DDynamics with Choice
Lev Kapitanski ∗ and Sanja ˇZivanovi´cDepartment of MathematicsUniversity of MiamiCoral Gables, FL 33124, USANovember 5, 2018 Abstract
Dynamics with choice is a generalization of discrete-time dynamics where insteadof the same evolution operator at every time step there is a choice of operators totransform the current state of the system. Many real life processes studied in chem-ical physics, engineering, biology and medicine, from autocatalytic reaction systemsto switched systems to cellular biochemical processes to malaria transmission inurban environments, exhibit the properties described by dynamics with choice. Westudy the long-term behavior in dynamics with choice. We prove very general re-sults on the existence and properties of global compact attractors in dynamics withchoice. In addition, we study the dynamics with restricted choice when the allowedsequences of operators correspond to subshifts of the full shift. One of practicalconsequences of our results is that when the parameters of a discrete-time systemare not known exactly and/or are subject to change due to internal instability, or astrategy, or Nature’s intervention, the long term behavior of the system may not becorrectly described by a system with “averaged” values for the parameters. Theremay be a Gestalt effect.
Mathematical setting for discrete dynamics is a space X and a map S : X → X . Thespace X is the state space, the space of all possible states of the system. The map S , ∗ The work was supported in part by the NIH Director’s Initiative, 1 P20 RR020770-01. a r X i v : . [ m a t h . D S ] J u l he evolution operator, defines the change of states over one time step: x ∈ X at time t = 0 evolves into S ( x ) at t = 1, S ( S ( x )) at t = 2, . . . , S n ( x ) at t = n , etc. If instead ofone operator, S , we have a choice of evolution operators, S , S , . . . , S N − , and at everytime step choose one of them, then we have a dynamics with choice . One way to visualizethe multitude of choice through time is to generate the infinite tree of choices. This is aninfinite rooted tree in which the root has N children, every child has N children, and soon. The root corresponds to t = 0, its children correspond to t = 1, the children of thechildren correspond to t = 2, etc. At every step, the children of each node are labeled 0through N −
1. Beginning at the root infinite branches (paths, strategies) represent thepossible choices: for example, in Figure 1 we choose the path w that starts with 011...(bold edges). For this choice, the first few points in the trajectory of a point x ∈ X are x = S ( x ), x = S ( x ) = S ( S ( x )), x = S ( x ) = S ( S ( S ( x ))), etc. It isnatural to encode the infinite paths (beginning at the root) by one-sided infinite words(strings, sequences) on N symbols. If w is such sequence, it is convenient to align it withthe set of non-negative integers Z ≥ and denote by w ( k ) the ( k + 1)-st letter of w , i.e., w = w (0) w (1) w (2) . . . . Thus, w (0) = 0, w (1) = 1, w (2) = 1, are the first three symbolsof the path w = 011 . . . .Figure 1: The tree of choices in the case of two operators • In this paper we study dynamics with choice, i.e., the dynamics of points and subsets of X along all possible paths simultaneously. We will explain what this means momentarily.Here we would like to emphasize that, from the point of view of long-term behavior,dynamics with choice, in general, is not the same as the union of trajectories along differentinfinite paths. We will return to this point later when we talk about the Gestalt effect.Let Σ be the one-sided shift on N symbols, [14]. This means that, first, Σ is the set2f all one-sided infinite strings w = w (0) w (1) w (2) . . . , where each w ( j ) is a symbol fromthe list of N symbols [0 , , . . . , N − σ : Σ → Σacting by erasing the first symbol, σ ( w ) = w (1) w (2) w (3) . . . . Given the state space X andoperators S , . . . , S N − , we define the corresponding dynamics with choice as the discretedynamics on the state space the product X = X × Σ with the following evolution operator: S : ( x, w ) (cid:55)→ ( S w (0) ( x ) , σ ( w )) . (1)One can think of a w ∈ Σ as a plan, a strategy, or as Nature’s intervention. Dynamics withchoice is a language to describe processes where different strategies could be applied orhappen. Most of mathematical models in natural sciences and engineering are expressedin terms of differential equations. Those equations are often continuous limits of discreteequations. Continuous case is easier for qualitative analysis. However, there are situationswhere discrete equations describe the processes better. Every realistic model comes withparameters. We are interested in situations where parameters may change due to, e.g.,internal instability or outside intervention. In an illustrative example in section 3, thecoefficients a and b are proportional to the biting rate of mosquitoes which depends, forexample, on temperature and humidity which may change from day to day and duringthe day.In this paper we study long-term regimes in dynamics with choice. More specifically,we define and study global compact attractors in dynamics with choice. By a globalcompact attractor we mean the minimal compact set that attracts all boundedsets , see section 2.1 for definitions and references. Thinking in terms of a model withparameters, assume we know that for each admissible fixed (in time) set of parametersthe system possesses a global compact attractor. What happens when the parametersswitch between admissible values? Is there an attractor? How is it related to attractorscorresponding to fixed parameters? Is there a Gestalt effect? These are the questions weaddress in this paper.There are many real life and engineered systems that switch between different modesof operation (the so-called hybrid systems). When the behavior in each mode is modeledusing continuous dynamics and the transitions are viewed as discrete-time events, suchsystems are called switching or switched. Analysis and especially control of switchingsystems is an area of intensive research, see, e.g., Liberzon’s book [20] and the survey byMargaliot [22]. There is a natural affinity between switching systems and dynamics withchoice (see, e.g., [15]), but we will not explore it at this time.Readers familiar with iterated function systems, [13, 7], may wonder if there is aconnection between iterated function systems and dynamics with choice. Indeed there is,but we have to establish it (in section 2.3.3).3 (general) Iterated Function System (IFS) can be viewed as a discrete dynamics onthe space 2 X (of subsets of X ). The operators S , S , . . . , S N − define evolution on 2 X by means of the Hutchinson-Barnsley operator: F : A (cid:55)→ F ( A ) := S ( A ) ∪ S ( A ) ∪ · · · ∪ S N − ( A ) . (2)Following a long-standing tradition, people studying dynamics are first of all interested infixed points. In the case of an IFS, those are the fixed points of the Hutchinson-Barnsleyoperator. As has been well illustrated by Barnsley, for many simple IFSs on the planeone can (use a computer to) plot their compact fixed points (sets) and obtain fascinatingfractals, see [7, 8]. Generating fractals is one of the main motivations in the study ofIFSs. In some papers a fractal is defined as the compact invariant set of an IFS, see [8]and references therein.To prove that an IFS does have a fixed point, the general definition should be mademore specific. One needs to specify the properties of the space X ; the space 2 X should benarrowed to an appropriate class of subsets; assumptions should be made on the operators S , S , . . . , S N − . As an example we state the original result of Hutchinson, [13, Section3]. Theorem (Hutchinson).
Let X be a complete metric space (with metric d ). Denote by ¯ B ( X ) the space of all non-empty closed bounded subsets of X . Assume that each operator S , S , . . . , S N − is a strict contraction (i.e., there is a number γ ∈ (0 , such that d ( S j ( x ) , S j ( y )) ≤ γ d ( x, y ) for every pair x, y ∈ X and for all j ). Define the evolutionoperator ¯ F : ¯ B ( X ) → ¯ B ( X ) by the formula ¯ F : A (cid:55)→ ¯ F ( A ) = S ( A ) ∪ S ( A ) ∪ · · · ∪ S N − ( A ) . (3) Then there exists a unique fixed point K ∈ ¯ B ( X ) of ¯ F . Viewed as a subset of X , the set K is compact. Also, K attracts every closed bounded subset of X in the sense that, forany C ∈ ¯ B ( X ) , d H ( ¯ F n ( C ) , K ) → as n → ∞ , where d H is the Hausdorff distance. The IFS with contractive operators S j are called hyperbolic. Over the years this resulthas been generalized in many different directions (different assumptions on X and/or S j ),see [4] for references.The iterated function systems with probabilities and the ‘chaos games’ in particularshow an apparent link to dynamics with choice. Recall that an iterated function systemwith probabilities is an IFS ( X ; S , . . . , S N − ) together with probabilities p , p , . . . , p N − We use the abbreviation IFS for single and IFSs for plural forms. S , S , . . . , S N − , where each p j > p + p + · · · + p N − = 1,[7]. The random iteration algorithm (aka the chaos game, [8]) starts with the choice ofinitial state x ∈ X . Next, define recursively x n +1 by choosing its value from the set { S ( x n ) , S ( x n ) , . . . , S N − ( x n ) } with respective probabilities p , p , . . . , p N − . The choiceof operators thus will be encoded in some strategy w = w (0) w (1) w (2) · · · ∈ Σ, i.e., x n +1 = S w ( n ) ( x n ). To show that the sequence ( x n ) is determined by x and w ∈ Σ, wewrite ( x n ( x , w )). Consider the averages of the delta-measures concentrated at the points x n . It turns out that (under certain conditions) the averages n − (cid:0) δ x + δ x + · · · + δ x n − (cid:1) converge weakly to the invariant measure of the IFS. More precisely, consider the followingMarkov operator on the space of probability measures on X : M : ν (cid:55)→ M ( ν ) = N − (cid:88) j =0 p j S j ( ν ) , where S j ( ν )( A ) = ν ( S − j ( A )) for measurable sets A ⊂ X . Hutchinson showed in [13]that under the assumptions of the Hutchinson Theorem there exists a unique fixedpoint µ of the Markov operator and the support of µ is the fixed point K of the IFS( X ; S , . . . , S N − ). The following theorem was proved by Elton, [10], with later simplifi-cations by Forte and Mendivil, [11]. Theorem (Elton).
