point entropy, pseudo-orbit, partial entropy, bundle-like entropy, 2000 Mathematics Subject Classification: Primary: 37B40, 37A35, 37B10, 37A05",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/69020.pdf",chapterXML:"https://mts.intechopen.com/source/xml/69020.xml",downloadPdfUrl:"/chapter/pdf-download/69020",previewPdfUrl:"/chapter/pdf-preview/69020",totalDownloads:203,totalViews:0,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:"May 14th 2019",dateReviewed:"August 5th 2019",datePrePublished:"September 23rd 2019",datePublished:null,dateFinished:null,readingETA:"0",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.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/69020",risUrl:"/chapter/ris/69020",book:{slug:"dynamical-systems-theory"},signatures:"Kesong Yan and Fanping Zeng",authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Topological entropy and pseudo-orbits",level:"1"},{id:"sec_2_2",title:"2.1 Topological entropy via open covers",level:"2"},{id:"sec_3_2",title:"2.2 Separated sets, spanning sets, and topological entropy",level:"2"},{id:"sec_4_2",title:"2.3 Topological entropy via pseudo-orbits",level:"2"},{id:"sec_6",title:"3. Pointwise preimage entropies for open covers and local Hurley’s inequality",level:"1"},{id:"sec_6_2",title:"3.1 Preimage branch entropy",level:"2"},{id:"sec_7_2",title:"3.2 Pointwise preimage entropies",level:"2"},{id:"sec_8_2",title:"3.3 Local Hurley’s inequality",level:"2"},{id:"sec_10",title:"4. Point entropy for open covers with pseudo-orbits",level:"1"},{id:"sec_11",title:"5. Partial entropy and bundle-like entropy for nonautonomous discrete dynamical systems",level:"1"},{id:"sec_11_2",title:"5.1 Topological entropy for nonautonomous discrete dynamical systems",level:"2"},{id:"sec_12_2",title:"5.2 Partial entropy and bundle-like entropy",level:"2"},{id:"sec_13_2",title:"5.3 Some relationships between \n\n\nh\ntop\n\n\nF\n\n\n and \n\n\nh\n\np\n,\nP\n\n\n\nF\n\n\n\n",level:"2"},{id:"sec_15",title:"6. Conclusion",level:"1"},{id:"sec_16",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'\nHurley H. On topological entropy of maps. Ergodic Theory and Dynamical Systems. 1995;15:557-568. DOI: 10.1017/S014338570000852X\n'},{id:"B2",body:'\nAdler RL, Konheim AG, McAndrew MH. Topological entropy. Transactions of the American Mathematical Society. 1965;114:309-319. DOI: 10.1090/S0002-9947-1965-0175106-9\n'},{id:"B3",body:'\nDinaburg EI. On the relations among various entropy characteristic of dynamical systems. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 1971;35:324-366\n'},{id:"B4",body:'\nBowen R. Entropy for group endomorphisms and homogeneous spaces. Transactions of the American Mathematical Society. 1971;153:401-414. DOI: 10.1090/S0002-9947-1971-0274707-X\n'},{id:"B5",body:'\nLangevin R, Walczak P. Entropie d’une dynamique [Entropy of a dynamic]. Comptes Rendus de I’ Académie des Sciences-Series I-Mathematics. 1991;312:141-144. (in French)\n'},{id:"B6",body:'\nGhys E, Langevin R, Walczak P. Entroie geoetrique des feuilletages [Geometric entropy of foliations]. Acta Math. 1988;160:105-142 (in French). DOI: 10.1007/BF02392274\n'},{id:"B7",body:'\nLangevin R, Przytycki F. Entropie de I’image inverse d’une application. [Entropy of the inverse of a mapping]. Bulletin de la Société Mathématique de France. 1992;120:237-250. DOI: 10.24033/bsmf.2185 (in French)\n'},{id:"B8",body:'\nNitecki Z, Przytycki F. Preimage entropy for mappings. International Journal of Bifurcation and Chaos. 1999;9:1815-1843. DOI: 10.1142/S0218127499001309\n'},{id:"B9",body:'\nBiś A. Entropies of a semigroup of maps. Discrete and Continuous Dynamical Systems. 2004;11:639-648. DOI: 10.3934/dcds.2004.11.639\n'},{id:"B10",body:'\nNitecki Z. Topological entropy and the preimage structure of maps. Real Analysis Exchange. 2003;29:9-43. DOI: 10.14321/realanalexch.29.1.0009\n'},{id:"B11",body:'\nYan K, Zeng F. Variational principles of partial pre-image entropy and conditional pre-image entropy. Scientia Sinica Mathematica. 