On the Asymptotic Properties of Piecewise Contracting Maps
We study the asymptotic dynamics of maps which are piecewise contracting on a compact space. These maps are Lipschitz continuous, with Lipschitz constant smaller than one, when restricted to any piece of a finite and dense union of disjoint open pieces. We focus on the topological and the dynamical properties of the (global) attractor of the orbits that remain in this union. As a starting point, we show that the attractor consists of a finite set of periodic points when it does not intersect the boundary of a contraction piece, which complements similar results proved for more specific classes of piecewise contracting maps. Then, we explore the case where the attractor intersects these boundaries by providing examples that show the rich phenomenology of these systems. Due to the discontinuities, the asymptotic behaviour is not always properly represented by the dynamics in the attractor. Hence, we introduce generalized orbits to describe the asymptotic dynamics and its recurrence and transitivity properties. Our examples include transitive and recurrent attractors, that are either finite, countable, or a disjoint union of a Cantor set and a countable set. We also show that the attractor of a piecewise contracting map is usually a Lebesgue measure-zero set, and we give conditions ensuring that it is totally disconnected. Finally, we provide an example of piecewise contracting map with positive topological entropy and whose attractor is an interval.
Instituto de Matemática y Estadística Rafael Laguardia, Universidad de la República,
Centro de Investigación y Modelamiento de Fenómenos Aleatorios Valparaíso, Universidad de Valparaíso,
Departamento de Matemáticas, Instituto Venezolano de Investigaciones Científicas,
Apartado 20632, Caracas 1020A, Venezuela
Instituto de Física, Universidad Autónoma de San Luis Potosí,
78000 San Luis Potosí, Mexico
Keywords: Piecewise contraction, Periodic points, Attractor, Recurrence.
MSC 2010: 37B25, 37B20, 54C08, 37N99
Piecewise continuous dynamical systems have been considered as an alternative to classical continuous models for the study of nonlinear phenomena from nature and engineering. Within this class, piecewise contracting systems have been used to model the dynamics of dissipative systems interacting in a nonlinear way. For instance, they appear in biological networks as discrete time models [6, 7, 17], or as Poincaré return maps of continuous time models defined by piecewise continuous vector fields [3, 5, 9, 18]. More generally, the dynamics of many dissipative hybrid systems can be described by a piecewise contracting map (see for instance [13, 21], the introduction of , and references therein). In some cases, piecewise continuous dynamical systems are more convenient for an exhaustive mathematical analysis. Nevertheless, the presence of discontinuities can produce phenomena that do not always exist in a continuous framework and which deserve detailed studies. Because of their good properties far from the discontinuities, piecewise contracting maps appear as suitable study models.
Several results have been obtained for particular families of piecewise contracting maps. Let us mention some of them which we consider representative. In , Gambaudo and Tresser gave a complete description of the possible asymptotic dynamics of quasi-contractions. These systems, which act on an arbitrary metric space, generalize one-dimensional piecewise contractions with two contracting domains by maintaining only their essential features. The characterization of the symbolic dynamics of these maps obtained in  shows that their limit set is either composed of at most two periodic orbits or is a Cantor set supporting a quasi-periodic dynamics. Some results in the same direction have also been obtained for piecewise contracting maps with more than two contraction pieces. In , Brémont studied contracting interval exchange transformations and proved, among other results, that the asymptotic dynamics is typically concentrated in a finite number of periodic attractors whose number can be bounded by a linear function of the number of monotonicity intervals of the transformation. This result was further refined by Nogueira and Pires in , where they established that the number of periodic attractors is at most equal to the number of monotonicity intervals. In larger dimensions, piecewise contracting affine maps have received a particular attention. We can mention the result by Bruin and Dean , according to which the asymptotic dynamics of piecewise-affine contractions of the complex plane is typically supported by a finite number of periodic orbits. This result was also proved in [6, 17] for a particular class of affine piecewise contractions on introduced in  as models of genetic regulatory networks. In addition it has been shown that the symbolic complexity of these systems (including non typical ones) grows sub-exponentially with time [8, 17]. Finally, general piecewise contracting maps on were studied in [3, 5, 21] and the generic character of a periodic asymptotic dynamics was proved in  under some separation property asumption (which is verified by injective maps, for instance).