Assume X is a compact metric space, the operators S , . . . , S N − are strict contractions on X , and p , p , . . . , p N − are the probabilities. Let µ be thecorresponding invariant probability measure. Then, for any continuous function f : X → R and any x ∈ X , lim n →∞ n n − (cid:88) i =0 f ( x i ( x , w )) = (cid:90) X f ( x ) dµ ( x ) , for almost all strategies w ∈ Σ with respect to the product probability measure on Σ =[0 , , . . . , N − N induced by the distribution [ p , p , . . . , p N − ] on each factor. Returning to dynamics with choice, we repeat that our interest has not been motivatedby fractals. We would like to understand the long-term behavior in dynamics with choice.We assume that X is a complete metric space (with metric d ), the operators S , . . . , S N − are continuous, and each of the (semi)dynamical systems ( X, d, S j ) possesses a globalcompact attractor. Consider the corresponding dynamics with choice as the dynamics onthe product metric space X = X × Σ generated by the operator S acting accordingto the rule (1). From general theory (see section 2.1) we know that a system ought to Σ can be equipped with a metric making it a compact metric space, see Section 2.3.1 for a specificchoice. We denote here by dist the corresponding product-metric on X × Σ. X, d, S j ) do have attractors, the system( X , dist , S ) will not have a global compact attractor. There are several reasons why. Onecounter-example we borrow from [3] (where it is used in the context of IFS). Take X = R with standard metric d and define two maps, S and S , as follows: S ( x ) = (cid:26) , if x ≤ , − x, if x > S ( x ) = (cid:26) − x, if x ≤ , , if x > X, d, S j ) has the global compact attractor, a singleton { } . At thesame time, the trajectory x n = S w ( n − ◦ S w ( n − ◦ · · · ◦ S w (0) ( x ) corresponding to theperiodic string w = 010101 . . . is unbounded for any initial point x (cid:54) = 0. Hence, there isno compact attractor attracting ( x , w ).The second example is infinite-dimensional. Let B = B ( p ) and B = B ( p ) be twodisjoint closed unit balls centered at p and p in an infinite-dimensional Banach space.Let X = B ∪ B . Define the maps S and S as follows: on B the map S is a contractionand it maps B to B ; the map S is a contraction on B and maps B to B : S ( x ) = (cid:26) p + ( x − p ) , if x ∈ B ,p + ( x − p ) , if x ∈ B S ( x ) = (cid:26) p + ( x − p ) , if x ∈ B ,p + ( x − p ) , if x ∈ B The system (
X, d, S ) does have the global compact attractor, { p } , and ( X, d, S ) doeshave the global compact attractor, { p } . The corresponding dynamics with choice,( X , dist , S ), does have the global closed attractor, namely, X , but does not have theglobal compact attractor.In the first example, the maps are compact (which is good), but they do not have ajoint bounded absorbing set (lack of dissipativity in ( X , dist , S )). In the second example,there is a joint bounded absorbing set, B ∪ B , but there is not enough compactness (themaps S j are not compact, not contracting, and, more generally, not condensing).These examples show what kind of situations do not allow global compact attractorsin the dynamics with choice. Thus, we make additional assumptions. First, we assumethat there exists a bounded absorbing set that absorbs every bounded set regardless ofthe strategy. In applications, absorbing set is usually a ball of the radius that dependson the parameters of the model. Our “dissipativity” assumption means that there is acommon estimate on the radius for different values of the parameters. Assumption 1.
There is a closed, bounded set B ⊂ X such that for every bounded A ⊂ X there exists m ( A ) > such that S w ( n − ◦ S w ( n − ◦ · · · ◦ S w (0) ( A ) ⊂ B for every word w = w (0) w (1) . . . w ( n − of length n ≥ m ( A ) . S j is condensingwith respect to a common measure of noncompactness. This assumption covers practicallyall situations encountered in applications: contractions, compact operators, and compactplus contractions. As their name suggests, measures of noncompactness measure how fara set is from being compact. There are several different measures of noncompactness inuse, [1]. For example, the Hausdorff measure of noncompactness of a set A is the infimumof (cid:15) > A has a finite (cid:15) -net. In this paper we use only very general propertiesshared by all popular measures of noncompactness, see Definition 7 in section 2.2 below.Let ψ be a measure of noncompactness (as in Definition 7). An operator S : X → X is condensing with respect to ψ iff ψ ( S ( A )) < ψ ( A ) for any non-compact set A , and ψ ( S ( A )) = ψ ( A ) = 0 if A is compact. Our second general assumption is this. Assumption 2.
Each operator S j is ψ -condensing. In section 2.3 we prove the following result.
Theorem 1.
Let X be a complete metric space and let S , S , . . . , S N − be continuous,bounded (i.e., take bounded sets to bounded sets) maps X → X . In addition, let assump-tions 1 and 2 be satisfied. Then the system ( X , dist , S ) has a global compact attractor, M . (That M is the global compact attractor means that M is the smallest compact in X attracting every bounded set in X .)The attractor M has the following properties.(1) M is (strictly) invariant: S ( M ) = M .(2) M is the union of all closed bounded sets A ⊂ X with the property A ⊂ S ( A ) .(3) M is the maximal closed set with the property A ⊂ S ( A ) ; in particular, M is themaximal (strictly) invariant closed set.(4) Through every point ( x, w ) ∈ M passes a complete trajectory. This means there ex-ists a two-sided sequence . . . , x − , x − , x , x , x , . . . of points in X and a two-sidedinfinite string . . . s ( − s ( − s (0) s (1) s (2) . . . such that x (0) = x and s (0) s (1) s (2) · · · = w (0) w (1) w (2) . . . and such that S s ( n ) ( x n ) = x n +1 for every integer n .(5) M is the union of all complete, bounded trajectories in X . Given the state space X and operators S , . . . , S N − , there are two ways of describingdynamics generated by the corresponding IFS. First, one can follow the trajectories ofbounded subsets of X under the iterations of the Hutchinson-Barnsley map ¯ F , see (3).We denote such system by ( X, d, ¯ F ). The notion of the global compact attractor as theminimal compact set that attracts all bounded sets, is well-defined for ( X, d, ¯ F ). The7econd possibility is to choose the space of closed bounded sets, ¯ B ( X ), as the state spaceof the system and study the dynamics of its points under the iterations of ¯ F . As a rule,¯ B ( X ) is equipped with the Hausdorff distance d H . Thus we obtain the second system,( ¯ B ( X ) , d H , ¯ F ). It turns out that from the point of view of global compact attractorsthe dynamical system ( ¯ B ( X ) , d H , ¯ F ) is not very interesting (because convergence in theHausdorff metric is too strong). It possesses an attractor (in the sense we use here)essentially only if the maps S j are contractions, so then the attractor is just one point in¯ B ( X ). For more general S j , it makes more sense to study the fixed points of ¯ F .In sections 2.3.4 and 2.3.5 we establish the following connection between the dynamicswith choice and the corresponding IFS. Theorem 2.