2019;49:681-698. DOI: 10.1360/SCM-2016-0381. (in Chinese)\n'},{id:"B12",body:'\nCheng W. Relations among conditional entropy, topological entropy and pointwise preimage entropy [thesis]. East Lansing: Michigan State University; 2004\n'},{id:"B13",body:'\nCheng W. Pre-image entropy. Ergodic Theory and Dynamical Systems. 2005;25:1091-1113. DOI: 10.1017/S0143385704000240\n'},{id:"B14",body:'\nCheng W. Forward generator for preimage entropy. Pacific Journal of Mathematics. 2006;223:5-16. DOI: 10.2140/pjm.2006.223.5\n'},{id:"B15",body:'\nCheng W. Two-point pre-image entropy. Discrete and Continuous Dynamical Systems. 2007;17:107-119. DOI: 10.3934/dcds.2007.17.107\n'},{id:"B16",body:'\nCheng W. On preimage entropy of a sequential mapping. Open Systems and Information Dynamics. 2008;15:345-357. DOI: 10.1142/S1230161208000237\n'},{id:"B17",body:'\nCheng W. Pre-image entropy for free semigroup actions. Chaos, Solitons and Fractals. 2016;91:286-290. DOI: 10.1016/j.chaos.2016.06.011\n'},{id:"B18",body:'\nFiebig D, Fiebig U, Nitecki Z. Entropy and preimage sets. Ergodic Theory and Dynamical Systems. 2003;23:1785-1806. DOI: 10.1017/S0143385703000221\n'},{id:"B19",body:'\nYan K, Zeng F, Zhang G. Equilibrium states for pre-image pressure. Nonlinear Dynamics and Systems Theory. 2010;10:409-416\n'},{id:"B20",body:'\nYan K, Zeng F, Zhang G. Differentiability properties of the pre-image pressure. Discrete Dynamics in Nature and Society; 2012. DOI: 10.1155/2012/951691, Article ID 951691, 14 pages\n'},{id:"B21",body:'\nZeng F, Yan K, Zhang G. Pre-image pressure and invariant measures. Ergodic Theory and Dynamical Systems. 2007;27:1037-1052. DOI: 10.1017/S0143385706000812\n'},{id:"B22",body:'\nZhu Y. Preimage entropy for random dynamical systems. Discrete and Continuous Dynamical Systems. 2007;18:829-851. DOI: 10.3934/dcds.2007.18.829\n'},{id:"B23",body:'\nZhu Y, Li Z, Li X. Preimage pressure for random transformations. Ergodic Theory and Dynamical Systems. 2009;29:1669-1687. DOI: 10.1017/S0143385708000758\n'},{id:"B24",body:'\nBlanchard F. Fully positive topological entropy and topological mixing. In: Walters P, editor. Symbolic Dynamics and its Applications. Contemporary Mathematics. Vol. 135. Providence: RI: American Mathematical Society; 1992. pp. 95-105. DOI: 10.1090/conm/135\n'},{id:"B25",body:'\nBlanchard F, Lacroix Y. Zero-entropy factors of topological flows. Proceedings of the American Mathematical Society. 1993;119:985-992. DOI: 10.1090/S0002-9939-1993-1155593-2\n'},{id:"B26",body:'\nBlanchard F. A disjointness theorem involving topological entropy. Bulletin de la Société Mathématique de France. 1993;121:565-578. DOI: 10.24033/bsmf.2216\n'},{id:"B27",body:'\nBlanchard F, Host B, Maass A, Martinez S, Rudolph D. Entropy pairs for a measure. Ergodic Theory and Dynamical Systems. 1995;15:621-632. DOI: 10.1017/S0143385700008579\n'},{id:"B28",body:'\nGlasner E, Weiss B. Topological entropy of extensions. In: Petersen Karl E, editor. Ergodic Theory and its Connections with Harmonic Analysis. Cambridge: Cambridge University Press; 1995. pp. 299-307. DOI: 10.1017/CBO9780511574818.011\n'},{id:"B29",body:'\nHuang W, Ye X. A local variational relation and applications. Israel Journal of Mathematics. 2006;151:237-279. DOI: 10.1007/BF02777364\n'},{id:"B30",body:'\nGlasner E, Ye X. Local entropy theory. Ergodic Theory and Dynamical Systems. 2009;29:321-356. DOI: 10.1017/S0143385708080309\n'},{id:"B31",body:'\nBowen R. Equilibrium States and the Ergodic Theory of Axiom Diffeomorphisms, Lecture Notes in Math. Second revised ed. Vol. 470. New York: Springer-Verlag, Berlin; 2008. 75 p. DOI: 10.1007/BFb0081284\n'},{id:"B32",body:'\nConley C. Isolated Invariant Sets and the Morse Index. Vol. 38. Providence: CBMS Regional Conference Series in Mathematics, American Mathematical Society; 1978. 89 p. DOI: 10.1090/cbms/038\n'},{id:"B33",body:'\nHammel SM, Yorke JA, Grebogi C. Do numerical orbits of chaotic dynamical processes represent true orbits? Journal of Complexity. 1987;3:136-145. DOI: 10.1016/0885-064X(87)90024-0\n'},{id:"B34",body:'\nHammel SM, Yorke JA, Grebogi C. Numerical orbits of chaotic processes represent true orbits. Bulletin of the American Mathematical Society. 