Available online at www.sciencedirect.com
Journal of Mathematical Economics 44 (2008) 423–444
The inverse limits approach to chaos Judy Kennedy a,b, David R. Stockman c,∗, James A. Yorke d a Departmentof Mathematical Sciences, University of Delaware, Newark, DE 19716, United States b Department of Mathematics, Lamar University, Beaumont, TX 77710, United States c Department of Economics, University of Delaware, Newark, DE 19716, United States d Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, United States Received 27 March 2006; received in revised form 31 October 2007; accepted 3 November 2007 Available online 22 November 2007
Abstract When analyzing a dynamic economic model, one fundamental question is are the dynamics simple or chaotic? Inverse limits,as an area of topology, has its origins in the 1920s and since the 1950s has been very useful as a means of constructing “pathological” continua. However, since the 1980s, there is a growing literature linking dynamical systems with inverse limits. In this paper, we review some results from this literature and apply them to the cash-in-advance model of money. In particular, we analyze the inverse limit of the cash-in-advance model of money and illustrate how information about the inverse limit is useful for detecting or ruling out complicated dynamics. © 2007 Elsevier B.V. All rights reserved.
JEL Classification: C6; E3; E4
Keywords: Chaos; Inverse limits; Continuum theory; Topological entropy; Cash-in-advance
1. Introduction
An equilibrium of a dynamic economic model can often be characterized as a solution to a dynamical system. One fundamental question when analyzing such a model is are the dynamics simple or chaotic? There are numerous definitions and measurements for chaos. Two of the more popular definitions are those of Devaney (2003) and Li and Yorke (1975). Given a definition of chaos, one still needs a means for establishing or ruling out chaos. For example, Li and Yorke (1975) show that for a continuous map on an interval, a three cycle is sufficient for establishing chaos. Other measures of complicated dynamics include topological entropy, measure-theoretic entropy, zeta function, Liapunov exponents and box dimension. In this paper, we focus on inverse limits as a method of detecting or ruling out chaotic behavior. As an area in topology, inverse limits has its origins in the 1920s and since the 1950s has been very useful as a means of constructing “pathological” continua. However, since the 1980s there is a growing literature that looks at the connection between inverse limits and topological dynamics. In this paper, we review some results from this literature and apply them to the cash-in-advance model of money. In particular, we analyze the inverse limit of the
∗ Corresponding author. E-mail address: [email protected] (D.R. Stockman).
0304-4068/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2007.11.001 424 J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 cash-in-advance model of money and illustrate how information about the inverse limit is useful for detecting or ruling out complicated dynamics. Inverse limits is a relatively new tool to economic theory that has recently been used to understand models with backward dynamics. Many nonlinear dynamical systems are single-valued moving forward, but multi-valued going backward. However, in economics we sometimes have just the opposite, namely dynamics that are multi-valued going forward, but are single-valued going backward. In such instances, we say the model (or dynamical system) has backward dynamics. Two economic models that may have backward dynamics include the overlapping generations (OLG) model and the cash-in-advance (CIA) model.1Medio and Raines (2007, 2006) use inverse limits to analyze the long-run behavior of an OLG model with backward dynamics. Even though the forward dynamics are multi-valued, they show that long-run behavior of equilibria in the model corresponds to an “attractor” of the shift map on the inverse limit space. Kennedy and Stockman (2007) use the inverse limit space to show that a model with backward dynamics is chaotic going forward in time if and only if it is chaotic going backward in time. Though inverse limits have been used to analyze models with backward dynamics, the results in this paper apply to models with forward dynamics as well. Heuristically, the inverse limit of a dynamical system is a subset of an infinite dimensional space (e.g. the Hilbert cube) where each point in the inverse limit corresponds to a backward solution (backward orbit) of the dynamical system. When considering the inverse limit space of a dynamic economic model, two questions naturally arise: (1) what does the structure of the inverse limit tell us about the underlying dynamics, and (2) what does the underlying dynamics imply for the inverse limit? Ingram and Mahavier (2004) explore the connections between these two aspects of dynamical systems for maps on the unit interval. Working within a family of one-dimensional maps, they illustrate that restrictions on parameters that lead to simple/complicated dynamics also lead to simple/complicated inverse limits. Consequently, by analyzing the inverse limit one can detect chaotic/complicated dynamics or rule out such behavior. The inverse limit approach offers an alternative method of exploring chaotic behavior that does not work directly with establishing a 3-cycle (or some other cycle and using Sarkovskii’s ordering on the integers) or calculating some measure of entropy. Moreover, when the topology of the inverse limit space changes there is a qualitative change in the dynamics. Thus the inverse limit space can be used to detect qualitative changes in chaotic behavior. The paper is organized as follows. We provide the necessary background material from dynamical systems and inverse limits in Section 2. In Section 3, we illustrate how inverse limits can be applied to the CIA model of money to establish or rule out complicated dynamics. We conclude in Section 4.
2. Dynamics, continuum theory, and inverse limits
The applicability of inverse limits to detecting or ruling out chaotic behavior in a dynamic economics model like the CIA model depends on the relationship between equilibria in the model, the inverse limit, continuum theory and dynamics. In this section we discuss the necessary background from dynamics, inverse limits and continuum theory for our application to the CIA model. We do not express definitions or theorems in their most general form, but at a level sufficient for our purposes.
2.1. Dynamics
In this subsection, we discuss some definitions and facts from dynamics that we will use later. Throughout this paper, we will suppose X is a compact metric space and f : X → X is continuous. We are interested in using the inverse limit space to establish or rule out complicated dynamics. What does one mean by complicated? One can say that the dynamics are complicated if the dynamical system is chaotic. There are several definitions of chaotic in the dynamical systems literature. Two properties that most definitions include are topological transitivity and sensitive dependence on initial conditions.
Definition 1. A map f : X → X is topologically transitive if whenever U and V are nonempty open subsets of X, there is some integer n such that f n(U) ∩ V =∅.
1 See Grandmont (1985) for the OLG model and Michener and Ravikumar (1998) for the CIA model. J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 425
Definition 2. A map f : X → X, where X is a metric space has sensitive dependence on initial conditions on an invariant closed subset H of X if there is some number r>0 such that for each point x in H and for each >0, there is a point y in H with d(x, y) < and an integer k ≥ 0 such that d(f k(x),fk(y)) ≥ r.
Two of the more commonly used definitions of chaotic are those of Li and Yorke (1975) and Devaney (2003).
Definition 3. The map f : X → X is chaotic in the sense of Li and Yorke (1975) if f has sensitive dependence on initial conditions on X. The map f is chaotic in the sense of Devaney (2003) if (1) f is topologically transitive, (2) the set of periodic points in X is dense in X, and (3) f has sensitive dependence on initial conditions.
Three more concepts used to indicate that a system has complicated dynamics are a homoclinic fixed point,a horseshoe and topological entropy.
Definition 4. Suppose that x is a fixed point of f. We say that y is homoclinic to the fixed point x if x = y such that f n(y) → x and that there is a choice of inverse images f −1(y), f −2(y),...with f −n(y) → x.Ifx is a periodic point of period n under f,wesayy is homoclinic to x if y is homoclinic to x under f n.
Definition 5. Let f : I → I be continuous where I := [a, b]. We say that f has a (one-dimensional) horseshoe if there n n are disjoint subintervals I0 and I1 of I and n0 ∈ N such that I0 ∪ I1 ⊂ f 0 (I0) and I0 ∪ I1 ⊂ f 0 (I1).