Most works on piecewise contracting maps have studied specifics (affine) maps, which allowed to obtain detailed results concerning their periodic or quasi-periodic asymptotic behaviours. In the present paper, we propose to study the asymptotic dynamics of piecewise contracting maps in a larger framework. Our purpose is on one hand to obtain properties that are shared by all piecewise contracting maps defined on a compact set, and in the other hand to illustrate the rich phenomenology emerging from the presence of discontinuities by providing numerous examples, most of them contrasting with the (quasi-)periodic case.
Our discussion is organized as follows. In Section 2, we give the principal definitions and we review some of the classical notions used in the study of asymptotic dynamics, as those of attractor, limit set, and non-wandering set, in order to adapt them to discontinuous maps. In Section 3, we show that when the attractor does not intersect the set of discontinuities, it is composed of a finite number of periodic orbits. This result is true for any piecewise contracting map defined on a compact set and thus is independent of the particularities of the map (dimension of the space, number and geometry of the contractions pieces, in particular). In the sequel of the paper, we focus on the case where the attractor contains discontinuity points (which justifies our generalization of the asymptotic sets introduced in Section 2). In this situation the strong recurrence properties of the periodic case are lost in general, since attractor, limit set and non-wandering can be different. In Section 4, we show that the discontinuities can give rise to an asymptotic behaviour which does not coincide with the dynamics of the map on the attractor. This is why we introduce the notion of ghost orbit, which are generalized orbits allowing to describe the asymptotic dynamics as well as to define recurrence and transitivity in the attractor. Section 5 deals with the dynamical complexity of piecewise contracting maps. There, we show that the total disconnectedness of the attractor is related to a low complexity of the dynamics and we give examples of piecewise contracting maps with a connected attractor. Our final example is a piecewise contracting map with positive topological entropy whose attractor is a Cantor set or an interval. In the concluding section we present our final comments and we point some directions for future research.
Piecewise contracting map. Let be a compact metric space. We say that a map is piecewise contracting if there exists a finite collection of non-empty open subsets , such that:
1) and for all .
2) There exists , called a contraction rate, such that for any we have
3) The set , where , is non-empty.
If the set is non-empty, then it contains all the discontinuity points of the piecewise contracting map. Let us note however that each restriction of to a piece admits a continuous extension which we call the continuous extensions of . In this paper we study the asymptotic dynamics of the points whose orbit never intersects , namely the points of . Nevertheless, we describe this dynamics by mean of compact asymptotic sets that can intersect the set . The set is dense in under mild hypothesis on (see Proposition A.1), and it is usually not compact.
Attractor . Let be a piecewise contracting map and consider the family of transformations defined for all by . We say that is an atom of generation if
for some . Denote by the set of all the atoms of generation , and for all let . We call the attractor of .
Note that any atom is a subset of an atom of the previous generation. It implies that for all . Since is non-empty, there exist a non-empty atom of any generation. The sets , and therefore , are non-empty and compact. Here, we use the terminology attractor in a formal analogy with the attractor of iterated functions systems . Indeed, can also be written as , where for any compact set , the map is defined by . The invariance of the attractor by implies that for any and such that . Unless it does not intersect , the attractor is not necessarily invariant by , but for any there is always a continuous extension such that .
We also consider other asymptotic sets traditionally used for continuous maps, but we adapt their definitions to deal with discontinuities.
Non-wandering set . A point is non-wandering if for all there exists a sequence going to infinity and such that the ball of center and radius satisfies:
We denote by , and call the non-wandering set, the set of all the non-wandering points. A point is wandering if .