1988;19:465-469\n'},{id:"B35",body:'\nBarge M, Swanson R. Rotation shadowing properties of circle and annulus maps. Ergodic Theory and Dynamical Systems. 1988;8:509-521. DOI: 10.1017/S0143385700004661\n'},{id:"B36",body:'\nMisiurewicz M. Remark on the definition of topological entropy. In: Lara-Carrero L, Lewowicz J, editors. Dynamical Systems and Partial Differential Equations (Caracas, 1984). Caracas: Equinoccio; 1986. pp. 65-67\n'},{id:"B37",body:'\nBarge M, Swanson R. Pseudo-orbits and topological entropy. Proceedings of the American Mathematical Society. 1990;109:559-566. DOI: 10.1090/S0002-9939-1990-1012923-7\n'},{id:"B38",body:'\nKolyada S, Snoha L. Topological entropy of nonautonomous dynamical systems. Random & Computational Dynamics. 1996;4:205-233\n'},{id:"B39",body:'\nZeng F. A note on topological entropy of maps. Northeastern Mathematical Journal. 1997;13:250-255. DOI: 10.13447/j.1674-5647.1997.04.012\n'},{id:"B40",body:'\nWalters P. An introduction to ergodic theory. In: Graduate Texts in Mathematics. Vol. 79. New York: Berlin: Springer-Verlag; 1982. 256 p. DOI: 10.1007/978-1-4612-5775-2\n'},{id:"B41",body:'\nKolyada S, Snoha L. On topological dynamics of sequences of continuous map. International Journal of Bifurcation and Chaos. 1995;5:1437-1438. DOI: 10.1142/S0218127495001125\n'}],footnotes:[],contributors:[{corresp:null,contributorFullName:"Kesong Yan",address:null,affiliation:'
School of Information and Statistics, Guangxi University of Finance and Economics, Nanning, Guangxi, PR China
School of Information and Statistics, Guangxi University of Finance and Economics, Nanning, Guangxi, PR China
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“The public debt is a burden on the back of our children and grandchildren…
\n
All debt is evil; public debt absolutely evil [1].”
\n\n
\n
1. Definition of public debts
\n
In the modern state perspective, the needs constantly increase; therefore, the state has to spend more to meet these needs. Public expenditures are generally met by ordinary public revenues such as taxes, duties, fees, parafiscal revenues, property and enterprise revenues, taxes, and penalties. However, the state is faced with the public sector deficit due to reasons such as large infrastructure investments, war, development financing, natural disasters, economic crises, budget deficits, as well as the ever-increasing ordinary public expenditures. To overcome this situation, they refer to borrowing.
\n
Borrowing is the taking of money and similar values for repayment after a certain period of time. Public borrowing refers to the legal obligation of the state to pay back the principal and interest to the holders of the predetermined rights in accordance with a certain schedule. Public credit and public borrowing referred as state borrowing in the economic literature mean debts taken by government or other public institutions [2].
\n
Governments in ancient and medieval ages required funding, as in modern states. But governments did not borrow “publically” in the concept of drawing funds from a large populace and paying principal and interest, as like deferred taxes [3].
\n
In the thirteenth century, public borrowing, including even the king’s borrowing, was first scientifically examined by Charles Davemont in 1710. Thereafter, economists such as David Hume, Adam Smith, D. Ricardo, Malthus, J.S. Mill, J.B. Say, A.P. Lerner, and A.G. Hart have worked on borrowing. Smith and Ricardo opposed public borrowing. In their view, borrowing can be spent irresponsibly because of being an easy income; so that causes deterioration in the functioning of economic life. In this context, the classics have advocated that capital is wasted, and the debt burden is transferred to the next generations due to the inefficiency of public expenditures [4]. In addition, classics have defended that borrowing could be in some case such as large infrastructure investment and war but emphasized that it should be limited and not be kept on.