The notion of topological entropy involves the concept of an (n, )-separated set. Let f : X → X be a continuous map on a compact metric space X with metric d. Let A, E ⊂ X. We say that E is (d, , A)-spanning if E is finite and f for every y ∈ A, there exists an x ∈ E such that d(x, y) < .Givenf, for n ∈ N we define a new metric dn on X given f i i f by dn (x, y):= max0≤i≤n−1d(f (x),f (y)). For n ∈ N and >0, let S(dn , ,A) denote the minimum cardinality of f f all (dn , ,A)-spanning sets. Heuristically, S(dn , ,A) denotes the number of initial conditions in A that an observer of the dynamical system f can distinguish given orbits of length n and an ability to measure the system with accuracy no greater than .
Definition 6. For A ⊂ X, we define = 1 f h(f, A , ): limsup( n )[log S(dn , ,A)]. n→∞ The topological entropy of f on A is defined by h(f, A ):= limh(f, A , ). →0 The topological entropy of f is defined by h(f ):= h(f, X ).
2.2. Inverse limits and continuum theory
In this subsection, we discuss the concept of a continuum along with a few of its topological properties. We also introduce the notion of an inverse limit and discuss some relevant theorems for our application to dynamic economic models.
Definition 7. A space is connected if and only if it is not the union of two disjoint, closed and nonempty sets.
Definition 8. A continuum is a compact, connected metric space. If X and Y are continua, and Y ⊂ X, then Y is a subcontinuum of X.IfY is a subcontinuum of X,butY = X, then Y is a proper subcontinuum of X.
Definition 9. An arc is a space homeomorphic to the unit interval [0,1]. An arc continuum is a continuum with the property that each of its proper subcontinua is homeomorphic to an arc. A finite graph is a connected union of finitely many arcs. 426 J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444
One simple example of a continuum is the unit interval I := [0, 1]. A subcontinuum of I is [1/2, 3/4]. An arc is of course an arc continuum. A more interesting example of an arc continuum is the Knaster bucket handle (discussed below). The topologist sine curve given by {x, sin(1/x):0 Definition 10. A continuum is decomposable if it is the union of two of its proper subcontinua, otherwise it is indecomposable. The unit interval I is a simple example of continuum that is decomposable. For example, I = [0, 1/2] ∪ [1/4, 1], and so I is decomposable. An indecomposable continuum is central to our investigation of the dynamics from a topological point of view so we want to spend some time developing intuition for these objects. When one starts to sketch continua on a piece of paper, it is hard to imagine a continuum that is not decomposable, and a reader not familiar with these objects might ask whether such continua exist. They do indeed, and are quite common occurrences in chaotic dynamical systems. Fortunately such continua have been studied and their properties explored by mathematicians since 1910.2 All indecomposable continua share certain structure in that each can be partitioned into composants. Definition 11. If X is an indecomposable continuum and x ∈ X, then the composant of x is defined by Com(x):={y ∈ X : ∃Y ⊂ X a proper subcontinuum with x, y ∈ Y}. The set of composants of an indecomposable continuum partitions the continuum into what can be shown must be an uncountable collection of mutually disjoint connected sets, each of which is dense in the continuum. Each composant is like a “highway” in the continuum. The continuum consists of a collection of highways, each close to any other but always separate. Indecomposable continua have been used to describe strange attractors of nonlinear dynamical systems. For example, the Smale horseshoe attractor is indecomposable and can be shown to be homeomorphic to the Knaster bucket handle continuum, one of the more famous indecomposable continua from topology.3Kuratowski (1968) gives the following constructive description of the Knaster bucket handle, a set of points in the plane we will denote by K. Let C be the Cantor middle-thirds set. Note that this set is “symmetric” in the interval, so that if x ∈ C there is a y ∈ C such that (x + y)/2 = 1/2. This allows one to draw semi-circles in the upper half of the plane (i.e., with non-negative y-coordinate) centered at [1/2, 0] that touch the x-axis only at points in C ×{0}. Attached to these semi-circles in the upper half of the plane are semi-circles from below in the lower half plane in the following way: connect “symmetric” points in the intervals [(2/3n), (1/3n−1)] ×{0} with semi-circles centered at [5/(2 · 3n), 0]. The union of all these semi-circles is the Knaster bucket handle K (see Fig. 1). This continuum has an uncountable number of composants. To see this, note that C is an uncountable set and K goes through each point in C. Since each composant goes through a countably infinite number points of C, there must be an uncountable number of composants. One composant that is easily constructed is the one that goes through all the end points of the Cantor set. This is the only composant that is the one-to-one image of a ray, i.e., it has an end-point.4 Starting at the origin, this composant goes up and down through the x-axis at points {0, 1, 2/3, 1/3, 2/9, 7/9, 8/9,...}×{0}. Being an indecomposable continuum, each composant of K must be dense in the continuum. To think about how they are dense, consider the end-point composant described above in K. This composant goes through all of the end points 2 See Kennedy (1993) for how indecomposable continua arise in dynamical systems and Kennedy (1995) for a brief history of indecomposable continua. 3 One interesting thing to note, is that even if indecomposable continua seem strange, Bing (1951) shows they are “normal” in the sense that most continua are indecomposable. 4 It is known that each of the other composants is a one-to-one image of the real line. This is a non-trivial result. J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 427 Fig. 1. Knaster bucket handle. This figure contains part of one of the uncountable number of composants of the Knaster bucket handle. This composant passes through the x-axis at all of the end points of the Cantor middle-thirds set, an countably infinite set. The figure drawn only goes through 32 points in this Cantor set. in C, but this set of end points is dense in C so it is relatively clear why this composant is dense in the K. Heuristically, each composant is “winding” through the continuum getting arbitrarily close to all other points in the continuum. Why is the Knaster bucket handle indecomposable? One can show that K is an arc continuum, i.e., the only proper subcontinua of K are arcs (connected segments of the composants with finite length). With two such arcs, one cannot cover a single composant yet alone the uncountable number of composants that make up K. Intuitively, a decomposable continuum can be broken into two connected pieces (think of a disk being broken into two connected pieces). If one tries to break K into two pieces, it will shatter into an uncountable number of disjoint pieces. One way to