Make the same assumptions on the space X and operators S , . . . , S N − asin Theorem 1. Then(1) The IFS ( X, d, ¯ F ) does have a global compact attractor, K .(2) The set K is the largest compact set in X which is invariant under the Hutchinson-Barnsley map ¯ F , K = ¯ F ( K ) .(3) The attractor M of the dynamics with choice has the following product structure: M = K × Σ . In the extensive literature on IFSs the main question is the existence of “the fractal”,i.e. the maximal compact set invariant under the Hutchinson-Barnsley operator ¯ F . Thiscorresponds to the second assertion of our Theorem 2. We believe that viewing “thefractal” of an IFS as the attractor of the dynamical system ( X, d, ¯ F ) is beneficial to thetheory of IFSs. This approach, in particular, points to the “right” assumptions on thespace X and the operators S j .Iterated function systems with compact (possibly multi-valued) operators have beenconsidered previously, see, e.g., [3]. The statement of Theorem 5.8 in [3] which establishesthe existence of a compact set invariant under ¯ F , needs some additional (dissipativity)assumption such as our Assumption 1, for example. The IFS with condensing (and multi-valued, in addition) operators have been considered by Le´sniak, [19] and Andres et al.,[4]. The assumptions of Theorem 3 in [4] require that the image ¯ F ( X ) of the whole space X be bounded. The word “minimal” referring to the “fractal” in [4, Theorem 3] shouldprobably be replaced by “maximal”, see also [19, Theorem 3].Our assumptions on the state space and the operators guarantee that, for every fixed j = 0 , . . . , N −
1, the discrete dynamics generated on X by S j does possess the globalcompact attractor (in X ). More generally, as we show in sections 2.3.4 and 2.3.5, itmakes sense to define individual attractors , A w corresponding to every string (infinite8ath in the tree of choices) w ∈ Σ. The attractors generated by each S j correspondto “constant” strings, w = jjj . . . . It is not hard to see that such attractors do notexhaust the attractor (fractal) K . There are situations when the union of all A w is K (this happens, in particular, when S j ’s are strict contractions). However, in general, theunion (cid:83) w ∈ Σ A w is strictly smaller than K . We give an example of this in section 2.3.5. Inthe cases when (cid:83) w ∈ Σ A w is strictly smaller than K we say that there is a Gestalt effect ,i.e., “the whole is greater than the sum of its parts.” This is a new phenomenon. It hasnot been observed in the framework of Iterated Function Systems because, as we show inLemma 15, the Gestalt effect cannot occur when operators S j are contractions.An important generalization of dynamics with choice is dynamics with restricted choice .The name should indicate that not all strategies (sequences w = w (0) w (1) · · · ∈ Σ) areallowed. In particular, we consider the sets in Σ that are closed and shift invariant,i.e., subshifts , see [21, 14]. Given a subshift Λ ⊂ Σ, we consider the dynamics on theproduct-space X Λ = X × Λ generated by the map S as in (1). Theorem 3.
Let the space X and the operators S ,. . . , S N − satisfy Assumptions 1 and2. Let Λ be a (one-sided) subshift of Σ . Consider the dynamical system ( X Λ , dist , S ) .(1) The dynamical system ( X Λ , dist , S ) does possess a global compact attractor, M Λ .(2) The attractor M Λ is invariant in the sense that S ( M Λ ) = M Λ . In fact, M Λ is themaximal invariant compact set in X Λ . Also, M Λ is an invariant compact subset ofthe global attractor M of the unrestricted dynamics ( X , dist , S ) .(3) Through every point ( x (0) , w ) ∈ M Λ passes a complete trajectory, i.e., there exista two-sided sequence of points . . . , x ( − , x (0) , x (1) , . . . and a two-sided symbolicsequence . . . w ( − w (0) w (1) . . . extending w (in the extension of the subshift Λ )such that S w ( i ) ( x ( i )) = x ( i + 1) for all integers i .(4) Let K Λ denote the projection of the attractor M Λ onto the X component. Theset K Λ is a compact subset of the set K of Theorem 2. There exist compact sets A , . . . , A N − such that K Λ = A ∪ A ∪ · · · ∪ A N − and K Λ = A ∪ A ∪ · · · ∪ A N − = S ( A ) ∪ S ( A ) ∪ · · · ∪ S N − ( A N − ) (4) (5) In general, the attractor M Λ is not a product. There may be infinitely many differentsets among the slices M ( w ) = { x ∈ X : ( x, w ) ∈ M Λ } . However, if the subshift Λ is sofic, the number of different slices is finite. K Λ is understood interms of the map from the code space to K as the image of Λ, [8, Theorem 4.16.3]. Thecorrespondence between the points of the code space, Σ, and the points of K is possiblebecause the maps are contractions (right away, or eventually). Our approach gives a newand more general view on restricted dynamics. We justify the name – attractor – andunveil attractors’ more subtle structure (assertion 5). This new approach allows us towork in a much more general setting and with transformations that are not contractions.We do not have and do not use a map from the code space into the attractor.We should mention the paper of Andres and Fiˇser, [2]. They use their result of [3] onthe existence of the fractal (the set K in our notation) for an IFS with compact opera-tors S j to conclude that fixed time solution operators of systems of ordinary differentialequations could play the role of maps generating the IFSs. As an illustration they usefive two-dimensional systems of ODEs to produce five operators (incidentally, contrac-tions, as noted in [2]) and plot the corresponding dragon-tail-like fractal set. Althoughtheir message is that IFSs and fractals can be generated by solution operators of ODEs,their examples can serve as an illustration for our dynamics with choice attractors (dueto Theorem 2(3)).Figure 2: Attractors for ( X, d, S ) (right) and ( X, d, S ) (left).In section 3 we apply the theory to a specific example of a discrete Ross-Macdonaldtype model of malaria transmission. The model can be viewed as a time discretization(with time step ∆ t ) of the ODE model, or as a pre-ODE form of the model. The reasonwe have chosen this model is because it is simple and we can visualize all the attractors.10he state space is the unit square X = { ( x, y ) : 0 ≤ x ≤ , ≤ y ≤ } . We usetwo sets of parameters, which define two operators, S and S . Those operators are notcontractions, but they are compact, because the system is finite-dimensional. The discretedynamical system generated on X by S has two fixed points, (0 , / , / S also has two fixed points,(0 ,
0) (unstable) and (7 / , /
12) (stable). The attractors of the systems (
X, d, S ) and( X, d, S ) are just the heteroclinic trajectories connecting the unstable and stable fixedpoints. They are depicted on figure 2. The effects of freedom of choice on the dynamicsare as follows. For the unrestricted dynamics with choice, we see (figures 4 and 5) thatFigure 3: The attractor slice K ;∆ t = 0 .
05. Figure 4: The attractor slice K ;∆ t = 0 . K , is a rather big set with the two individual attractors corresponding to S and S , respectively, forming parts of the boundary of K (the right and left sides). Theremaining part of the boundary is quite irregular when ∆ t is relatively large, figure 3. Forsmaller ∆ t , this part of the boundary becomes much smoother and looks like a smoothcurve, figure 4. In the limit ∆ t →
0, the set K retains its two-dimensional fullness. Itis not an attractor of an ODE with averaged parameters. In fact, all such attractors areone-dimensional (each is a heteroclinic trajectory connecting two fixed points) and lieinside of K .We consider also the dynamics with restricted choice corresponding to the golden meansubshift. We exhibit two different slices in the global attractor. Their projections on thestate space do overlap, and their union is smaller than the set K for the full shift.Finally, we remark that more general compact and condensing operators will be neededin the study of dynamics with choice related to nonlinear dissipative partial differentialequations, which we plan to address at a later time. The structure of the paper.
This Introduction is followed by two chapters. Inthe long chapter 2 we address theoretical questions. In section 2.1 we give definitionsand state the basic results related to global compact attractors. Section 2.2 deals withmeasures of noncompactness. In section 2.3 we study dynamics with choice. Dynamics11ith restricted choice is studied in section 2.4. In chapter 3 we analyze a simple exampleillustrating some of our theoretical results. Despite its simplicity, this example shows thatthe result of dynamics with choice is larger than the sum of its parts.
We start by collecting the basic facts about attractors. There are several books such as[5, 12, 17, 18, 26] devoted to this subject. Our presentation is closer to [17]. We presentonly the results that we need. For the proofs of Theorems 5 and 6 see the books quotedabove.Let Y be a complete metric space with metric dist and let Φ : Y → Y be a continuousmap. Iterations, Φ n , of Φ define a discrete (semi)dynamical system on ( Y, dist ). It is usefulto consider not only the dynamics of individual points under the action of Φ, but, moregenerally, the dynamics of bounded sets. Denote by B ( Y ) the collection of all boundedsubsets of Y . We say that the set A ∈ B ( Y ) attracts the set B ∈ B ( Y ) if (cid:126) dist (Φ n ( B ) , A ) → n →∞ , where the one-sided distance between two sets, (cid:126) dist ( C, A ), is understood as sup y ∈ C dist ( y, A ). Definition 4.
We call a set M ⊂ Y the global compact attractor of the system ( Y, dist , Φ) if • M is compact, • M attracts every bounded subset of Y , • M is the minimal set with these two properties. For a system to possess a global compact attractor, it should enjoy certain properties,namely, some form of compactness and some dissipativity. Here is the basic existence(and uniqueness) result.
Theorem 5.
The semidynamical system ( Y, dist , Φ) has a global compact attractor if andonly if it enjoys the following two properties:1. (“compactness”) For every bounded sequence ( y k ) in Y and every increasing sequenceof integers n k → + ∞ , the sequence Φ n k ( y k ) has a convergent subsequence. . (“dissipativity”) There exists a bounded set B ⊂ Y which absorbs every bounded setin the sense that for every A ∈ B ( Y ) there exists m ( A ) > such that Φ n ( A ) ⊂ B for all n ≥ m ( A ) . Some of basic properties of a global compact attractor are collected in the followingtheorem.