Contrarily to continuous maps, the existence for all of a such that and have a non-empty intersection is not equivalent with the existence for all of a sequence such that satisfies (3). This equivalence is true if , but not necessarily otherwise (see the final comment of Example 3.2). This why we use the sequence in order to ensure a sufficiently strong recurrence property to the non-wandering points belonging to . However, as for continuous maps, the set of the wandering points is open an thus is closed.
Limit set . We say that is an -limit point of if there exists a sequence going to infinity and such that . We denote by , and call -limit set of , the set of all the -limit points of . We call the limit set of .
Since and is compact, for all and .
Note that two piecewise contracting maps that coincide on share the same asymptotic sets, since the attractor, the non-wandering set and the limit set are entirely determined by the orbits of the points of . In order to study the recurrence and transitivity properties on the attractor, when it intersects , we will need to introduce generalized orbits (see Section 4).
3 On the asymptotic sets
As a starting point, we state a general property of piecewise contracting maps concerning the periodicity of the dynamics in the attractor.
Let be the set of discontinuities of a piecewise contracting map and be its attractor. If or if , then is a finite union of periodic orbits.
We prove Theorem 3.1 in Section A.2. The finite periodicity of the limit set has been reported for special classes of piecewise contracting maps: one-dimensional injective maps , two-dimensional affine maps , arbitrary dimensional affine maps with convex continuity pieces [6, 8, 17] and more general real maps satisfying a so-called separation property [3, 4] or with two contraction pieces . In each of these works, the specificity of the maps allows to obtain additional results, such as the genericity of the asymptotic periodicity or a bound on the number of periodic orbits. Nevertheless, one does not need more than the hypothesis of Theorem 3.1 to ensure the finite periodicity of the attractor of piecewise contracting maps as general as those of Definition 2.1.
The principal idea of the proof is the same as in the particular cases. However, for the maps we consider, the successive iterations (namely, the atoms in Definition 2.2), may have infinitely many connected components. This situation cannot occur for specific maps as considered before, for example, if all the pieces are convex and embedded in and the map is an affinity in each piece. When the atoms have an infinite number of connected components, there is no guarantee a priori, that each connected component of the atom has an iterate which returns inside itself. This avoids an immediate reduction of the proof of Theorem 3.1 using the Banach fixed point theorem, for instance.
When the attractor of a piecewise contracting map does not contain discontinuity points, the effect of the discontinuities is the emergence of a finite number of periodic orbits in the attractor. Therefore, the discontinuities increase the complexity of the dynamics compared with pure contractions, but in both cases the dynamics is regular. From now on we are interested in the situation where there are always orbits in arbitrarily close to discontinuity points. We will see that most of the characteristics of the periodic case are lost and a rich phenomenology appears.
Let be a piecewise contracting map. Then the chain of inclusions holds.111It is actually true for any piecewise continuous map, disregarding whether it is piecewise contracting. Besides, if is locally connected, then any non-wandering point in the interior of is periodic.
We prove Proposition 3.2 in Section A.3. An obvious consequence of Theorem 3.1 is that the limit set, the non-wandering set and the attractor are finite and coincide when the attractor does not intersect the set of discontinuities of the map. As shown by the following examples, when the attractor contains discontinuity points, this equality does not hold in general. In Example 3.1, the non-wandering set and the limit set are equal, but the attractor contains an infinite number of wandering points in the interior of .
Consider the square and the two pieces and with the parabolic boundary . Let and define on the map and on the map . Let be such that for all (see the left frame of Figure 1.) One can check that and . Thus, the attractor is countably infinite, but the non-wandering and the limit sets are finite.
Now, in Example 3.2 all the points of the attractor are non-wandering but not all are limit points.
Let and consider the three open pieces , and (see right frame of Figure 1). The set of discontinuities is thus
Let and define on the maps , and , where . Then for any piecewise contracting map such that for all , we have
Thus, the attractor is countably infinite, all its points are non-wandering and the limit set is finite. Also, the points of are not in and therefore the hypothesis of the last assertion of Proposition 3.2 does not hold. Since the points of are not periodic, the conclusion also fails. As a final comment, note that if we take and as in Example 3.1, then the intersection of with any sufficiently small ball centered in one of the point , with , has an image which intersects itself only once. These points became therefore wandering.