construct indecomposable continua is through inverse limits. We turn now to our discussion of inverse limits. Again, we want to note that inverse systems and inverse limits can be defined for a much broader class of factor spaces and indexing sets, but the definitions below are sufficient for our purposes. Definition 12. Let I := [0, 1] (a nonempty compact metric space) and suppose f : I → I is a continuous function. Let Q := [0, 1]∞ be the Hilbert cube. The space I is called the factor space and the function f is called the bonding map. The pair (I, f ) is called an inverse system. The set of points lim(I, f ):={x = (x ,x ,...) ∈ Q | x = f (x + ) for i ∈ N}, ← 1 2 i i 1 is the inverse limit of the inverse system (I, f ).5 The topological properties of inverse limits have been studied at least since the 1950s and much is known about them. Background theorems we need about their properties are given below. The theorem statements and many of their proofs can be found in Ingram (2000), Nadler (1992), and other books. A nice, well-written introduction to inverse limits on an interval with one bonding map is given in Ingram and Mahavier (2004), along with an investigation of the relationship between the complexity of the topology of the inverse limit space and the complexity of the dynamics of the bonding map on the factor space.6 5 More generally, one has a sequence of factor spaces {X1,X2,...}, where Xi is a nonempty metric space, and a sequence of bonding maps {f1,f2,...} such that fi : Xi+1 → Xi for i ∈ N. In this case, the collection (Xm,fm) is an inverse system, and the inverse limit is defined in an analogous way. So more carefully, we would say that our inverse system (I, f ) should be written as (Xm,fm) where Xm = I and fm = f for all m ∈ N. 6 See also the appendix to Medio and Raines (2007) for a brief survey intended for economists. 428 J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 Theorem 1. (Ingram (2000), Theorems 2.4, 2.13 and 2.14). If (I,f) is an inverse system, then lim[I, f ] is a nonempty ← continuum contained in Q. We need to be able to talk about subsets of lim(I, f ), especially closed subsets of lim(I, f ). If m ∈ N, the map ← ← π : lim(I, f ) → I defined by π ((x ,x ,...)) = x is called the projection map (or, if specificity is required the mth m ← m 1 2 m The next two theorems deals with the projection maps πm and how these maps are related to each other via the bonding map f). Theorem 2. (Ingram (2000), Theorem 1.4). If (I,f) is an inverse sequence and X := lim(I, f ), then π : X → I is ← m continuous. Theorem 3. (Ingram (2000), Theorem 1.6). If (I,f) is an inverse system, and X := lim(I, f ) is the inverse limit, then ← n−m for each m lim(π (A),f):={x = (x ,x ,...) ∈ Q | x = f (x + ) for i ∈ N,x+ ∈ π + (A)} ← i i 1 2 i i i 1 i 1 i 1 Proposition 1. Suppose f : I → I is continuous. If A is a nonempty subset of the inverse limit space Y := lim(I, f ), ← then lim(π (A),f|π + (A)) is a nonempty subset of Y and A ≡ lim(π (A),f|π + (A)). Furthermore, if A is a closed, ← n n 1 ← n n 1 nonempty subset of the inverse limit space Y := lim(I, f ), then lim(π (A),f|π + (A)) is a closed, nonempty subset ← ← n n 1 of Y and A ≡ lim(π (A),f|π + (A)). ← n n 1 Proof. See Appendix A.1. The next theorem establishes that if the bonding maps from two inverse systems are “conjugate,” then the inverse limits are homeomorphic. Theorem 4. (Ingram (2000), Theorem 6.5). Suppose f : X → X and g : X → X are conjugate (i.e., there is a home- omorphism h : X → X such that h ◦ f = g ◦ h), then their inverse limits lim(X, f ) and lim(X, g) are homeomorphic. ← ← Sometimes part of an inverse limit can be described as a ray.Atopological ray is a locally compact, connected metric space R containing a point O such that R \{O} is connected, and if p ∈ R,butp = O, then R \{p} is the union of two disjoint connected sets. Two examples of rays are (1) the set [0,1) in R and (2) the graph of sin(1/x) on (0,1] in R2. The next theorem gives sufficient conditions for a union of arcs to be a ray. Theorem 5. (Ingram (2000), Theorem 2.15). If α1,α2,...is a sequence of arcs each of which is a proper subset of a continuum X such that α1 ⊂ α2 ⊂ α3 ⊂···, the point O is a common endpoint of α1,α2,..., R = α1 ∪ α2 ∪ α3 ∪···, and no point of αn belongs to R \ αn+1, thenRisaray. Theorem 6. (Ingram (2000), Theorem 6.1). Suppose (I, f ) is an inverse limit sequence. If, for each m, Km is a subcontinuum of I and f (K + ) = K , then lim(K ,f|K + ) is a subcontinuum of lim(I, f ). m 1 m ← m m 1 ← Theorem 7. (Ingram (2000), Theorem 1.13). Let X = lim(I, f ). For x = (x ,x ,...) ∈ X, define F(x) = ← 1 2 F((x1,x2,...)) = (f (x1),f(x2),...) = (f (x1),x1,x2,...). Then F : X → X is a homeomorphism. We call F the induced homeomorphism (by f)onX. The inverse σ := F −1 of F, is then defined by σ(x) = σ((x1,x2,...)) = (x2,x3,...). The induced map σ is called the shift homeomorphism. Thus, the pair (X, F ) forms a dynamical system, one that runs both forward and backward. Note that in general f and F are not conjugate since F is always a homeomorphism and f need not be. However these systems are closely related.7 7 For more on this relationship, see Li (1992) and Canovas (1999). J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 429 2.3. Dynamics and inverse limits The next theorem is our first to discuss the relationship between the bonding map f and lim(I, f ). It provides ← sufficient conditions for the inverse limit to be extremely simple (topologically). Theorem 8. (Ingram (2000), Theorem 2.1). If (I,f) is an inverse limit system such that f is a homeomorphism, then the inverse limit of the inverse system is homeomorphic to an arc, i.e., is homeomorphic to the space [0,1]. Barge and Martin (1985) show that if the dynamics of f are complicated then the inverse limit contains indecom- posable subcontinua. Theorem 9. (Barge and Martin (1985), Theorem 1). Let I = [a, b], f : I → I be continuous, X := lim(I, F ) and ← F : X → X be the induced homeomorphism. Suppose k and n are integers with k ≥ 0, n ≥ 1 and that f has a periodic point of power 2k(2n + 1), i.e., not a power of 2. Then X has an indecomposable continuum that is invariant under k+1 F 2 . They also establish a partial converse under certain conditions on f. Definition 13. Let I = [a, b] and suppose f : I → I is continuous and onto. We say f has finitely many turning points if there exists a finite set {a1,a2,...,aN }, a = a1 Theorem 10. (Barge and Martin (1985), Theorem 10). Suppose f : I → I is continuous and onto with finitely many turning points. If lim(I, f ) is indecomposable, then f has a periodic point whose period is not a power of 2. ← Theorem 11. (Barge and Martin (1985), Theorem 4). Let I = [a, b], f : I → I be continuous and X := lim(I, f ). ← If f has a point homoclinic to a periodic point then X contains an indecomposable subcontinuum. Barge and Diamond (1994) prove the following. Theorem 12 (Barge and Diamond (1994), p. 774). Suppose f : X → X is continuous with finitely many turning points, and X is a finite graph (e.g.,X = [a, b]). Then the following are equivalent: (a) h(f ) > 0. (b) lim(X, f ) contains an indecomposable continuum. ← (c) f has a horseshoe. (d) There exists r, M ∈ N such that for m ≥ M, f has a periodic point of prime period rm. The theorems below from Ingram (1995, 2002, 2003) deal with properties of the bonding map f and the associated inverse limit or periodic points of f. The properties of the bonding map include monotone, unimodal, and type (1).We define these properties next. Definition 14. A map of a continuum to itself is monotone provided each point inverse is a continuum. A map f of an interval [a, b] onto itself is unimodal provided f is not monotone, and there is a point c in (a, b) such that f (c) ∈{a, b} and f |[a, c] and f |[c, b] are both monotone. The map f is a type (1) unimodal map if f (b) = a. Now on to the theorems. Theorem 13. (Ingram (1995), Theorem 2). Suppose f is a type (1) unimodal mapping of an interval [a, b] onto itself with critical point c, and q is a point in (c, p] such that f 2(q) = q and f (a) = q. Then the inverse limit of the inverse system ([a, b],f) is the union of two Knaster bucket handle continua intersecting at a point or an arc. See Fig. 2 for part of one composant from this continuum. Theorem 14. (Ingram (1995), Theorem 7). Suppose f is a type (1) unimodal mapping of an interval [a, b] onto itself and q is the first fixed point for f 2 in [c, b]. Then lim([a, b],f) is indecomposable if and only if f (a) Fig. 2. Two Knaster bucket handles joined at a point: each bucket handle contains one composant that is homeomorphic to a ray. These ray composants are joined at a point. This figure contains part of each of these composants. Theorem 15. (Ingram (2002), Theorem 11). Suppose f :[a, b] → [a, b] is a continuous mapping, a is periodic of period n ≥ 3 under f and b is in O+(a). If k is an integer such that f k(a) is the first member of O+(a) \{a}, and n and k are relatively prime, then lim([a, b],f) is an indecomposable continuum. (Note: f k(a) is the first member of ← O+(a) \{a} means first relative to the order on the interval [a, b].) Type (1) unimodal maps go up and then come down; maps from the CIA model that we will investigate in Section 3 go down and then come up. This is not a problem: we can “flip” our map over so as to more easily use the results in the literature. We can also translate to [0,1] from the interval [x, x¯] ⊂ (0, ∞). Proposition 2. Suppose f :[a, b] → [a, b] has the following properties: (1) There is some c in (a, b) such that f |[a, c] is strictly decreasing, and f maps [a, c] onto [a, b]. (2) The map f |[c, b] is strictly increasing. Then f is conjugate to a Type (1) unimodal map on [0,1]. Proof. See Appendix A.2. 2.4. Models with backward dynamics The discussion above deals with the topological complexity of the inverse limit and the forward dynamics of f. If the dynamic economic model has backward dynamics, one is really interested in the backward dynamics of f or, equivalently, the forward dynamics of f −1. To apply these results to f −1 one would need a connection between the dynamics of f −1 and f. Kennedy and Stockman (2007) prove the following theorem about f −1 and f. Theorem 16. (Kennedy and Stockman, 2007). Suppose X is a compact metric space and f : X → X is continuous, Then f −1 is chaotic (in the sense of Devaney) on X if and only if f is chaotic on X. Now consider topological entropy. Let f : X → X be a continuous onto map on a compact metric space X with metric d.8 Recall that the topological entropy of f is defined to be = 1 f h(f ): lim limsup log S(dn , ,X) . →0 n→∞ n Note that since f is a single-valued map, we can associate each finite orbit of length n from our dynamical system f (x1,x2,...,xn) with its initial condition x1. Intuitively our set E is (dn , ,X)-spanning if for every orbit y under f of 8 Again, the assumption of onto is useful because we want f and f −1 to have the same domain, which we can only do if f is onto. J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 431 length n there is some orbit x under f of length n starting in E such that each point in time xi and yi are no more than f apart. If one cannot distinguish orbits that are never more than apart then S(dn , ,X) tells us the number of distinct orbits there are of length n. An alternative (and equivalent) formulation, motivated by the inverse limits approach to dynamical systems, is the following. Let n lim{X, f, n}={(x ,x ,...,x ) ∈ X | x = f (x − ),i= 2,...,n}. → 1 2 n i i 1 Note that this is just the truncation of the first n coordinates of the direct limit space ∞ lim{X, f } :={(x ,x ,...) ∈ X | x + = f (x ),i∈ N}. → 1 2 i 1 i We can define a metric D on Xn given by D (x, y):= max ≤ ≤ d(x ,y). Then we call a subset C ⊂ lim{X, f, n} a n n 1 i n i i → (D , ,lim{X, f, n})-spanning set if C is finite and for each y ∈ lim{X, f, n}, there exists an x ∈ C such that D (x, y) < . n → → n If C ⊂ lim{X, f, n} isa(D , ,lim{X, f, n})-spanning set then the projection of the first coordinate E := π (C) ⊂ X → n → 1 f f is (dn , ,X)-spanning. Conversely, if E ⊂ X isa(dn , ,X)-spanning set then there exists a (unique) finite subset C ⊂ lim{X, f, n} such that C is (D , ,lim{X, f, n})-spanning and E = π (C). So the notion of spanning can be → n → 1 applied to either a subset E ⊂ X (“initial conditions”) or a subset of orbits C ⊂ lim{X, f, n}. The difference here → is entirely superficial: since f is single-valued, there is a bijection between orbits and initial conditions. Given the equivalence of the cardinality of these sets, we can use the direct limit space approach to give an equivalent definition of topological entropy for f: 1 h(f ):= lim limsup log S(Dn, ,lim{X, f, n}) . →0 n→∞ n → When f −1 is multi-valued, we can no longer associate a unique orbit to each initial condition. However, we can still talk about the space of orbits under the action of f −1 with length n and what it would mean for a finite subset of this space to form a spanning set. Let n lim(X, f, n):={x ∈ X | x − = f (x ),i=, 2,...,n}. ← i 1 i This set is just the truncation of the first n coordinates of the inverse limit space lim(X, f ). Let D be a metric on ← n lim(X, f, n) as defined above (this makes sense since both lim(X, f, n) and lim{X, f, n} are subsets of Xn). We say that ← ← → a finite set C¯ ⊂ lim(X, f, n)is(D , ,lim(X, f, n))-spanning if for every y¯ ∈ lim(X, f, n), there exists some x¯ ∈ C¯ such ← n ← ← that D (y¯, x¯) < . Having defined an (D , ,lim(X, f, n))-spanning sets for f −1, h(f −1) can be defined as above: n n ← −1 1 h(f ):= lim limsup log S(Dn, ,lim(X, f, n)) . →0 n→∞ n ← The next theorem generalizes the well-known result for entropy when f is a homeomorphism. Theorem 17. Let f : X → X be a continuous onto map on a compact metric space X (not necessarily a homeomor- phism). Then h(f ) = h(f −1). Proof. See Appendix A.3. 