Theorem 6.
Assume that M is the global compact attractor of the semidynamical system ( Y, dist , Φ) . Then1. M is the union of all possible limits of sequences of the form Φ n k ( y k ) , where y k is abounded sequence in Y and n k → ∞ .2. M is (strictly) invariant: Φ( M ) = M .3. M is the union of all closed bounded sets A with the property A ⊂ Φ( A ) .4. M is the maximal closed set with the property A ⊂ Φ( A ) ; in particular, M is themaximal (strictly) invariant closed set.5. Through every point y ∈ M passes a complete trajectory, i.e., there exists a two-sidedsequence . . . , y − , y − , y , y , . . . of points in M such that y = y and y m +1 = Φ( y m ) for all integers m .6. M is the union of all complete, bounded trajectories in Y . In applications, people do not verify the “compactness” property of Theorem 5 directly.Instead, they use one of the known sufficient conditions that imply it. Two of the mostuseful sufficient conditions are: • Φ is a compact map (i.e., Φ : Y → Y is continuous and maps bounded sets intorelatively compact sets); • in the case Y is a Banach space, Φ is a sum of a compact operator and a strictcontraction.Compact Φ arise, e.g., in the finite-dimensional dynamics described by differential or dif-ference equations, or, in the infinite dimensional case, in dynamics described by parabolicequations. The “compact + contraction” Φ appear, e.g., in hyperbolic problems withdamping. Each of the two sufficient conditions implies that Φ is condensing with respectto some measure(s) of noncompactness. Since measures of non-compactness and condens-ing operators are not widely known, below we give a brief account of the facts we needand refer to [1] for more details. 13 .2 Measures of noncompactness Measures of noncompactness assign real non-negative numbers to bounded sets with value0 assigned exclusively to relatively compact sets. The basic examples are the Kuratowskimeasure of noncompactness α and the Hausdorff measure of noncompactness χ . Bydefinition, α ( A ) is the infimum of numbers (cid:15) > A admits a finite cover bysets of diameter less than (cid:15) . The number χ ( A ) is the infimum of those (cid:15) > A possesses a finite (cid:15) -net in Y . In this paper we adopt the following definition of a generalmeasure of noncompactness (our definition differs from that in [1]). Definition 7.
A function ψ assigning non-negative real numbers to bounded subsets of(a complete metric) space Y will be called a measure of noncompactness iff it has thefollowing properties:(i) ψ ( A ) = 0 if and only if A is relatively compact;(ii) If A ⊂ A , then ψ ( A ) ≤ ψ ( A ) ;(iii) ψ ( A ∪ A ) = max { ψ ( A ) , ψ ( A ) } ;(iv) There exists a constant c ( ψ ) ≥ such that | ψ ( A ) − ψ ( A ) | ≤ c ( ψ ) d H ( A , A ) , where d H is the Hausdorff distance, d H ( A , A ) = max { (cid:126) dist ( A , A ) , (cid:126) dist ( A , A ) } . Both α and χ enjoy all these properties. Note that property (iv) implies that the measuresof noncompactness of a bounded set and its closure are equal:(v) ψ ( A ) = ψ ( A ) . Definition 8.
A continuous bounded map
Φ : Y → Y is called condensing with respectto the measure of noncompactness ψ (we also say Φ is ψ -condensing) iff ψ (Φ( A )) ≤ ψ ( A ) for any bounded A , and ψ (Φ( A )) < ψ ( A ) if ψ ( A ) > (i.e., if A is not compact). Theorem 9.
Consider the system ( Y, dist , Φ) . Assume that Φ is condensing with respectto some measure of noncompactness ψ and that there exists a bounded set B which absorbsevery bounded set. Then ( Y, dist , Φ) possesses a global compact attractor. In the case ψ is the Kuratowski measure of noncompactness this result is proved byˇSeda, [25]. In Lemma 10 below we establish a more general result.14 .3 Dynamics with choice Fix an integer
N >
1. Using the integers 0 , . . . , N − alphabet , construct strings(words) of finite length and (one-sided) strings of infinite length. Denote by Σ ∗ the setof all finite length strings (words), and denote by Σ the set of all (one-sided) infinitestrings. The word of length 0 is the empty word. The set of non-empty words is denotedΣ + . Given a string w ∈ Σ ∗ ∪ Σ, w (0) is the first letter of w , and w ( k ) is the ( k + 1)-stletter of w . The length of w is denoted | w | . If w is a finite string and u ∈ Σ ∗ ∪ Σ, theirconcatenation is denoted w.u ; if | w | = m , then ( w.u )( m + k ) = u ( k ) for k = 0 , , . . . . Fora w ∈ Σ ∗ and s ∈ Σ ∗ ∪ Σ, we write w (cid:64) s if w is the beginning of the string s , i.e., if thereexists u ∈ Σ ∗ ∪ Σ such that s = w.u . For an infinite string, s , its first n letters form aword denoted s [ n ], i.e., s [ n ] = s (0) s (1) . . . s ( n − m willbe denoted by Σ ∗ m .Equip the space Σ ∗ ∪ Σ with the metric d Σ , where d Σ ( u, v ) = 2 − m if u [ m −
1] = v [ m − u [ m ] (cid:54) = v [ m ]. It is well-known, [8], that both Σ ∗ ∪ Σ and Σ with metric d Σ arecompact. The shift operator, σ , acts on infinite strings by deleting the first letter, i.e., σ ( u ) = u (1) u (2) . . . . The shift operator maps Σ onto itself. It is continuous; in fact, d Σ ( σ ( u ) , σ ( v )) ≤ d Σ ( u, v ). Let X be a complete metric space with metric d , and let S , S , . . . , S N − be continuous,bounded maps X → X . Define the product metric space X = X × Σ with metric dist , dist (( x, u ) , ( y, v )) = d ( x, y ) + d Σ ( u, v ) . The skew-product dynamics on X is generated by the map S : X → X acting accordingto the rule S ( x, u ) = ( S u (0) ( x ) , σ ( u )) . This map is obviously continuous and bounded. Because we will consider iterations of S such as S n ( x, u ) = ( S u ( n − ◦ · · · ◦ S u (1) ◦ S u (0) ( x ) , σ n ( u )) , we introduce the notation S w = S w ( n − ◦ · · · ◦ S w (1) ◦ S w (0) , where w is a word of length n . Thus, we can write S n ( x, u ) = ( S u [ n ] ( x ) , σ n ( u )). Assumption 1.
Assume there is a closed, bounded set B ⊂ X such that for every bounded A ⊂ X there exists m ( A ) > such that S w ( A ) ⊂ B for every word w of length n ≥ m ( A ) . B is usually a closed ball of radius that depends on the parameters ofthe model. Showing that for different values of the parameters there is a common estimateon the radius is enough to verify Assumption 1.]Let ψ be a measure of noncompactness as in Definition 7. Assumption 2.
Assume that each operator S j is ψ -condensing. We are going to apply Theorem 5 to prove the existence of the attractor. For this towork we need to justify the following fact:
For every bounded sequence ( x k , u k ) ∈ X and every increasing sequence of integers n k → + ∞ , the sequence S n k ( x k , u k ) has a convergent subsequence. Thus, pick a bounded sequence x k ∈ X and any sequence u k ∈ Σ. Pick an increasing to+ ∞ sequence n k . Since Σ is compact, the sequence σ n k ( u k ) has a convergent subsequence.So, we may assume from the very beginning that σ n k ( u k ) → s ∈ Σ. Denote w k = u k [ n k ].We have S n k ( x k , u k ) = ( S w k ( x k ) , σ n k ( u k )) . Since the Σ-component converges, we need to show that the sequence S w k ( x k ) in X has a convergent subsequence. Because Σ ∗ ∪ Σ is compact, we can choose a convergentsubsequence from w k . We will assume that w k itself converges to some w ∈ Σ. Lemma 10.
Under the Assumptions 1 and 2, let w n be a sequence of finite words ofincreasing lengths | w n | → + ∞ and w n → w ∈ Σ . For any bounded sequence x n thesequence S w n ( x n ) has a convergent subsequence. Proof.