\n
The public borrowing policies over the world have especially experienced a turning point with the World War I (1914–1918) and the Great Depression (1930s). During the period in question, John Maynard Keynes had proposed public borrowing as a war financing to England and argued that it would be useful. In the process that started with this proposal, public borrowing became an indispensable source of financing for the states. This situation does not mean that states participated in Keynesian theory. While public borrowing becomes an indispensable source of financing, it also brings the debt-interest cycle, poverty, and crises. The result of public borrowing leaves a great burden on the next generations. This situation has justified the classics [4].
\n
Especially after the World War II, public borrowing indicated both significant increase and structural changes due to on the one hand the repair works of the countries affected by the war, on the other hand, the financing needs of developing countries [5]. In the following period, the borrowing process are no longer interstate and have started to gain a new dimension by establishing international organizations such as International Monetary Fund (IMF), World Bank (WB), International Finance Corporation (IFC), International Development Association (IDA), European Investment Bank (EIB), and Islamic Development Bank (IDB).
\n
In the process of globalization, the mobility of capital has increased; and serious financial competition has emerged in global markets.
\n
In particular, developing countries have sought to use them to development financing by attracting international short-term capital movements to their countries through various incentive instruments (such as low taxes, high interest rates, etc.). However, both the sudden fluctuation in capital movements and the implemented incentive mechanisms have dragged the developing countries to the external debt spiral.
\n
\n
\n
2. Classification of public debts
\n
Public debts are classified into various types according to their characteristics. When the public debt literature is analyzed, it is classified into three main groups according to maturity, resources, and voluntariness [6, 7, 8, 9] (Figure 1):
Public debts according to maturities: short-, medium-, and long-term public debts
Short-term public debts (floating debts) refer to debts up to 1 year. In short-term borrowing, treasury bills and treasury guaranteed bond are used.
Medium-term public debts refer to debts ranging from 1 to 5 years.
Long-term public debts refer to debts more than 5 years. The instrument of long-term borrowing is the government bond. These debts are provided from the capital markets and have a higher interest rate than the interest rate of short-term borrowing. Long-term debts are classified as redeemable debts and irredeemable debts.
Public debts according to sources: internal debts and external debts
Internal borrowing refers to a country’s borrowing from own national resources. This borrowing has no effect on increasing or decreasing national income.
External borrowing refers to the resources provided from a foreign country that is repaid with principal and interest at the end of a certain period. External debt has an increasing effect on national income when it is taken and vice versa has a decreasing effect on national income when it is paid.
Public debt as a voluntary basis: voluntary debts and obligatory debts
Voluntary debts refer the debts that are lent to the state by its own will and desire.
Obligatory debts refer to the debts which are lent by forcing to take the bonds issued by the government. These debts are applied in times of war, natural disaster, or economic crises. In itself, it is classified as the debts taken by full compulsion, the debts taken by the threat of forcing, the debts taken by creating the necessary savings, and the liabilities taken by the moral coercion.
\n
Productive and unproductive debts are also available. If the debts are used in construction, such as railways, power stations, and irrigation projects, which contribute to the productive capacity of the economy, they denote to productive debts. By this way, productive debts provide a constant flow of income to the state. The state generally pays the interest and principal debt amount from these projects’ revenues. If the debts are used in the area such as war, famine relief, social services, etc., which do not contribute to the productive capacity of economy, they denote to unproductive debts. The state generally pays the interest and principal debt amount from taxes; therefore, these debts are a burden on the society [10, 11, 12] (Figure 1).
\n
Figure 1.
Classification of public debts. Note: Prepared by the author by utilizing from literature.
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Today, rapidly increasing international relations have increased the importance of external debts. The less developed and developing countries have to refer to external borrowing for the realization of their economic development. The lack of adequate capital markets for development in these countries and the insufficient number of technical materials and personnel required external resources. As a matter of fact, these are the main reasons for applying to external borrowing in the Ottoman Empire and the Turkish Republic periods before internal borrowing [13].
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Internal and external borrowing amounts are adversely progressed in less developed and developed countries. According to this, the debts of developed countries are predominantly internal debts; the debts of less developed and developing countries are mostly external debts. Because in developed countries, the state can easily provide the debts needed by own internal sources. It is also important from where and how the sources of funding are provided in a country’s economy as well as how these resources are channeled back into the economy [13].
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As it is known, external borrowing has an increasing effect on national income when taken and has a decreasing effect on national income when paid. Because of these features, it is important to use for what purpose external borrowing. For instance, the development credits, that are provided in order to investing in economic development and increasing the existing investments, contribute to the economy by using the programs and projects included in the development plans. Development credits are dealt within four groups [6, 9, 13]:
Project credits and program credits
Project credits are the credits that provided for the purpose of realizing the investment projects in the development plans of the countries. Countries that request credit provide projects with detailed information to countries or organizations that will give credit. Project credits are accredited/opened for the financing of eligible projects. Thus, less developed and developing countries are forced to use credits in a productive area, while creditor countries or organizations have the opportunity to control their credits. The biggest drawback of these credits is the long time for the preparation, submission, and creditor approval of the project, which makes the efficient use of credits more difficult.