3. An application of inverse limits The topological complexity of the inverse limit space of a dynamical system f : I → I is an indication of the complexity of the dynamics of f on I. A simple inverse limit space (e.g., an arc) implies simple dynamics, whereas a complicated inverse limit space (e.g., a Knaster bucket handle) implies (under certain conditions) complicated dynamics. Understanding the topological structure of the inverse limit space is also important because there is a huge difference (topologically) between being, say, an indecomposable arc continuum and an hereditarily indecomposable continuum which is a continuum with the property that each subcontinuum is indecomposable. Though both may be associated with chaotic dynamical systems, they are very different types of chaos. 432 J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 In this section, we apply the results from Section 2 to the cash-in-advance model (described in Section 3.1). We show that the inverse limit space can be as simple as an arc, so that dynamics are simple. It can be a double topologist’s sine curve (see Fig. 5). A double topologist’s sine curve is not an arc, but is a topologically simple continuum (very close to being an arc) so the dynamics are simple in this case as well. The inverse limit space can be the union of two indecomposable continua intersecting in a point or an arc. Many things can happen; this list is not exhaustive. 3.1. The CIA model In this subsection, we briefly describe the cash-in-advance model with particular focus on the model possibly having backward dynamics and being chaotic. The model is the standard endowment CIA model of Lucas and Stokey (1987). We closely follow the exposition of Michener and Ravikumar (1998), hereafter [MR]. It is an endowment economy with both cash and credit goods. There is a representative agent and a government. The government consumes nothing and sets monetary policy using a money growth rule. The household has preferences over sequences of the cash good (c1t) and credit good (c2t) represented by a utility function of the form ∞ t β U(c1t,c2t), (1) t=0 with the discount factor 0 <β<1. { }∞ ≥ The household seeks to maximize (1) by choice of c1t,c2t,mt+1 t=0 subject to the constraints c1t,c2t,mt+1 0, ptc1t ≤ mt, (2) mt+1 ≤ pty + (mt − ptc1t) + θMt − ptc2t, (3) { }∞ taking as given m0, y, θ and pt,Mt t=0. θMt represents a lump-sum transfer of cash from the government. The money supply {Mt} is controlled by the government and follows a constant growth path Mt+1 = (1 + θ)Mt, where θ is the growth rate and M0 > 0 given. [MR] make assumptions on the function U so that the solution to this problem will be interior and the solution to the first-order conditions and transversality condition will be necessary and sufficient. 2 2 Assumption 1. ([MR], p. 1120) The function U : R+ → R is C with U1 > 0,U2 > 0 and the Hessian matrix negative definite. Both c1t and c2t are assumed to be normal goods. Further, to guarantee interior solutions we will assume limU1(c, y − c) = lim U2(c, y − c) =∞, (4) c→0 c→y and that U1(y, 0) < ∞ and U2(0,y) < ∞. { }∞ A perfect foresight equilibrium is defined in the usual way as a collection of sequences c1t,c2t,mt t=0 and { }∞ = + Mt,pt t=0 satisfying the following: (1) the money supply follows the stated policy rule: Mt+1 (1 θ)Mt; (2) markets clear: mt = Mt and c1t + c2t = y; and (3) the solution to the household optimization problem is given by { }∞ c1t,c2t,mt+1 t=0. Let xt := mt/pt denote the level of real money balances. [MR] show that an equilibrium sequence for {xt} must satisfy B(xt) = A(xt+1), (5) where B(x):= xU2(min[x, c],y− min[x, c]), β A(x):= xU (min[x, c],y− min[x, c]), 1 + θ 1 and c is the unique solution to U1(x, y − x) = U2(x, y − x). Whether or not the dynamics going forward are single- valued depends on whether or not A(·) is invertible. J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 433 [MR] use two more assumptions in their paper which we include here for completeness and briefly describe what they imply for the model. Assumption 2. ([MR], p. 1125). There exists a b ∈ [0,c) such that xU1(x, y − x) is increasing in the region [0,b) and decreasing in the region (b, c]. This assumption is putting additional restrictions on the utility function so that the function A(·) is either hump-shaped or monotonically decreasing on [0,c]. Assumption 3. ([MR], p. 1125). (a) (1 + θ) >βand (b) b These conditions guarantee the existence of a solution x∗ > 0toA(x∗) = B(x∗) and that this intersection of the two functions occurs when A(x) is decreasing. One can show that there is a one-to-one mapping between equilibria in the model and non-negative sequences {xt} that satisfy the difference Eq. (5) and transversality condition t lim β U1(min[xt,c],y− min[xt,c])xt = 0. (6) t→∞ Since the discount factor β is assumed to be strictly between 0 and 1, any solution to the difference Eq. (5) that is bounded from above and from below by a strictly positive constant will satisfy the transversality condition. Consequently, in the discussion that follows, solutions to (5) that satisfy 0 9 Type II with A(0) > 0 is essentially the same, but one has to be more careful because solutions for xt+1 to A(xt+1) = B(xt) may not exists for low values of xt. 10 Kennedy et al. (2007) explore the topological nature of the inverse limit for case I.A only. 434 J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 Fig. 3. The figure illustrates the generic possible shapes for f : J → J in each of cases I.A–I.C when 0 For the type I case, note that if a sufficiently large x0 is chosen, then the requirement that A(x1) = B(x0) forces x1 >x0, and x1 is unique. Moreover, the continuation (x0,x1,...) for that initial condition is unique with limt→∞xt = ∞. Likewise, if a sufficiently small positive x0 is chosen, then x1 Proposition 3. For cases I.A–I.C and II.D, if (x0,x1,...) is a solution to A(xt+1) = B(xt) such that xˆt [a ] for t ≥ ˆt, the choice of xt+1 is unique, i.e., xt+1 such that A(xt+1) = B(xt) is unique; [b ] lim∞ lim x = 0 and x >x+ >x+ > ···. → t ˆt ˆt 1 ˆt 2 Proposition 4. For case I.A–I.C, if (x0,x1,...) is a solution to A(xt+1) = B(xt) such that xˆt > x¯ for some ˆt, then the choice of xˆt+1 may not be unique, but either [a ] lim lim xt =∞and eventually xt b Proposition 5. For cases II.A–II.D, if (x0,x1,...) is a solution to A(xt+1) = B(xt) such that xˆt > x¯ for some ˆt, then [a ] for t ≥ ˆt, the choice of xt+1 is unique, i.e., xt+1 such that A(xt+1) = B(xt) is unique; [b ] lim lim xt =∞and xˆt Fig. 4. The figure illustrates the generic possible states for f |J : J → J in each of the cases II.A–II.D when x b Proposition 6. For case II.A–II.C, if (x0,x1,...) is a solution to A(xt+1) = B(xt) such that xˆt [a ] lim and eventually xt It follows from Propositions 3–6 that in cases I.A–I.C and II.A–II.D a solution containing a member not in J would be locked into one behavior—either its members would eventually increase (monotonically) without bound, or they would eventually decrease (monotonically) to 0. From a dynamics perspective, these trajectories are not very interesting. From an economics perspective, they may be very interesting provided they constitute an equilibrium (the transversality condition may be violated). If the transversality condition is satisfied in these cases, then such equilibria 11 are referred to as self-fulfilling