Pick a bounded sequence x k . By Assumption 1, when the length of the word w n is sufficiently large, S w n ( { x k } ) ⊂ B . Dropping the first few terms if necessary, wewill assume that S w n ( { x k } ) ⊂ B for all n . Next, since w n → w , for arbitrarily large m ∗ we have w n [ m ∗ ] = w [ m ∗ ] for all sufficiently large n . Choosing m ∗ large enough, we have S w [ m ∗ ] ( { x k } ) ⊂ B . Writing w n = w [ m ∗ ] .s n , we see that S w n ( { x k } ) = S s n (cid:0) S w [ m ∗ ] ( { x k } ) (cid:1) and s n → s = σ m ∗ ( w ). Use Assumption 1 to find m ( B ) and assemble the set˜ B = (cid:91) v ∈ Σ ∗ m ( B ) S v ( B ) . The set ˜ B has the property that S v ( ˜ B ) ⊂ B for all finite words v . If we choose m ∗ abovesufficiently large (first choose it to guarantee S w [ m ∗ ] ( { x k } ) ⊂ B and then increase it by m ( B )), the set S w [ m ∗ ] ( { x k } ) will be inside of ˜ B . We will reformulate our problem now.16e have a sequence y k = S w [ m ∗ ] ( x k ) in ˜ B and a sequence s k → s . We need to show thatthe sequence S s k ( y k ) is relatively compact.Consider the positive (infinite) trajectory of ˜ B under the Hutchinson-Barnsley evolu-tion: C = ˜ B ∪ (cid:91) n ≥ (cid:91) v ∈ Σ ∗ n S v ( ˜ B ) . Define inductively C n +1 = (cid:91) j =0 ,...,N − S j ( C n ) . We have C ⊃ C ⊃ C ⊃ . . . .Introduce a collection H of all sets A ⊂ B that can be represented in the form A = (cid:91) n ≥ A n , where A n is a finite (or empty) subset of C n . We show next that every set A ∈ H is relatively compact. Because the sequence { S u n ( y n ) } is in H , our lemma will be proved. The argument that follows is a modification of a partof [1, Lemma 1.6.11].First, we claim that there exists a set A ∗ ∈ H such that ψ ( A ∗ ) = sup A ∈ H ψ ( A ) . This follows from Lemma 1.6.10 of [1]. Although their lemma is stated for the Hausdorffmeasure of noncompactness, their proof uses only the properties (i), (ii), and (iii) ofDefinition 7 and, therefore, works for our ψ .Now, A ∗ = (cid:83) n ≥ A ∗ n , where each A ∗ n is a finite (or empty) subset of C n . For every p ∈ A ∗ n with n ≥ q ∈ C n − and a j ∈ [0 , , . . . , N −
1] such that S j ( q ) = p . Denotethe resulting subset of C n − by A n − and define A = (cid:83) n ≥ A n . Clearly, A ∈ H . Hence, ψ ( A ) ≤ ψ ( A ∗ ). Now, consider the set F ( A ) = S ( A ) ∪ · · · ∪ S N − ( A ). Clearly, F ( A ) ∈ H and F ( A ) ⊃ (cid:83) n ≥ A ∗ n . Hence, ψ ( A ∗ ) = ψ ( (cid:91) n ≥ A ∗ n ) ≤ ψ ( F ( A )) = max { ψ ( S ( A )) , . . . , ψ ( S N − ( A )) } ≤ ψ ( A ) . The properties of ψ used here are, from left to right: the first equality uses properties (i)and (iii), the first inequality follows from (ii), the second equality follows from (iii), thesecond inequality is due to the assumption that each operator S j is ψ -condensing. Now, if A is not relatively compact, we must have ψ ( S j ( A )) < ψ ( A ) for every j . This would imply17 ( A ∗ ) < ψ ( A ) which contradicts the fact that ψ ( A ) ≤ ψ ( A ∗ ). Thus, ψ ( A ∗ ) = ψ ( A ) = 0,and so every set in H is relatively compact. This concludes the proof of Lemma.Applying now Theorems 5 and 6, we immediately obtain Theorem 1. We next proceedto the proof of Theorem 2. Consider the IFS dynamics (
X, d, ¯ F ). Having made Assumption 1 we will prove theexistence of a global compact attractor K ⊂ X if we show that for every bounded sequence( x k ) ⊂ X and every increasing sequence n k → + ∞ , the sequence ¯ F n k ( x k ) is relativelycompact. But this follows from Lemma 10. Thus, the existence of the global compactattractor for the IFS is established. The global compact attractor, K , comes with all theproperties listed in Theorem 6. In particular, K is the maximal compact set invariantunder ¯ F .To prove that M = K × Σ we start by showing that the slices of the attractor corre-sponding to different strings are all the same, i.e., the set { x ∈ X : ( x, s ) ∈ M } does notdepend on s . All slices are equal . Recall that every point ( x, s ) in M is a limit of some sequence S n k ( x k , s k ) with bounded ( x k ) ⊂ X and σ n k ( s k ) converging to s . As we argued above,we can write S n k ( x k , s k ) = ( S w k ( x k ) , σ n k ( s k )) , where w k is a prefix of length | w k | = n k of the string s k , i.e., s k = w k .σ n k ( s k ). Thesequence S w k ( x k ) converges to x and σ n k ( s k ) converges to s . The limit of the pair willnot change if we replace s k by w k .s . Clearly, for any string u ∈ Σ, we havelim ( S w k ( x k ) , σ n k ( w k .u )) = ( x, u ) . This proves that M = A × Σ, with a compact set A ⊂ X .Since Σ = 0 . Σ ∪ . Σ ∪· · ·∪ ( N − . Σ and since S ( M ) = M , we get S ( A × Σ) = ( S ( A ) ∪ S ( A ) ∪ · · · ∪ S N − ( A )) × Σ = A × Σ. In other words, A = S ( A ) ∪ S ( A ) ∪ · · · ∪ S N − ( A ).Because K is the maximal compact in X with this property, we have A ⊂ K . On theother hand, S ( K × Σ) = K × Σ. Since A × Σ is the maximal compact in X with thisproperty, we have K ⊂ A , and hence, A = K . This completes the proof of Theorem 2. Every fixed strategy also generates a dynamics on X : if w ∈ Σ is the (fixed) strategy, thenan x ∈ X moves to S w (0) ( x ), then to S w (1) (cid:0) S w (0) ( x ) (cid:1) , then to S w (2) (cid:0) S w (1) (cid:0) S w (0) ( x ) (cid:1)(cid:1) , etc.18enote this dynamics by ( X, d, w ). This is not a (semi)dynamical system, but we shouldnot worry about names. Certain important notions related to the long-term behavior withnatural adjustments still make sense. For example, the individual, i.e., corresponding toan individual strategy w , trajectory of a set B is the union B ∪ S w [1] ( B ) ∪ S w [2] ( B ) ∪ . . . . We define the individual ω -limit set of a bounded set B as ω ( B, w ) = { y ∈ X : y = lim S w [ n k ] ( y k ) for some sequence ( y k ) in B } . By analogy with Definition 4, we say that a set A is the global compact attractor ofsystem ( X, d, w ) if it is the minimal set with the following two properties: A is compactand A attracts every bounded set under the strategy w , i.e., for any bounded B , we havelim n →∞ (cid:126) dist ( S w [ n ] ( B ) , A ) = 0.Next theorem establishes the existence of individual compact attractors, A w , of sys-tems ( X, d, w ). Along the way we establish various properties of the ω -limiting sets ω ( B, w ). Theorem 11.
Under the Assumptions 1 and 2, every system ( X, d, w ) has the globalcompact attractor, which we denote by A w . This attractor is the intersection of the closuresof the tails of the trajectory of the absorbing set ˜ B , A w = (cid:92) n ≥ (cid:91) k ≥ n S w [ k ] ( ˜ B ) . The attractor, A w , is the union of all ω ( B, w ) with bounded B . Proof.