Program credits are the credits that received for the purpose of importation of raw materials, semifinished goods, finished goods, and spare parts required for development targets in general. It is more flexible to use because it is not connected to any project. Thus, with the help of program credits, import bottlenecks are eliminated and the economy is kept in working condition. Therefore, it is a credit that is demanded more by developing countries.
Tied loans and soft loans
Tied loans refer to the credit that should be used in the country which gave the credit. In this case, the debtor country does not have the authority to spend the credit on its own request. Nowadays this feature of loans granted by developed countries generates to establish the system that is working in favor of the lender countries. Thus, the creditor country has provided advantages such as new foreign market, export growth, employment increase, and technology transfer. In terms of the debtor country, the situation is not bright at all. The borrower country is exposed to practices that increase the real cost of a loan such as buying a product from creditors at a much more expensive price than the normal market rate, transporting the goods through the creditor country’s transport system and insuring them by the creditor country’s insurance system. Hence, countries that receive loans remain under heavy debt burden.
Soft loans are credits that granted the free use of developing countries. Thus, the debtor countries can provide the goods and services necessary for development financing from the international market in the cheapest way.
Debt postponement and refinancing credits
Debt postponement is to postpone debt payment for an expired credit to a later date in return for a lower rate of interest compared to the first interest rate.
Refinancing credits are to pay an expired debt by the creditor country with the same amount of a new loan (a new debt). The main reason for creditor countries to accept debt postponement and refinancing credits is to enable them to accept some new commitments to the debtor country with these instruments. Thus, the creditor country can direct the debtor country’s economic policies in line with its own advantages.
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3. Effects of public debts
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Borrowing has an important place among the public revenues, so its political, economic, and social impacts have great importance. The political effects of public borrowing are handled within the framework of political business cycle theory. According to theory, public expenditures increase during the election period. The government with the vote worry increases the public investments, but prefer to finance these public expenditures with internal borrowing instead of tax or emissions. In the short and medium term, governments, that do not want to seem repellent for voters, transfer the debt principal and interest payments to the next governments in the long term. This situation brings along the debt burden, which is often worsening with an alternative to closing debt with debt in developing countries [15].
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Economic and social effects of borrowing take place in different ways in the following condition [4, 8]:
To be long- or short-term maturity
To spend or to keep the source that provided from borrowing
To borrow from internal and external sources
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The long-term or short-term maturity of public borrowing determines the duration of the contraction or expansionary effects. In this respect, the short-term borrowing changes the economic conjuncture frequently, because of the more liquidity and monetization feature of short-term debt instruments. If the resources obtained by borrowing are expended, it causes an expansionary effect; if the resources obtained by borrowing are not expended, it causes a contractionary effect [4].
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In order to provide the expected results of debt policies (i.e., borrowing methods, credit instruments, and payment and redemption methods), the economic effects of borrowing should be well known and analyzed. At this point, the source of the public debt and the place where it is used gain importance [2, 4, 5, 8, 14, 16]:
The effect of public debt on the general level of prices: It is true that borrowing will create a deflationary effect only when it is considered as a bond sale. Because the private sector uses its own resources for buying public bonds, therefore, the private demand and the total demand are decreasing. This situation causes deflation by reducing the general level of prices. However, the state purchases goods and services with the resources that are collected from the sale of bonds or bills; thus the total demand increases due to public demand. This situation causes inflation by increasing the general level of prices as a result of the operation of various mechanisms.
The effect of public debt on income distribution: The effect of public debt on income distribution depends on which income groups burden with debt costs and depends on which income groups are the obtained debt sources transferred to. This effect usually occurs during the principal and interest repayments. In particular in the internal borrowing, if the taxpayers and the lenders to the government are the same person or organization, there will be no inequality in the income distribution. However, vice versa, if the principal and interest payments related to public debt are paid by taxes collected from the middle- to low-income groups, then there is a transfer of resources from the middle- and low-income groups to the high-income group. This situation causes income distribution, the detriment of the middle- and low-income group, to deteriorate. In terms of external borrowing, the income distribution to favor of those beneficiaries from public expenditures in the period which they were taken was effected by the external debts positively. On the other hand, the external borrowing will affect the income distribution for next generation due to the debt burden adversely (such as the reduction of public expenditures and excessive tax payment). The effect of public debts on income distribution also points to the social impact of public borrowing.