inflations (xt → 0) and self-fulfilling deflations (xt →∞). For cases I.A, I.B, II.A, II.B and II.D, f |J : J → J is onto, so if xt ∈ J there is a point xt+1 ∈ J such that xt = f (xt+1). However, in cases I.C and II.C, the map f |J : J → J is not onto, this implies that moving forward in time, some points must be thrown out of J implying (eventually) monotonic dynamics. Since our focus is on potentially interesting dynamics (e.g., chaotic), we remove these points from J. Let K be the collection of points not removed (this is nonempty since the steady state solution x∗ ∈ J). Since f |J is monotonic, one can show that K must be of the form K := [z, z¯] ⊂ J with z ≤ z¯. Note that f |K : K → K is onto and the picture looks similar to I.B or II.B if z < z¯. ∗ However, it is possible for z = z¯, in this case the only bounded solution is xt = x . To simplify notation, we will denote K by J and always consider f |J : J → J. We close this subsection with a characterization theorem for type I.A backward maps in the CIA model. By type I.A, we mean f :[x, x¯] → [x, x¯] with 0 Theorem 18. In the CIA model, f is a type I.A backward map generated by a utility function U(c1,c2) satisfying Assumption 1 with 0 < β<˜ 1 if and only if f is a type I.A map satisfying 1. for x ≥ x2, f is linear with slope 0 < β<˜ 1 with x2 = c/¯ β˜ , 1 2. f is C on [x, x¯] \{x1, c,¯ x2}, where x − + f x f x ( 1 ) = ( 2 ) + − , f (x1 ) f (x2 ) and 4. f (x)x Proof. See Appendix A.4. In the next two subsections, we analyze the inverse limit of the CIA model in cases I.A–I.C, II.B and II.C. 3.2. Simple inverse limits and dynamics The next proposition deals with cases I.B, I.C, II.B, and II.C and shows that the dynamics are simple in these cases. 11 See Woodford (1994) for a careful discussion of these cases. 436 J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 Proposition 7. Consider cases I.B, I.C, II.B, and II.C. Let (J, f ) be the inverse system. The inverse limit is an arc or a point. Proof. In these cases, f is a homeomorphism so by Theorem 8, the inverse limit is homeomorphic to J. Note that we still have a model with backward dynamics (A is non-invertible), however in these cases, the possible equilibria are not very complicated. There may be self-fulfilling inflations and deflations, but the dynamics restricted to J are simple. Note this result significantly adds to the result in Michener and Ravikumar (1998), Proposition 5, p. 1132. The next theorem proves that even though the dynamics (restricted to J) are multi-valued going forward in case I.A of the CIA model, the dynamics may be relatively simple. In particular, we give conditions under which the inverse limit space is an arc or a double topologist’s sin(1/x) curve. The dynamics are therefore simple, even though forward in time the corresponding map from our family is multi-valued. Theorem 19. (Kennedy et al., 2007). Suppose 0 (a) f ([a, b]) = [e, d], (b) f |[c, 1]:[c, 1] → [0, 1] is one-to-one, onto, and decreasing, (c) f |[0,c] is increasing, and (d) f (c) = 1,f(0) = d = f (a),f(b) = e, f (1) = 0. Then lim([0, 1],f) is either an arc or a double topologist’s sin(1/x) curve (see Fig. 5). ← The next two examples give maps satisfying the hypotheses in Theorem 19. They also (with a little added curvature) satisfy Theorem 18, i.e., these maps can be generated from the CIA model with a utility function satisfying Assumption 1 and 0 <β/(1 + θ) < 1. In the first case, the inverse limit is an arc, and for the second, the inverse limit is a double topologist’s sin(1/x) curve. Example 1. (Arc inverse limit). A map g satisfying the hypotheses of Theorem 19 exists such that the inverse limit space Z is an arc. Let m Fig. 5. Double topologist’s sin(1/x)curve: two topologist’s sin(1/x) curves joined together. J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 437 Fig. 6. A function on an interval that would yield simple dynamics and simple topology. The inverse limit for this map is an arc. where m = 1, n = 2, c = m + (3/8)(n − m), h = m + (3/8)(c − m), n2 =−3.8, Δn =−n2(c − h), Δm = (m − n) − Δn, m2 =−Δm/(h − m), m1 = n − m2m, n1 = m − n2c, d2 = 1.2n, p2 = m/(c + 2.4n), p1 = 2.4p2n. This map (with added curvature and “smoothed out” at h) is consistent with a type I.A from the CIA model. The flipped map g :[m, n] → [m, n] given by g(x):= (m + n) − f (m + n − x)isgivenby ⎧ ⎨⎪ p˜ 1 + p2xx∈ [m, c˜] g(x):= n˜ 1 + n2xx∈ [˜c, h˜] (8) ⎩⎪ m˜1 + m2xx∈ [h,˜ n] wherec ˜ = m + n − c, h˜ = m + n − h,˜p1 = (m + n)(1 − p2) − p1,˜n1 = (m + n)(1 − n2) − n1 andm ˜ 1 = (m + 2 n)(1 − m2) − m1. The “flipped” map g in Fig. 6 satisfies the hypotheses of Theorem 19. Note that g ([1, 1.2]) = g([1.8988, 1.9312]) = [1.0535, 1.0787] ⊂ [1, 1.2]. and |g(x)|=0.1619 for x ∈ [1, 1.2]. Also, g2([1.86, 2]) = g([1.0, 1.1089]) = [1.8988, 1.9165] ⊂ [1.86, 2] and g(x) =−0.778 for x ∈ [1.86, 2]. We | 2 |= ∈ ∪ =∩∞ 2n also have g (x) 0.1260 for x [1, 1.2] [1.86, 2]. Hence, I1 : n=0g ([1, 1.2]) consists of one point, as does =∩∞ 2n =∅ I2 : n=0g ([1.86, 2]) . Since lim(I, f ) does not contain an indecomposable continuum, by Theorem 12 f has zero topological entropy and ← does not have a horseshoe. Furthermore, by Theorem 11, we know f does not have a homoclinic fixed point. Example 2. (Double topologist’s sin(1/x) curve inverse limit). Let m Fig. 7. A function g on an interval that would yield simple dynamics and simple topology. The inverse limit for this map is a double topologist’s 2 sin(1/x)curve. Also included is the second iterate of this map g along with two invariant subsets I1 := [a1,b1] and I2 := [a2,b2]. wherec ˜ = m + n − c, h˜ = m + n − h,˜p1 = (m + n)(1 − p2) − p1,˜n1 = (m + n)(1 − n2) − n1 andm ˜ 1 = (m + n)(1 − m2) − m1. The “flipped” map g in Fig. 6 satisfies the hypotheses of Theorem 19. Fig. 7 contains a plot of g and g2. Then g is continuous on [0, 1] and satisfies the hypotheses of Theorem 19. Let I1 := [a1,b1] and I2 := [a2,b2], where b1 = (h˜ − p˜ 1)/p2 (so g(b1) = h˜), a1 = g(h˜) = n˜ 1 + n2h˜ (so a1 = 2 2 2 2 g (b1)), a2 = g (h˜) and b2 = g (a2). The points {a1,b1} form a two cycle for g . The points {a2,b2} also form a 2 2 2 two cycle for g . In the figure, we see that g |I1 is one-to-one and onto and g |I2 is one-to-one and onto. Thus, =∩∞ 2n =∩∞ 2n I1 n=0g (I1) is an interval in this case. And I2 : n=0g (I2) is an interval, too. Then the inverse limit space for this example is a double topologist’s sin(1/x) curve (again see Fig. 5 for a picture of this continuum). Again, since lim(I, f ) does not contain an indecomposable continuum, by Theorem 12 f has zero topological entropy and does not ← have a horseshoe. We also conclude by Theorem 11, f does not have a homoclinic fixed point. 