We use some notation and keep in mind the argument from the proof of Lemma10. Due to Assumption 1, every bounded set eventually finds itself in the set ˜ B and afterthat stays there. Step 1. The ω -limit sets of bounded sets are not empty. Pick a point x ∈ X and follow its trajectory, x n = S w [ n ] ( x ). There will be a time n such that x n ∈ ˜ B ⊂ C , and then inevitably x n +1 ∈ C , x n +2 ∈ C , and so on. ByLemma 10, the sequence ( x n ) is relatively compact. Thus, ω ( { x } , w ) (cid:54) = ∅ . Because ω ( { x } , w ) ⊂ ω ( B, w ) if x ∈ B , we have ω ( B, w ) (cid:54) = ∅ . Step 2. ω ( B, w ) is the intersection of the closures of the tails of its trajectory,hence ω ( B, w ) is closed. Note that ω ( B, w ) can be characterized as follows. ω ( B, w ) is the set of all y ∈ X suchthat for every (cid:15) > k ≥ x ∈ B and n > k so that19 w [ n ] ( x ) ∈ O (cid:15) ( y ) (where O (cid:15) ( y ) is the (cid:15) -neighborhood of y ). Yet another way to describe ω ( B, w ) is to consider the trajectory of B and its tails: D = B ∪ S w [1] ( B ) ∪ S w [2] ( B ) ∪ . . . , D n = (cid:91) m ≥ n S w [ m ] ( B ) . Clearly, D ⊃ D ⊃ D ⊃ . . . . It turns out that ω ( B, w ) = (cid:92) n ≥ D n . (5)Indeed, inclusion ⊂ is obvious. To prove the “ ⊃ ” part, pick a y in the intersection ofthe tails and set (cid:15) n = 2 − n . In D there is a point y = S w [ m ] ( x ), x ∈ B , such that d ( y , y ) ≤ (cid:15) . In D m +1 there is a point y = S w [ m ] ( x ), m > m , such that d ( y , y ) ≤ (cid:15) ,and so on. The limit of y n belongs to ω ( B, w ), i.e., y ∈ ω ( B, w ). Step 3. ω ( B, w ) is compact. Compactness of ω ( B, w ) will follow from the fact that the intersection of the closuresof the sets C n in the proof of Lemma 10 is compact, because, thanks to Assumption1, (cid:84) n ≥ D n ⊂ (cid:84) n ≥ C n . Denote C ∗ = (cid:84) n ≥ C n . Since ω ( B, w ) is not empty, C ∗ is notempty as well. And it is closed. If the set C ∗ is not compact, then there exist (cid:15) > y n ) ⊂ C ∗ such that d ( y n , y m ) ≥ (cid:15) for all n and m (cid:54) = n .Since y n ∈ C n (in fact, the whole sequence lies in every set C n ), there exists a sequence y nk ∈ C n that converges to y n as k → ∞ . For every (cid:15) > k n such that d ( y nk n , y n ) ≤ (cid:15) for all n . When (cid:15) < (cid:15) /
2, the Hausdorff distance between the sets { y nk n } and { y n } is not greater than (cid:15) . Using property (iv) of the measure of noncompactness ψ ,we obtain | ψ ( { y n } ) − ψ ( { y nk n } ) | ≤ c ( ψ ) (cid:15) . Now, ψ ( { y nk n } ) = 0 by Lemma 10. Then ψ ( { y n } ) ≤ c ( ψ ) (cid:15) . Since this is true for any (cid:15) , we obtain ψ ( { y n } ) = 0, a contradiction.This proves that C ∗ is compact. Step 4. ω ( B, w ) attracts B . To show that for every (cid:15) > m such that S w [ n ] ( B ) ⊂ O (cid:15) ( ω ( B, w )) for all n ≥ m , we argue by contradiction. Assume there exists an (cid:15) > S w [ n ] ( B ) doesnot lie inside O (cid:15) ( ω ( B, w )) for infinitely many n . This means that there is a sequence x k in B and a sequence n k → + ∞ such that S w [ n k ] ( x k ) / ∈ O (cid:15) ( ω ( B, w )). But we alreadyknow that S w [ n k ] ( x k ) must have a convergent subsequence whose limit must be in ω ( B, w ).A contradiction.
Step 5. A w = ω ( ˜ B , w ) = (cid:83) bounded B ω ( B, w ).Because every bounded set is eventually absorbed by the set ˜ B , we have ω ( B, w ) ⊂ ω ( ˜ B , w ). Thus, ω ( ˜ B , w ) attracts every bounded set. It is compact and minimal, hence,it is the global compact attractor. The theorem is proved.20 .3.5 Interplay between individual attractors Recall, that (with Assumptions 1 and 2) the global attractor M of ( X, dist , Σ) is a product M = K × Σ.We start with a few simple observations.
Lemma 12. A w ⊂ F ( A w ) ⊂ K , where F is the Hutchinson-Barnsley operator. Proof.
Pick a point, x , in A w . Then x = lim S w [ n k ] ( x k ) for some bounded sequence ( x k )in X and n k → ∞ . Among the last letters of the words w [ n k ] there is at least one, say, j ,that repeats infinitely many times. Sparse the sequence so that every w [ n k ] has the lastletter j . Then, S w [ n k ] ( x k ) = S j (cid:0) S w [ n k − ( x k ) (cid:1) . The sequence S w [ n k − ( x k ) has a convergent subsequence by Lemma 10, and the limit isin A w . Thus, x ∈ S j ( A w ). Lemma is proved. Lemma 13. A w ⊂ A σ ( w ) . Proof.
Again, if x ∈ A w , then x = lim S w [ n k ] ( x k ). Clearly, S w [ n k ] ( x k ) = S σ ( w )[ n k − (cid:0) S w (0) ( x k ) (cid:1) . The sequence ( S w (0) ( x k )) is bounded and σ ( w )[ n k − → σ ( w ). Lemma is proved. Corollary 14.
If the string w is periodic, then A w = A σ ( w ) . The union of individual attractors A w lies inside of K , (cid:91) w ∈ Σ A w ⊆ K . (6)There are many important cases when this union equals K . Lemma 15.
We have (cid:83) w ∈ Σ A w = K in each of the following cases:a) Operators { S j } are eventually strict contractions, i.e., there exist a < γ < andan integer M ≥ such that for any finite word w ∗ of length ≥ M the operator S w ∗ is a contraction with factor γ . (This condition is automatically satisfied if each S j is a strict contraction.)b) S − j ( K ) ⊇ K for j = 0 , . . . , N − .c) Each operator S j is invertible on K . roof. The inclusion (6) is obvious. To prove the equality in the special cases a) andb), pick an x ∈ K . There exists a sequence of points { x k } ⊂ K , and a sequence w n k of lengths n k increasing to infinity such that x = lim k →∞ S w nk ( x k ). We claim that x ∈ A u ,where u = w n .w n . . . w n k . . . . Denote u [ m k ] = w n .w n . . . w n k . The lengths of the words u [ m k ] go to infinity.In the case a) , for every k and any y ∈ K we have d ( S w nk ( x k ) , S u [ m k ] ( y )) = d ( S w nk ( x k ) , S w nk S u [ m k − ] ( y ))= d ( S w nk ( x k ) , S w nk ( z k ))where z k = S u [ m k − ] ( y ). Then, d ( S w nk ( x k ) , S w nk ( z k )) ≤ γ l k d ( x k , z k ) ≤ γ l k diam( K ), where l k is the round down of n k /M . Therefore, d ( S w nk ( x k ) , S u [ m k ] ( y )) →
0, as k → ∞ . Since,lim k →∞ S u [ m k ] ( y ) ∈ A u , and lim k →∞ S u [ n k ] ( y ) = lim k →∞ S w nk ( x k ) = x , it follows that x ∈ A u andthe inclusion K ⊂ (cid:83) w ∈ Σ A w is proved.In the second case, since S − j ( K ) ⊇ K , for every y ∈ K there exist z j ∈ K with y = S j ( z j ), j ∈ { , , . . . , N − } . Therefore, for every k , we can find y k ∈ K suchthat S u [ m k − ] ( y k ) = x k . Then, S u [ m k ] ( y k ) = S w nk S u [ m k − ] ( y k ) = S w nk ( x k ). It follows that x ∈ A u .Finally, c) is a special case of b) . This concludes the proof. Remark 16.
The case c) may seem too restrictive. However, there are many situationswhere the operators S j are not invertible on X but are invertible on the attractor K . Thiswas first observed by Ladyzhenskaya in the case of Navier-Stokes equations, [16]. The factis due to the invariance of K and, what is called, backward uniqueness property of certainparabolic-like equations. Although K equals the union of individual attractors in many cases, there are situ-ations when K is strictly larger than that union. This is what we call a Gestalt effect.This is a new phenomenon. As we have shown in Lemma 15, the Gestalt effect cannotoccur when operators S j are contractions. Example of a Gestalt effect.
In this example the state space X will be the space Σ of one-sided infinite strings of0’s and 1’s. There will be two operators, S and S , defined as follows: S ( v ) = v (2) .v , S ( v ) = v (1) .v for all v = v (0) v (1) v (2) v (3) · · · ∈ X . The conditions of Theorem 1 are satisfied, so let M = K × Σ be the global compact attractor of the corresponding dynamics with choice.22Note that the global compact attractor of the system generated by S is the set of allstrings with period 3, and the attractor of the system generated by S is the set of allstrings with period 2.]We claim that the sequence u = 000100 is in K but not in A w for any w ∈ Σ .Let v = 001 .σ ( v ), i.e., the first three symbols of v are 001, and let w k = 000 ... k zeros before 1. Then, for every k, S w k ( v ) = 0 . ...