A continuous system such that or has some topological expansion, or at least the lack of a contracting rate, in some part of the space. For instance for bifurcating diffeomorphisms on compact manifolds that exhibit a homoclinic tangency of a dissipative saddle periodic point . As said above, piecewise contracting systems may also exhibit in spite of the uniform contracting rate in their pieces. Hence, the set of discontinuities acts as a generator of a peculiar topological expansion, and piecewise contracting maps have a topological flavor of partial hyperbolicity. In fact, we provide in Example 5.1 a piecewise contracting map whose discontinuities produce a chaotic attractor contained in .
4 Dynamics on the attractor
We are now interested in the dynamical properties of the attractors of piecewise contracting maps. As mention earlier, depending on the definition of the map on the set of discontinuities the attractor may fail to be invariant. This makes non trivial to define a dynamics in the attractor which is representative of the asymptotic behaviour of the points of . One of the goals of this section is to define such a dynamics on the attractor by introducing the concept of ghost orbit. It will also allow to adapt the notions of recurrence and of transitivity for attractors containing discontinuity points.
In some case it is possible to define a representative dynamics in the attractor by multi-defining the map in the intersection of the attractor with the discontinuities, using its continuous extensions222Recall that the attractor is invariant under these transformations.. The following example illustrates this fact, and also shows that when the attractor intersects the discontinuities it is not necessarily of infinite cardinality.
Consider the rectangle and let . Define the set and let , , , and , (see left frame of Figure 2).
Let and and be defined by and . For all let . Take and suppose from now on that is such that . Then, we have , for all and . Now, let and with and , where . Then, we have , and the condition on ensures that and .
Consider a piecewise contracting map such that for all , then one can show that
Alternatively, if we denote the periodic sequence such that for all , , for all and , then can be written as the orbit of by a multi-valued map:
Moreover, there is a neighborhood of such that for any point in . Together with (4), this implies that all the points of follow the orbit of by , which covers the whole attractor and is periodic of period . Note that and . Therefore, the asymptotic behaviour of the points of is described by a periodic sequence which is not an orbit of the map (but the orbit of a multi-valued map) and this, independently of the definition of in . See right frame of Figure 2.
Example 4.1 shows that the asymptotic dynamics of the point of is well represented by an orbit of a multi-valued map constructed with the continuous extensions of the original map. But in the forthcoming examples of this section, a multi-valued map will not be enough to describe the asymptotic behaviours, and this is why we introduce the concept of ghost orbit. A ghost orbit is a generalized orbit: it can be a usual orbit, the orbit of a multi-valued map, or a more sophisticated set whose points are ordered with time sets of order type larger than .
Ghost orbit. Let be a well-ordered infinite set with smallest element , and consider the map defined by (by convention we suppose that ). Let and . A map is said to be a ghost orbit of a piecewise contracting map , if
1) For all there exists such that .
2) For all such that there exists such that for a .
3) There exists such that for all there is a point whose orbit is -close to , that is, for an increasing map with and , we have
for all such that .
Note that if is bounded, then conditions and inequality (7) do not apply for . In particular, if , then is a ghost orbit of if and only if, it is the orbit of a point of , or, it is an orbit of a point of obtained by successively applying continuous extensions of and such that for all there exists a point of which orbit remains at distance smaller than of all its points (as the ghost orbit (5) of Example 4.1). The forthcoming Examples 4.2 and 4.3 require ghost orbits whose time set is endowed with the lexicographic order.