3.3. Complicated inverse limits and dynamics The next three examples give maps conjugate to maps satisfying Theorems 13, 14 and 15. Again, with a little added curvature, these maps will satisfy Theorem 18 and so are consistent with the CIA model. Example 3. (Two Knaster bucket handles inverse limit). Let f : I → I where ⎧ ⎨⎪ m1 + m2xx∈ [a, d] f (x):= n1 + n2xx∈ [d, c¯] , (11) ⎩⎪ p1 + p2xx∈ [¯c, b] with I = [a, b], a = 1, b = 2,c ¯ = a + (2/7)(b − a), d = (2/3)a + (1/3)c, n2 =−1, m2 =−(b − a − c + d)/(d − a), m1 = b − m2a, n1 = a − n2c, p2 = a/(c + 24/5), p1 = (24/5)p2. We see in Fig. 8 that this map (“flipped”) satisfies Theorem 13 so the resulting inverse limit is two Knaster bucket handles joined at a point or an arc. Since lim(I, f ) contains an indecomposable continuum, by Theorem 12 f has ← positive topological entropy and f has a horseshoe. Example 4. (Theorem 14). Here we construct an example that is conjugate to a map satisfying Theorem 14 so the inverse limit is indecomposable. Let ⎧ ⎨⎪ m1 + m2xx∈ [a, d] f (x):= n1 + n2xx∈ [d, c¯] , (12) ⎩⎪ p1 + p2xx∈ [¯c, b] with I = [a, b], a = 1, b = 2,c ¯ = a + (1/6)(b − a), d = (2/3)a + (1/3)c, n2 =−1, m2 =−(b − a − c + d)/(d − a), m1 = b − m2a, n1 = a − n2c, p2 = a/(c + 24/5), and p1 = (24/5)p2. We see in Fig. 9 that this map (“flipped”) satisfies Theorem 14 so the resulting inverse limit is an indecomposable continuum. This map does not have a period 3 (it does have a period 5 orbit). Let 0 <μ<1 and considerc ¯ = J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 439 Fig. 8. Two Knaster Bucket Handles: We see that f “flipped” satisfies Theorem 13. a + μ(b − a). When μ = 1/5, we have f (a) = q,soTheorem 13 implies the inverse limit is two Knaster bucket handles joined at a point or by an arc. For slightly smaller μ,wehavef (a) Example 5. (Theorem 15). Here we construct an example that is conjugate to a map satisfying Theorem 15 so the inverse limit is indecomposable. Let f : I → I where ⎧ ⎨⎪ m1 + m2xx∈ [a, d] f (x):= n1 + n2xx∈ [d, c¯] , (13) ⎩⎪ p1 + p2xx∈ [¯c, b] Fig. 9. Indecomposable continuum: we see that f and f 2 “flipped” satisfy Theorem 14. 440 J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 Fig. 10. Indecomposable continuum: we see that f “flipped” satisfies Theorem 15. with I = [a, b], a = 1, b = 2,c ¯ = a + (0.255)(b − a)p2 = 0.7, p1 = a − p2c¯, d = (1.05)ac1 = p1 + p2b, c2 = p1 + p2c1, n2 =−(¯c − a)/(¯c − c2), n1 = c¯ − n2c2, Δn =−n2(¯c − d), Δm = (b − a) − Δn, m2 =−Δm/(d − a), and m1 = b − m2a. We want the point a = 1 to be part of an n-cycle with n ≥ 3 with an integer k (relative prime to n) with f k(a) being the first element in O+(a) \{a}. We see in Fig. 10 that this map (“flipped”) satisfies Theorem 15: a = 1 is a 5-cycle (5 ≥ 3) and f (a) is the first element in O+(a) \{a} (k = 1 is relative prime to n = 5). Consequently, the resulting inverse limit is indecomposable. Again, by Theorem 10, since lim(I, f ) is indecomposable, f must have periodic point ← not of power 2, and by Theorem 12 f has positive topological entropy and a horseshoe as well. 4. Conclusion In this paper, we demonstrate how inverse limits can be used to explore chaotic behavior in dynamic economic models (with or without backward dynamics). The inverse limit approach is useful in at least two ways. First, the inverse limit can be useful for detecting or ruling out complicated dynamics. This offers an alternative method that does not work directly with establishing the (non)existence of a 3-cycle (or other cycles and using Sarkovskii’s ordering on the integers), nor require the numerical calculation of topological entropy. Second, the inverse limit approach offers a way of thinking about different types of chaos based upon the topological structure of the inverse limit. When the topology of the inverse limit space changes there is a qualitative change in the dynamics. Thus whenever the inverse limit changes topologically, the behavior of the dynamical system changes in a significant qualitative manner. We have shown that the inverse limit spaces from the cash-in-advance model can behave quite differently, both topologically and dynamically. The topology and dynamics can be quite simple when the inverse limit space is an arc, or the space may be indecomposable – which automatically means the presence of chaotic behavior, although not necessarily on the entire continuum. There are many more questions one could ask and possibilities to explore, even about members of the family of maps we have studied, not to mention cases II.A and II.D (which we did not consider at all). Barge et al. (1996) analyze a one-parameter family of tent maps given by ⎧ ⎪ λ − 1 ⎨ λx + (2 − λ), 0 ≤ x ≤ λ fλ(x):= , ⎩⎪ λ − 1 −λx + λ, ≤ x ≤ 1 λ for λ ∈ [1, 2]. They prove that for a topologically large set of parameters (λ ∈ [1, 2]), the inverse limit space lim([0, 1],f ) is not only indecomposable, it also contains homeomorphic copies of lim[0, 1],f for each λ ∈ [1, 2]. ← λ ← λ J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 441 Thus, not only is this space complex, it has “sublayers” that are themselves complex. This continuum must con- tain uncountably many topologically different indecomposable proper subcontinua. We conjecture that there is some backward map from the CIA model conjugate to one of these tent maps. There are also dynamic economic models where the dynamics are multi-valued going forward and backward in time (e.g., Christiano and Harrison, 1999). One can think of this system as represented by a set-valued function R : X → 2X. New tools will need to be developed to handle such dynamical systems. One possible path is to pursue a strategy that is similar to the inverse limit approach. Fortunately, Ingram and Mahavier (2006) have already started exploring inverse limits using upper semi-continuous set-valued functions as the “bonding map.” Acknowledgements We would like to thank two anonymous referees for helpful comments and suggestions. Kennedy and Yorke would like to thank the NSF for financial support. Stockman would like to thank the Lerner College of Business & Economics for its generous summer research support. Appendix A. Proofs A.1. Proof of Proposition 1 We prove the proposition for a closed subset A; the proof is similar for an arbitrary nonempty set A. Suppose A is a closed nonempty subset of Y. Then πn(A) =∅and πn(A) is closed in X for each positive integer n. Fur- thermore, fn := f |πn+1(A) maps πn+1(A) into πn(A). If fn does not map πn+1(A) onto πn(A), there is a point xn in πn(A) \ fn(πn+1(A)). But xn ∈ πn(A) means that there is some point y ∈ A such that the n th coordinate of y is xn, and yn+1 ∈ πn+1(A), and f (yn+1) = fn(yn+1) = yn = xn. This is a contradiction, so fn is onto. If z is a point of A, then for each n, z ∈ π (A), so z ∈ lim(π (A),f|π + (A)). Thus, A ⊂ lim(π (A),f|π + (A)) and n n ← n n 1 ← n n 1 Aˆ := lim(π (A),f|π + (A)) is a closed, nonempty subset of Y. ← n n 1 For each n, let An ={x ∈ Y : for i ≤ n, xi ∈ πi(A)}. Then each An is a closed nonempty subset of Y, A1 ⊃ A2 ⊃···, ∩∞ = ˆ ∈ ≤ n and n=1An A.Ifx An, then d(x, y) 1/2 , where y is a point of A whose n th coordinate is xn (which means that yi = xi for each i ≤ n). Then A = Aˆ . A.2. Proof of Proposition 2 First we translate. Let d = f (b). Then h :[a, b] → [0, 1] defined by h(x) = (x − a)/(b − a) is a homeomorphism = ◦ ◦ −1 = = = = c−a and g h f h is a map from [0, 1] onto [0, 1]. Note that h(a) 0 and h(b) 1. Let cg h(c) b−a . The map g is strictly decreasing on [0,cg] and strictly increasing on [cg, 1]. If f |[c, b] is linear, then g|[cg, 1] is linear. Thus, f is conjugate to the map g :[0, 1] → [0, 1]. Now, we flip the map. Let H(x):= 1 − x for x ∈ [0, 1], then H :[0, 1] → [0, 1] is a homeomorphism such that H−1 = H. Let k = H ◦ g ◦ H. Note that H(0) = 1, H(1) = 0, and H(cg) = 1 − cg. The map k|[1 − cg, 1] is strictly decreasing and k|[0, 1 − cg] is strictly increasing. If f is linear on [c, b], then k is linear on [0, 1 − cg]. A.3. Proof of Theorem 17 Given our definition for h(f −1), the proof is as trivial as the case when f is a homeomorphism. The result follows from the symmetry of the sets lim(X, f, n) and lim{X, f, n}. To see this, let C :={x1,...,xM}⊂lim{X, f, n}. Then C ← → → is (D , ,lim{X, f, n})-spanning if and only if C¯ :={x¯1, x¯2,...,x¯M}⊂lim(X, f, n)is(D , ,lim(X, f, n))-spanning n → ← n ← ¯ j = j ≤ ≤ ≤ ≤ where (after possibly re-ordering the points in C and C) we have xi x¯n+1−i for 1 i n and 1 j M.It then follows that S(D , ,lim(X, f, n)) = S(D , ,lim(X, f, n)) for each n ∈ N and >0. Taking limits we have n → n ← h(f ) = h(f −1). 