001 with 001 repeating k times. Therefore, S w k ( v ) → u as k → ∞ , i.e., u ∈ K . To show that u does notbelong to the union (cid:83) w ∈ Σ A w , we argue by contradiction. If u ∈ A s , then there exists asequence v k ∈ Σ such that lim k →∞ S s [ n k ] ( v k ) = u , where n k (cid:37) ∞ . Therefore, we can find l , such that S s [ n l ] ( v l ), S s [ n l +1 ] ( v l +1 ), . . . , S s [ n l +8 ] ( v l +8 ), all begin with 000100100 ... . Since S s [ n l +1 ] ( v l +1 ) = S s ( n l +1 ) ...S s [ n l ] ( v l +1 ) = 0001001001 ... , and the action of operators S and S depends only on the first three symbols in the strings, it follows that v l [3] (cid:54) = v l +1 [3],because if v l [3] = v l +1 [3], then S s [ n l +1 ] ( v l +1 ) starts with at least 4 zeros, i.e., 0000100100 ... ,which is impossible. Similarly, v l + k [3] (cid:54) = v l + j [3] for j, k = 0 , . . . , j (cid:54) = k . But there canbe only 8 different three-letter words in 2 symbols. A contradiction. Hence, u does notbelong to the (cid:83) w ∈ Σ A w . As in section 2.3.1, Σ denotes the space of one-sided infinite strings on N symbols, and d Σ is the metric on Σ. Let Λ be a subshift of Σ, i.e., Λ is a closed subset of Σ and σ (Λ) = Λ.Dynamics with restricted choice is defined on the space X Λ = X × Λ by the operator S : ( x, w ) (cid:55)→ ( S w (0) ( x ) , σ ( w )), where the strings w are now taken from Λ only.We assume that X and S , . . . , S N − satisfy our Assumptions 1 and 2. The existenceof the global compact attractor, M Λ , then follows from the abstract result, Theorem5. The assertions 2 and 3 of Theorem 3 are among the general properties of globalcompact attractors, see Theorem 6. Denote by K Λ the projection of M Λ onto the X component. Clearly, K Λ is compact. Also, K Λ is a subset of the slice K correspondingto the full shift Σ, as in Theorem 2. Because of the invariance property of M Λ , forevery point y ∈ K Λ there is a j , one of the symbols 0 , . . . , N −
1, and a point x ∈ K Λ such that y = S j ( x ). Define the sets A j = { x ∈ K Λ : S j ( x ) ∈ K Λ } . It is easy to seethat each A j is compact and K Λ = A ∪ A ∪ · · · ∪ A N − . By construction, we have A ∪ A ∪ · · · ∪ A N − = S ( A ) ∪ S ( A ) ∪ · · · ∪ S N − ( A N − ).To analyze the slices M Λ ( s ) = { x ∈ X : ( x, s ) ∈ M Λ } , we follow the argument of thecorresponding part of section 2.3.3.Every point ( x, s ) ∈ M Λ is the limit of the form( x, s ) = lim n k →∞ ( S w k ( x n k ) , σ n k ( s n k )) , x n ) is a bounded sequence in X , ( s n ) is a bounded sequence in Λ, and w k is theprefix of s n k , s n k = w k .σ n k ( s n k ). Because M Λ is invariant under S and we know that theunrestricted dynamics has the global compact attractor M = K × Σ, the sequence ( x n )can be taken from the compact K , and we may assume that x n k → x ∗ ∈ K . Also, we mayassume that the words w k converge (to some infinite string w ∗ ∈ Λ). The strings σ n k ( s n k )converge to s . Consider all strings u ∈ Λ such that w k .u is a string in Λ for infinitelymany k . For every such u we will have x ∈ M Λ ( u ).We see that the number of different slices of the attractor M Λ may depend on thesequence x n k , but more importantly, it depends on what strings can be attached to con-vergent sequences of finite words in Λ.With every sequence ( w k ) of finite words in Λ we associate the set s (( w k )) of one-sidedinfinite strings u ∈ Λ such that w k (cid:96) .u ∈ Λ for some subsequence w k (cid:96) . In order to provethe third assertion of Theorem 3 we will show that, if Λ is a sofic shift, the number ofdifferent sets among all s (( w k )) is finite. The argument will be similar to the proof ofTheorem 3.2.10 in [21].Recall that Λ is a sofic shift if it has a presentation by a finite labeled graph, see [21].This means that there is a directed graph, G = ( V, E ), with a finite number of vertices, V , and edges, E ; the edges are labeled by the symbols 0 , , . . . , N −
1; from every vertexbegins at least one infinite directed path; the labels of the edges in the infinite directedpaths form infinite one-sided strings that exhaust exactly all strings in Λ.
Lemma 17. If Λ is a one-sided sofic subshift of Σ , then the number of different setsamong all s (( w k )) is finite. Proof.
Let G = ( V, E ) be a labeled graph presenting Λ. Let ( w k ) be a sequence offinite words allowed in Λ. For each word w k pick a finite directed path in G presentingit. We can find a subsequence, ( w k (cid:96) ), such that all the words w k (cid:96) have the same terminalvertex in their presentation. If T is such vertex, then w k (cid:96) .u ∈ Λ for all infinite paths u starting at T . Because the number of vertices is finite, we are done. (cid:3) Remark 18.
Even if the number of different sets among all s (( w k )) is > , the attractor M Λ may be a product, M Λ = K Λ × Λ , with the same slice for every string in Λ . Indeed, let N = 2 and let Λ consist of the periodic string u = 100100 . . . and its shifts σ ( u ) = 00100 . . . and σ ( u ) = 0100 . . . . If ( w k ) consists of words ending in 00, then theonly string that can be attached to w k is u . If ( w k ) consists of words ending in 1, thenthe only string is σ ( u ), and for words ending in 10 the only string is σ ( u ). Thus, we havethree different sets of the form s (( w k )). At the same time, the individual attractors A u , A σ ( u ) , and A σ ( u ) , are all equal, as we argue in Corollary 14.24 B0 C 0 1 0golden+even shift
A1 B0 1 C 0 1 0animated golden+even shift
Figure 5: The golden+even shift and its animationOne may ask whether M Λ is always a product. The answer is no, as the followingexample shows. Let Λ be the intersection of the one-sided golden mean shift with theeven shift. In other words, Λ consists of all sequences of 0s and 1s such that between anytwo 1s there are two or a larger even number of 0s. A graph presenting Λ is given onFigure 5. We will animate this graph to define the dynamics. First, identify the nodeswith three distinct points A , B , and C in R , see Figure 5 left, and define X = { A, B, C } .Second, define the maps S and S acting on points as shown by the directed edges labeledcorrespondingly; for example, S ( A ) = B , S ( A ) = A , and S ( C ) = B .Now consider the set Λ + of non-empty finite words (blocks) of Λ. We divide Λ + inthree classes and correspondingly divide the strings in Λ into three classes. The first classof words in Λ + consists of the words ending in 1. Such words can serve as prefixes ofstrings starting with an even (or infinite) number of 0s. Denote these classes by Λ + A and A Λ. The second class of finite words consists of the words ending in odd number of 0s.The strings for which such words can serve as prefixes are the strings starting with anodd number of 0s. These classes are denoted by Λ + B and B Λ. The last class in Λ + consistsof words ending in even number of 0s. The corresponding strings are those starting with1 or with an even number of 0s. These are denoted by Λ + C and C Λ. By looking at thepicture of the animated shift, it is easy to identify the possible limits of sequences S w k ( x k )when w k belong to a particular class, while x k ∈ { A, B, C } . We see that if w k ∈ Λ + A , thenthe limit set is { A, B } . If w k ∈ Λ + B , then the limit set is { B, C } . Finally, if w k ∈ Λ + C ,then the limit set is again { B, C } . Thus, there are two different slices in the attractor M Λ . One slice is { A, B } , and the other is { B, C } . We have M Λ ( u ) = { A, B } if u ∈ A Λ,and M Λ ( u ) = { B, C } if u ∈ B Λ ∪ C Λ. The global attractor M Λ is a union of the sets { A, B } × A Λ, { B, C } × B Λ, and { B, C } × C Λ.Another example of different slices appears in numerical results reported in the nextsection. 25
Example
The simplest mathematical model of malaria transmission goes back to Ross and Mac-donald. The state of the human-mosquito interaction system is described by the portionof infected humans, x , and the portion of infected mosquitoes, y . The change in time isdescribed by the following simple system of ordinary differential equations:˙ x = a y (1 − x ) − r x ˙ y = b x (1 − y ) − m y (7)The nature of the positive coefficients a , b , r , and m is discussed in [27]. In particular, thecoefficients a and b are proportional to the biting rate and the transmission efficiencies(infected human to mosquito and infected mosquito to human), r is the recovery rate (inhumans), and 1 /m is the average mosquito life-span. In practice, it is hard to measurethese parameters. Also, there are many factors that affect their values, see [27], page 8,and the values may change in time.The state space for the model (7) is the closed square X = { ( x, y ) : 0 ≤ x ≤ , ≤ y ≤ } . For initial conditions in X the solution stays in X for all t . If the quantity R = abrm is ≤
1, all trajectories starting in X converge to the origin, and the globalcompact attractor consists of a single point, P = (0 , R >
1, the equilibrium P becomes unstable and there emerges the second fixed point, P = ( x ∗ , y ∗ ), inside thesquare X , x ∗ = ab − rmb ( a + r ) , y ∗ = ab − rma ( b + m ) . (8)This second equilibrium is stable, and the global compact attractor of the system consistsof the two equilibria, P and P , and of the heteroclinic trajectory connecting them (andstaying entirely inside X ). The number R , known as the basic reproductive number,detects the emergence of epidemics: when R > x ( t + ∆ t ) = x ( t ) + ∆ t ( a y ( t ) (1 − x ( t )) − r x ( t )) y ( t + ∆ t ) = y ( t ) + ∆ t ( b x ( t ) (1 − y ( t )) − m y ( t )) . (9)The time step map ( x ( t ) , y ( t )) (cid:55)→ ( x ( t + ∆ t ) , y ( t + ∆ t )) maps X into itself provided∆ t < min { a + r , b + m } . (10)The fixed points for (9) are the same as for (7). As in the continuous case, if ab > rm andthe time step satisfies (10), the global attractor for (9) consists of the two fixed points, P and P , and the heteroclinic trajectory connecting them.26igure 6: Three individual attrac-tors A w for ∆ t = 0 .