The effect of public debts on savings volume and investments: As long as the government canalizes to investing the savings that are collected by the way of internal borrowing, national income will increase, and personal income and personal savings tendency will increase. If the government transfers to budget deficit or consumption of the resources which recollected by internal borrowing, it will reduce the private sector investment amount by affecting the private sector’s total savings volume. This event is called crowding out. The slowdown of national income growth as a result of the decrease in investments shows the real burden of the financing with borrowing instead of tax on the next generation. When debt is used to finance public expenditure, its real cost to society is the sacrifice in private sector production [17].
The effect of public debts on economic development: if the funds provided through borrowing for economic development can be canalized to infrastructure investments (such as dams, roads, ports, mining, agriculture), they increase the new investments through multiplier effect. As a result, national income and employment increase; and accordingly economic development is ensured. Nowadays, less developed and developing countries, which make the development effort, resort to external borrowing due to insufficient internal financing sources. If the aforementioned countries do not use external financing sources in the required fields, this situation may turn into debt financing by debt. This situation also shows the importance of debt management.
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4. Current reflections of public debts
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The phenomenon of globalization, which extends to geographical discoveries by origin, has gained momentum with the process of commercial and financial liberalization in the last quarter of the twentieth century. In this process, the globalization of capital, in particular, has dragged the developing countries, which have entered into a growth effort based on foreign capital, to the competition of encouragement (with high real interest rates, low exchange rates, and low tax rates). Increasing tax competition among developing countries led to a decrease in tax rates. The inadequate tax revenues for the financing of the increased public expenditures in these countries brought the need for new borrowing for the agenda. The application of high real interest rates to pay new debt principal and interest led to a rapid rise in the borrowing costs of developing countries and consequently a vicious cycle of debt-interest. Thus, the external borrowing process that developing countries started to finance development has undergone structural change. In this process, the method of closing the old debt with new debt (Ponzi-type financing) was adopted [18] (Figure 2).
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Figure 2.
Public debt (total and domestic debt) of all economies, as a percentage of GDP 1900–2010 [21].
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Developing countries, which cannot overcome the lack of resources, have faced severe crises due to their fragile market structures. While the crises experienced until the 1980s stemmed from the balance of payment problems, in the globalization process, the nature of the crises has changed and has become the external debt crises and financial market crises (1982 Mexico, 1992–1993 ERM, 1994 Mexico, 1997 Asia, 1998 Russia, 1999 Brazil, 2000–2001 Turkey and Argentina) [19]. Thus, the volume of financial transactions in the global economy was only 15.3 times larger than the nominal GDP in the 1990s, while in the 2000s, it was 73.5 times larger. Equities, bonds, and foreign exchange spot transactions have nearly doubled the nominal GDP worldwide [20].
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The world debt crises, which began with the declaration of the moratorium by Mexico in 1982 and spread by domino effect, caused the creditors to halt the supply of credit in a panic. Thus, developing countries, whose external debt burden has become more severe, had to implement the stabilization policies proposed by the IMF in order to get new loans or to delay debt. The process followed brought with inequalities in income distribution, unemployment, inflation, and poverty. On the other hand, developed countries have faced the problem of slowing capital accumulation due to the decrease in profit rates since the 1970s. This stagnation in developed countries has been tried to be overcome by lending excessive credit to developing countries through credit mechanism. Thus, both idle funds were evaluated in developed countries, and new markets were found for export increase [18].
\n
For developed countries to sell the surplus production, firstly, it was necessary to create a consumer society. This is only possible by changing the basic habits of society. Intangible concepts such as cultural values, traditions, beliefs, etc. have gained importance as the basis for the use of the method of power based on economic power. Developed countries consciously use these power tools and encourage other countries to have their own culture and economic structure. In this way, it can be said that especially the developing countries are transformed into a consumer society, and thus they are continuously indebted for both public and private sectors [22].
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In the globalization process, when the total external debt stock of developing countries is analyzed, it can be seen as a notable increase in short-term external debts and private sector external debts. Therefore, the process is called the rapid privatization of external debts. This phenomenon brings the important problems for developing countries. Short-term debts, which are not paid by the private sector (especially commercial banks) in times of crises, are taken under the state guarantee by IMF regulations. In the end, the short-term private sector debt was turned into long-term public debt by consolidating. After the 2001 crises in Turkey, we can see that the debt burden of bankrupt banks had transferred to the Treasury. As a result, Turkey’s public debt has increased rapidly [18].