442 J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 A.4. Proof of Theorem 18 (⇒): Let f :[x, x¯] → [x, x¯] given by f (x):= B−1(A(x)) be a type I.A backward map. We havex ¯ = f (x), f (¯c) = x with x < c<¯ x¯. f is continuous since A and B−1 are continuous. f |[x, c¯] one-to-one and onto [x, x¯], and f |[¯c, x¯] one-to-one. Recall we have B(x):= xU2(min[x, c¯],y− min[¯c, x]), A(x):= βxU˜ 1(min[x, c¯],y− min[¯c, x]). Note there exists a unique x Forc ¯ ≤ x ≤ x2,wehaveA(x) = β˜ U¯ 1x so B(f (x)) ≡ A(x) = β˜ U¯ 1x or f (x)U2(f (x),y− f (x)) ≡ β˜ U¯ 1x. This implies U2(f (x),y− f (x)) ≡ β˜ U¯ 1x/f (x) > 0, and f (x)x − f (x) U22(f (x),y− f (x)) − U21(f (x),y− f (x)) = β˜ U¯ 1 . f (x)f (x)2 2 Since D U is negative definite and both the cash and credit goods being normal goods, we have U22(f (x),y− f (x)) − U21(f (x),y− f (x)) < 0 implying that we must have f (x)x 1 since f (x) >f(x)x and f (x) > 0 for x ∈ I3. w is continuous since f is C on I3. To recover w(·), note that for x ∈ [y − c,¯ y − x], we have f˜ (y − x) w (x) = β˜ U¯ . 1 y − x Then for x ∈ [y − c,¯ y − x]wehave x w(x):= Cw + w (z)dz. y−c¯ Also for x ∈ [x, c¯] define B(x):= xw (y − x), then by construction we have B(f (x)) ≡ A(x) for x ∈ I3. Note that we now have B(x) defined on [x, x¯]. On I1 ∪ I2, define A(x):= B(f (x)) and define u (·)onI1 ∪ I2 by u (x):= A(x)/(βx˜ ) > 0. Then for x ∈ [x, c¯] x u(x):= Cu + u (z)dz. x 1 1 Note that A is C on I1 and on I2 with A (x) = B (f (x))f (x) < 0. Consequently, u (·)isC on I1 and on I2 with u(x) = [A(x)βx˜ − βA˜ (x)]/(βx˜ )2 < 0. For u to be continuous we need A to be continuous so we need to check − = + + − = − + − = whether or not A (x1 ) A (x1 ). By construction, we have B (¯c )/B (¯c ) f (x2 )/f (x2 ). At x1,wehaveA (x1 ) − − = + − + = + + = − + B (f (x1 ))f (x1 ) B (¯c )f (x1 ) and A (x1 ) B (f (x1 ))f (x1 ) B (¯c )f (x1 ). Taking ratios we have − − A x B c+ f x ( 1 ) = (¯ ) ( 1 ) = + − + 1 A (x1 ) B (¯c )f (x1 ) + − = − + ∪ since by the hypothesis of the theorem we have f (x1 )/f (x1 ) f (x2 )/f (x2 ). So u is continuous on I1 I2.By construction B(f (x)) ≡ A(x)onI1 ∪ I2. So u :[x, c¯] → R with u > 0 and u < 0 and w :[y − c,¯ y − x] → R with w > 0 and w < 0. Both of these function can be extended to R+, call the extensionsu ˜ and w˜ so that U˜ (c1,c2):= u˜(c1) + w˜ (c2) satisfies Assumption 1. References Barge, M., Brucks, K., Diamond, B., 1996. Self-similarity in inverse limit spaces of the tent family. Proceedings of the American Mathematical Society 124, 3563–3570. Barge, M., Diamond, B., 1994. The dynamics of continuous maps of finite graphs through inverse limits. Transactions of the American Mathematical Society 344 (2), 773–790. Barge, M., Martin, J., 1985. Chaos, periodicity, and snakelike continua. Transactions of the American Mathematical Society 289, 355–365. Bing, R.H., 1951. Concerning hereditarily indecomposable continua. Pacific Journal of Mathematics 1, 43–51. Canovas, J.S., 1999. On topological sequence entropy and chaotic maps on inverse limit spaces. Acta Mathematica Universitatis Comenianae 68 (2), 205–211. Christiano, L.J., Harrison, S.G., 1999. Chaos, sunspots and automatic stabilizers. Journal of Monetary Economics 44 (1), 3–31. Devaney, R.L., 2003. An introduction to chaotic dynamical systems, 2nd ed. Westview Press, Boulder, Colorado. Grandmont, J.-M., 1985. On endogenous competitive business cycles. Econometrica 53, 995–1045. Ingram, W.T., 1995. Periodicity and indecomposability. Proceedings of the American Mathematical Society 123 (6), 1907–1916. Ingram, W.T., 2000. Inverse limits, Aportaciones Matematicas,´ Investigacion´ 15. Sociedad Matematica´ Mexicana, Mexico.´ Ingram, W.T., 2002. Invariant sets and inverse limits. Topology and its Applications 126, 393–408. Ingram, W.T., 2003. Families of inverse limits. Topology Proceedings 27, 189–201. Ingram, W.T., Mahavier, W.S., 2004. Interesting dynamics and inverse limits in a family of one-dimensional maps. American Mathematical Monthly 111 (3), 198–215. Ingram, W.T., Mahavier, W.S., 2006. Inverse limits of upper semi-continuous set valued functions. Houston Journal of Mathematics 32 (1), 119–130 (electronic). Kennedy, J.A., 1993. How indecomposable continua arise in dynamical systems. In: Papers on General Topology Applications, Seventh Summer Conference at the University of Wisconsin, vol. 704, Annals of the New York Academy of Science, pp. 180–201. Kennedy, J.A., 1995. A brief history of indecomposable continua. In: Cook, H., Ingram, W., Kuperberg, K., Lelek, A., Minc, P. (Eds.), In: Modern Methods in Continuum Theory, Lecture Notes in Pure and Applied Math, vol. 179. Marcel Dekker, New York, NY, pp. 103–126. Kennedy, J.A., Stockman, D.R., 2007. Chaotic equilibria in models with backward dynamics. Journal of Economic Dynamics & Control, in press. 444 J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 Kennedy, J.A., Stockman, D.R., Yorke, J.A., 2007. Inverse limits and an implicitly defined difference equation from economics. Topology and its Applications 154, 2533–2552. Kuratowski, K., 1968. Topology II. Academic Press, New York, NY. Li, S.H., 1992. Dynamical properties of the shift maps on the inverse limit spaces. Ergodic Theory and Dynamical Systems 12 (1), 95–108. Li, T.-Y., Yorke, J.A., 1975. Period three implies chaos. American Mathematical Monthly 82, 985–992. Lucas, R.E., Stokey, N.L., 1987. Money and interest in a cash-in-advance economy. Econometrica 55, 491–513. Medio, A., Raines, B., 2006. Inverse limit spaces arising from problems in economics. Topology and its Applications 153, 3439–3449. Medio, A., Raines, B., 2007. Backward dynamics in economics. the inverse limit approach. Journal of Economic Dynamics and Control 31, 1633–1671. Michener, R., Ravikumar, B., 1998. Chaotic dynamics in a cash-in-advance economy. Journal of Economic Dynamics and Control 22, 1117–1137. Nadler Jr., S.B., 1992. Continuum Theory: An Introduction. Marcel Dekker, New York. Woodford, M., 1994. Monetary policy and price level determinacy in a cash-in-advance economy. Economic Theory 4, 345–380.