05: left: w =111 ... ; middle two: w = 1010 ... ;right: w = 000 ... . Figure 7: Three individual attrac-tors A w for ∆ t = 0 . w =111 ... ; middle two (very close to-gether): w = 1010 ... ; right: w =000 ... .We choose two sets of parameters, pset = { a = 4 , b = 6 , r = 1 , m = 2 } andpset = { a = 2 , b = 10 , r = 3 , m = 2 } , and denote the corresponding time step maps by S and S . These sets of parameters are not related to any real-life situation but ratherchosen to better visualize the attractors. The fixed point P for pset is (11 / , /
16) andfor pset it is (7 / , / t . On figures 6 and 7, the left line (theheteroclinic trajectory) is the (global compact) attractor for the discrete system ( X, S ),and the right line is the attractor of ( X, S ). The two lines between them form theindividual attractor A w corresponding to the periodic string w = 1010 ... (on figure 7the two line are very close). For our example of dynamics with choice, Σ is the spaceof one-sided infinite strings of symbols 0 and 1. According to Theorem 2, the globalcompact attractor for ( X , Σ) has one slice, i.e., M = K × Σ. The set K for ∆ t = . t = .
005 are depicted on figures 6 and 7, respectively. We have also looked atthe dynamical systems corresponding to convex combinations of the parameter sets pset and pset and plotted their global attractors. The result is different from K , see figure 6where the “convex combination” is superimposed onto the set K .When ∆ t →
0, the upper part of the boundary of K becomes smooth. Note that thelimit set is not an attractor of any system (9) with a fixed, averaged set of parameters a, b, r and m . It would be interesting to understand whether the limit set can be obtainedas a union of the attractors of the systems ( X, S t ), where the operator S t corresponds toa certain parameter set pset t for some curve connecting pset with pset in the space ofparameters.Next, we consider restricted dynamics associated with the golden mean subshift Λ(made of one-sided strings of 0s and 1s such that each 1 is necessarily followed by 0). Thegraph representing the golden mean shift is shown on figure 10.27igure 8: “Convex combination”superimposed over K ; ∆ t = 0 . t =0 . a 0 b1 0 Figure 10: The golden mean shift.Our analysis in section 2.4 shows that the global attractor of the restricted dynamics,( X , Λ) may have at most two different slices: one corresponding to sequences of wordsending in 1 (the red slice), and the other one corresponding to sequences of words endingin 0 (the blue slice). Our computation shows that the attractor of the restricted dynamics( X , Λ) indeed has two slices. The slices are shown on figures 11 and 12.As point sets on the plane, the slices overlap. Their union is plotted on figure 9.28igure 11: The red slice; ∆ t = 0 .
05. Figure 12: The blue slice; ∆ t =0 . References [1] Akhmerov, R. R.; Kamenski˘ı, M. I.; Potapov, A. S.; Rodkina, A. E.; Sadovski˘ı,B. N. Measures of noncompactness and condensing operators. Translated from the1986 Russian original by A. Iacob. Operator Theory: Advances and Applications, 55.Birkhuser Verlag, Basel, 1992.[2] Andres, J.; Fiˇser, J.; Fractals generated by differential equations. Dynam. SystemsAppl. (2002), no. 4, 471–479[3] Andres, Jan; Fiˇser, Jiˇr´ı: Metric and topological multivalued fractals. Internat. J. Bifur.Chaos Appl. Sci. Engrg. 14 (2004), no. 4, 1277–1289[4] Andres, J.; Fiˇser, J.; Gabor, G.; Le´sniak, K.; Multivalued fractals. Chaos SolitonsFractals (2005), no. 3, 665–700[5] Babin, A. V., Vishik, M. I.: Attractors of evolution equations.
Translated and revisedfrom the 1989 Russian original by Babin. Studies in Mathematics and its Applications,25. North-Holland Publishing Co., Amsterdam, 1992.[6] Bandt, Christoph Self-similar sets. I. Topological Markov chains and mixed self-similarsets. Math. Nachr. (1989), 107–123.[7] Barnsley, M. F.
Fractals everywhere.
Second edition. Academic Press Professional,Boston, MA, 1993.[8] Barnsley, M. F.:
Superfractals.
Cambridge University Press, Cambridge, 2006.[9] Barnsley, M. F.; Demko, S. G.; Elton, J. H.; Geronimo, J. S.: Invariant measures forMarkov processes arising from iterated function systems with place-dependent prob-abilities, Ann. Inst. H. Poincar Probab. Statist. 24 (1988), no. 3, 367–394; Erratum:Ann. Inst. H. Poincar Probab. Statist. 25 (1989), no. 4, 589–590.2910] Elton, John H.: An ergodic theorem for iterated maps. Ergodic Theory Dynam.Systems (1987), no. 4, 481–488[11] Forte, B.; Mendivil, F. A classical ergodic property for IFS: a simple proof. ErgodicTheory Dynam. Systems (1998), no. 3, 609–611[12] Hale, Jack K.: Asymptotic behavior of dissipative systems.
Mathematical Surveysand Monographs, 25. American Mathematical Society, Providence, RI, 1988.[13] Hutchinson, J. E.: Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), no.5, 713–747.[14] Kitchens, B. P.:
Symbolic dynamics. One-sided, two-sided and countable stateMarkov shifts.
Universitext. Springer-Verlag, Berlin, 1998.[15] Kloeden, P. E.: Nonautonomous attractors of switching systems. Dyn. Syst. 21(2006), no. 2, 209–230[16] Ladyzhenskaya, O. A.: The dynamical system generated by the Navier-Stokes equa-tions. (Russian) Boundary value problems of mathematical physics and related ques-tions in the theory of functions, 6. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst.Steklov. (LOMI) 27 (1972), 91–115; translation in J. Soviet Math. 3 (1975), no. 4,458–479[17] Ladyzhenskaya, O. A.: Attractors of nonlinear evolution problems with dissipation.(Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 152 (1986),Kraev. Zadachi Mat. Fiz. i Smezhnye Vopr. Teor. Funktsii18, 72–85, 182; translationin J. Soviet Math. 40 (1988), no. 5, 632–640[18] Ladyzhenskaya, O. A.:
Attractors for semigroups and evolution equations.
LezioniLincee. [Lincei Lectures] Cambridge University Press, Cambridge, 1991.[19] Le´sniak, K.: Infinite iterated function systems: a multivalued approach. Bull. Pol.Acad. Sci. Math. 52 (2004), no. 1, 1–8[20] Liberzon, Daniel:
Switching in systems and control. Systems & Control: Foundations& Applications.
Birkhuser Boston, Inc., Boston, MA, 2003.[21] Lind, D., Marcus, B.:
An introduction to symbolic dynamics and coding.
CambridgeUniversity Press, Cambridge, 1995.[22] Margaliot, Michael: Stability analysis of switched systems using variational princi-ples: an introduction. Automatica J. IFAC 42 (2006), no. 12, 2059–20773023] Mauldin, R. D., Urba´nski, M.:
Graph directed Markov systems. Geometry and dy-namics of limit sets.
Cambridge Tracts in Mathematics, 148. Cambridge UniversityPress, Cambridge, 2003.[24] Mauldin, R. Daniel; Williams, S. C.: Hausdorff dimension in graph directed con-structions. Trans. Amer. Math. Soc. 309 (1988), no. 2, 811–829.[25] ˇSeda, Valter: On condensing discrete dynamical systems. Math. Bohem. 125 (2000),no. 3, 275–306; A remark to the paper: ”On condensing discrete dynamical systems”Math. Bohem. 126 (2001), no. 3, 551–553[26] Sell, G. R.; You, Y.:
Dynamics of evolutionary equations.
Applied MathematicalSciences, 143. Springer-Verlag, New York, 2002.[27] Smith, David L., McKenzie, F. Ellis: Statics and dynamics of malaria infection inAnopheles mosquitoes. Malaria Journal,3