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5. Conclusions
\n
The state faced a financing deficit, when the ever-increasing needs in the social development process were not met by the state’s ordinary public revenues (such as taxes, duties, fees, parafiscal revenues, property and enterprise revenues, taxes, and penalties). In addition to these expenditures, the state had to resort to borrowing due to major infrastructure investments, war, development financing, natural disasters, economic crisis, and budget deficits. From the borrowing of the king representing the country in the thirteenth century to the current debt crises, public borrowing has served different purposes in this process. This process has brought different debt types according to maturities (short, medium, long term), resources (internal and external debts), and voluntariness (voluntary and compulsory debts). Thus, borrowing overreached to be an extraordinary public revenue and has started to be perceived as a means of intervention to the economy. For example, the state uses internal debts as a tool to reduce demand in inflationary periods and to increase demand in deflationary periods.
\n
In this section, which focuses on the political, economic, and social impacts and current reflections of borrowing in the context of public debt theory, the transformation in the external debt structure in the globalization process is emphasized. Initially external assistance and debts taken by developing countries as development financing have been used as a means to eliminate the stagnation by the developed countries in their economies. The given debts to the developing countries were used to increase the exports of developed countries, especially through tied loans. This situation has brought the new market, technology transfer, and economic power to the developed countries, while it has caused consumption society, external dependency, debt-interest spiral, and ultimately external debt crises in the developing countries. The restrictions of the structural adjustment programs have been introduced by IMF to improve the fragile economies of developing countries against external debt crises. In this period, IMF has started to use this structural adjustment programs, which include strict structural reforms, by determining the economic and social policies of the debtor countries as a tool to serve the purpose of developed countries.
\n
\n\n',keywords:"public debt, internal debt, external debt, crowding out, globalization",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/65684.pdf",chapterXML:"https://mts.intechopen.com/source/xml/65684.xml",downloadPdfUrl:"/chapter/pdf-download/65684",previewPdfUrl:"/chapter/pdf-preview/65684",totalDownloads:1862,totalViews:0,totalCrossrefCites:2,dateSubmitted:"November 23rd 2018",dateReviewed:"November 26th 2018",datePrePublished:"February 15th 2019",datePublished:"October 23rd 2019",dateFinished:null,readingETA:"0",abstract:"From the ancient ages to today, administrations needed continuous financing and met this financing with various sources. The process of social development necessitated public borrowing for different purposes ranging from creation of a consumer society to sell the surplus of developed countries to postwar human relations and from the development financing of developing countries to the payment of debt by debt. Particularly after World War II (1941–1945), the developed countries provided the external resources to developing countries for development financing. As a result of the increase in the mobility of capital in the process of globalization (especially short-term speculative capital investments), developing countries were dragged to the debt-interest helix problem and the external debt crises. The stabilization programs proposed by the IMF led to government guarantee of private sector external debts in the developing countries and led to a rapid increase in the public debt stock.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/65684",risUrl:"/chapter/ris/65684",signatures:"Sibel Aybarç",book:{id:"7598",title:"Public Economics and Finance",subtitle:null,fullTitle:"Public Economics and Finance",slug:"public-economics-and-finance",publishedDate:"October 23rd 2019",bookSignature:"Bernur Açıkgöz",coverURL:"https://cdn.intechopen.com/books/images_new/7598.