In the following, we are interested in ghost orbits with values in the attractor of piecewise contracting maps. Note first that if is a ghost orbit, then for any , the map defined for all by , where and are seen as ordinal numbers and is the ordinal addition, is a ghost orbit as well. We say that belongs to the basin of attraction of a ghost orbit , if for all there exists and such that is -close to . We say that a ghost orbit is stable, if there exists such that for all there is an open set such that the orbit of any point of is -close to . A ghost orbit is said to be unstable if it is not stable and repelling if the only points with orbit -close to belong to .
In Example 4.2, if then the attractor is invariant. Also, if we suppose then the attractor can be written as . Any point with has the point as -limit set, however is not stable, since any perturbation of changes the -limit set of . Also, although being contained in a contraction piece, is a repelling orbit: for any neighborhood of , there exists such that for all with , we have for some .
Actually, the orbits of the points of with exhibit a more complicated (stable) asymptotic behaviour than the orbit of the points and . As we can see in the right frame of Figure 3, and as we are going to show in the proof of Proposition 4.2, after a transient time the orbit of a point with get close to the point , follows the orbit of this point during a finite time before going back closer to , and so on. There is no way to define the map in in order to create an orbit in the attractor with such a recurrent behaviour. However, we can describe this asymptotic dynamics using a ghost orbit. More precisely, we have the following proposition:
Let be defined by for all and in , where is a piecewise contracting map of Example 4.2. If is endowed with the lexicographic order, then is a stable ghost orbit whose basin of attraction are all the points with . If then belongs to the basin of attraction of the fixed point , which is an unstable (but not repelling) ghost orbit.
If is endowed with the lexicographic order, then where for all . Also and for all and . It is easy to check that verifies and of Definition 4.1. Now, we show that there exists such that for all there is an open set such that the orbit of any point of is -close to . This will end to prove that is a ghost orbit and will show in the same time that it is stable. For sake of simplicity, and without loss of generality, we make the proof supposing in order to have (as in Figure 3).
Let and take . Since , the orbit of visits infinitely many times, and since , it does not stay more that one time step in . In other words, the sequence defined by
exists and is such that for all . Now, consider the sequence defined by and for all . This is an increasing sequence which gives the first return times of the orbit of in . Also the map of Definition 4.1 will be defined as for all .
For any , let , where denotes the distance induced by the infinite norm of . Then, for all , we have
Now, using the fact that , one can prove that if . This implies that for all , or equivalently for all . It follows that
On the other hand, for all , if , then . It follows that
if , since for all , if .
Let , and . Then by inequalities (8) and (9) any point is -close to , and therefore is a stable ghost orbit. On the other hand, any point with eventually enter in , since . These points belong thus to the basin of attraction of . Finally, it is easy to verify that the fixed point is an unstable ghost orbit, which basin of attraction are all the point with and thus is not repelling. ∎
For continuous maps, a point is recurrent if it belongs to its -limit set. Following this definition, the attractor of Example 4.2 does not have recurrent points (except for the point in the special case where ). In the case of Example 4.1, although the dynamics in the attractor being described by a periodic sequence, following the same definition, the attractor contains points that are not recurrent (these points depend on the definition of in (0,0)). Once introduced the concept of ghost orbit we can propose a generalized definition of recurrence and of transitivity.
Recurrent point. A point is recurrent if there exists a ghost orbit and such that and for all .
Transitivity. A compact set is transitive if there exists a ghost orbit such that .
Under this definition, the attractors of Example 4.1 and 4.2 are transitive. The respective attractors of the Example 3.1 and Example 3.2 are also transitive. The dynamics on the respective attractors is described by the same ghost orbits: the orbit of the unstable fixed point , the orbit of the stable fixed point and the stable ghost orbit defined by and for all . However, the fixed points are the only recurrent points.