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"43498",title:"Dr.",name:"Bernur",middleName:null,surname:"Açıkgöz",slug:"bernur-acikgoz",fullName:"Bernur Açıkgöz"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:[{id:"286689",title:"Dr.",name:"Sibel",middleName:null,surname:"Aybarç",fullName:"Sibel Aybarç",slug:"sibel-aybarc",email:"sibel.aybarc@cbu.edu.tr",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Definition of public debts",level:"1"},{id:"sec_2",title:"2. Classification of public debts",level:"1"},{id:"sec_3",title:"3. Effects of public debts",level:"1"},{id:"sec_4",title:"4. Current reflections of public debts",level:"1"},{id:"sec_5",title:"5. Conclusions",level:"1"}],chapterReferences:[{id:"B1",body:'Samuelson PA. Economics: An Introductory Analysis. 10th rev ed. USA: McGraw-Hill Inc.; 1976. p. 896. ISBN-10: 0070856419. ISBN-13: 978-0070856417\n'},{id:"B2",body:'Ulusoy A. Devlet Borçlanması. Trabzon: Aksakal Press; 2013. p. 358. ISBN: 975-98986-1-6\n'},{id:"B3",body:'Salsman RM. The Political Economy of Public Debt Three Centuries of Theory and Evidence, New Thinking in Political Economy Series. UK: Edward Elgar Publishing Limited; 2017. p. 336. ISBN-10: 1785363379\n'},{id:"B4",body:'Sugözü Hİ. Devlet Borçları IMF-Dünya Bankası ve Türkiye. Ankara: Nobel Press; 2010. p. 384. ISBN: 978-605-395-383-8\n'},{id:"B5",body:'Yaşa M. Devlet Borçları. Ankara: Ankara University Law Faculty Press; No: 421; 1978. p. 87\n'},{id:"B6",body:'Açba S. Devlet Borçlanması. Afyon: Pınar Press; 1994. p. 246\n'},{id:"B7",body:'Corina CR. Public Debt: Structure and Characteristics Romania’s Case. Annals of the “Constantin Brâncuşi” University of Târgu Jiu. Economy Series. 2013;3:30-35. ISSN: 1844-7007\n'},{id:"B8",body:'Erdem M. Devlet Borçları. 7th ed. Bursa: Ekin Press; 2016. p. 169. ISBN: 975-7338-07-9\n'},{id:"B9",body:'Günay Bekar A. Devlet Borçlarının Sınıflandırılması. In: Tüğen K, Egeli H, Tandırcıoğlu H, editors. Devlet Borçları. İzmir: Kitabana Press; 2018. pp. 49-88. ISBN: 978-605-9515-27-6\n'},{id:"B10",body:'JMPC. Module 3: Public Expenditure and Public Debt [Internet]. 2018. Available from: www.jmpcollege.org/downloads/module3.pdf [Accessed: Oct 11, 2018]\n'},{id:"B11",body:'Saman RM. Productive and unproductive debt spending of microloans taken by urban borrowers (Johannesburg) [thesis]. South Africa: University of Pretoria; 2007\n'},{id:"B12",body:'Muley R. Public Debt: Meaning, Classification and Method of Redemption [Internet]. Available from: www.economicsdiscussion.net/debt-2/public-debt-meaning-classification-and-method-of-redemption/17472 [Accessed: Oct 10, 2018]\n'},{id:"B13",body:'İnce M. Devlet Borçlanması (Kamu Kredisi). 5th ed. Ankara: Seçkin Press; 1996. p. 376. ISBN: 975-347-090-8\n'},{id:"B14",body:'Işık AK, Karayılmazlar E. In: Organ İ, Işık H, editors. Devlet Borçları. Bursa: Ekin Press; 2005. p. 162. ISBN: 975-8768-677\n'},{id:"B15",body:'Demirhan H. Devlet Borçlarının Etkileri. In: Tüğen K, Egeli H, Tandırcıoğlu H, editors. Devlet Borçları. İzmir: Kitabana Press; 2018. ISBN: 978-605-9515-27-6\n'},{id:"B16",body:'Bülbül D. Devlet Borçlanmasının Sosyal ve Ekonomik Etkileri. Ekonomik Yaklaşım [Internet]. Available from: http://docplayer.biz.tr/735044-Devlet-borclanmasinin-sosyal-ve-ekonomik-etkileri.html [Accessed: Oct 11, 2018]\n'},{id:"B17",body:'Sharp AM. Review: A general theory of public debt. The American Journal of Economics and Sociology. 1959;19(1):99-111\n'},{id:"B18",body:'Bingöl Ö. Dış Borç Sorunu Küreselleşme Sürecinde Gelişmekte Olan Ülkeler ve Türkiye. İstanbul: Derin Press; 2014. p. 374. ISBN-10: 6054993232\n'},{id:"B19",body:'Kovancılar B, Bursalıoğlu Aybarç S. Küresel Finansal Kriz ve Tobin Vergisi. Vol. 2. International Social Sciences Congress in the Balkans. pp. 907-924. ISBN: 978-975-7988-43-4\n'},{id:"B20",body:'Schulmeister SA. General Financial Transaction Tax: A Short Cut of the Pros, the Cons and a Proposal. WIFO Working Paper. No: 344 [Internet]. Available from: www.econstor.eu/bitstream/10419/128890/1/wp_344.pdf [Accessed: Oct 11, 2018]\n'},{id:"B21",body:'Reinhart CM. Rogoff KS. The Forgotten History of Domestic Debt. The Economic Journal, 121. [Internet]. 2011. pp. 319-350. Available from: https://onlinelibrary.wiley.com/doi/epdf/10.1111/j.1468-0297.2011.02426.x [Accessed: Oct 11, 2018]\n'},{id:"B22",body:'Korkmaz B. Bir iktidar/güç araci olarak diş yardim ve diş borçlarin kullanilmasi: Osmanlı devleti’nden günümüze Türkiye’nin diş borçlan(dırıl)ma serüveni [thesis]. Turkey: Manisa Celal Bayar University (Supervisor. Aybarç, S.); 2018\n'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Sibel Aybarç",address:"\n",affiliation:'
Public Finance, Faculty of Economics and Administrative Sciences, Manisa Celal Bayar University, Turkey
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