We have presented countable attractors so far, but it is known that piecewise contractive maps can exhibit Cantor limit sets. Also, it has been reported that the limit set of some piecewise contracting maps of the plane can be the disjoint union of a cantor set , supporting a minimal dynamics, with a finite set of periodic points . Such limit sets are decomposable, since they have at least two transitive components. In the following Example 4.3, we show that there exist transitive limit sets which are a disjoint union of the form where is countable. This example illustrates also a possible effect of the discontinuities on the characteristics of the recurrent points. Indeed, all the points of are isolated (non-periodic) recurrent points, whereas for continuous maps such points are necessarily periodic. On the other hand, for a continuous map defined on a compact set , if a point satisfies , where is a minimal Cantor set and is a scattered subset of , then is dense in . Example 4.3 shows also that this result does not hold for piecewise contracting maps.
Let , , and . Let and define on the map and on the map . Consider a map such that for any . Then, there exist uncountably many for which the limit set of is a minimal Cantor set , and this Cantor set satisfies (see [7, Theorem 6.1] for an explicit formula to compute such and ).
Let , be some continuous and increasing function, and consider the four open pieces: , , and (see left frame of Figure 4).
Let be defined by , and be defined by , where .
There exist a continuous and increasing function , an open set , and a contracting homeomorphism , such that for any map satisfying , and for any , the set is dense, the limit set of satisfies
and for any . Moreover, is transitive and all its points are recurrent.
The proof of Proposition 4.5 is given in Section A.4. The proof also shows that any point with and whose orbit visits is in the basin of attraction of the stable ghost orbit defined by and for all and , where has an orbit by which is dense in . The right frame of Figure 4 shows a numerical simulations of the orbit of and of the orbit of a point of in the basin of attraction of . This orbit accumulates on by getting every time closer to the orbit of and of .
5 Disconnectedness and complexity
All the attractors we have encountered until now were totally disconnected. In this section, we give general conditions ensuring the attractor to have this property, but we also provide examples of piecewise contracting maps with a connected attractor. We will see that the existence of a connected attractor needs a sufficiently fast growth of the complexity to counter-balance its contracting characteristic. However, as shown by the following theorem, in and under a reasonable hypothesis on the set of the discontinuity points, the Lebesgue measure of the attractor remains always null.
Let be a compact subset of and denote by the -dimensional Lebesgue measure. If is a piecewise contracting map such that , then .
We prove Theorem 5.1 in Section A.5. The condition that the Lebesgue measure of the set is null may possibly be loosen. But the measurability of this set seems to be important in order to avoid paradoxical decompositions that can lead to the expansion of the Lebesgue measure by a piecewise contracting map .
Theorem 5.1 has an immediate consequence on the connectedness of the attractor of a piecewise contracting map defined on a compact subset of : If the set of the discontinuity points is countable, then the attractor is totally disconnected. Indeed, if is countable then and it follows from Theorem 5.1 that . Therefore has empty interior, which in implies total disconnectedness. In higher dimension or in general metric spaces, the Lebesgue measure (when it exists) does not give information about the connectedness of the attractor. The following theorem gives sufficient conditions for the total disconnectedness of the attractor in compact metric spaces.
If a piecewise contracting map satisfies at least one of the following hypothesis:
1) there exists such that for all , if and then ,
2) the number of atoms of generation , , and the contraction rate satisfy ,
then its attractor is totally disconnected.
We prove Theorem 5.2 in Section A.6. We have referred earlier to condition 1) as separation property and it is satisfied by any map whose continuous extensions are injective. As a side comment, we mention that similar conditions to 2) allow to obtain an upper bound for the Hausdorff dimension of the attractor (see Proposition A.6). Note that if , the condition 2) is satisfied, i.e. strongly contracting maps have totally disconnected attractors. For some conformal piecewise contracting maps with polytope pieces, the number of atoms of generation is a sub-exponential function of . This function is polynomial in some cases where the contraction rate is sufficiently small . It implies that these maps always satisfy the condition and thus have totally disconnected attractors. As shown by the following example inspired by [14, 15], non-conformal piecewise contracting maps do not always satisfies condition 2) and can exhibit a connected attractor. This example also permits to discuss the optimality of the hypothesis of Theorem 5.2.
Consider in the compact triangle with corners in , and . We denote by and the two half open triangles, respectively bellow and above the line (here ). Let