Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map
Abstract
The article's purpose is twofold. First, we wish to draw attention to the insufficiently known field of continuous-time difference equations. These equations are paradigmatic for modeling complexity and chaos. Even the simplest equation , easily leads to complex dynamics, its solutions are perfectly suited to simulate strong nonlinear phenomena such as large-to-small cascades of structures, intermixing, formation of fractals, etc. Second, in the main body of the article we present a small but very important part of the theory behind the above equation marked by . Just as the discrete-time analog of this equation induces the one-dimensional dynamical system on some interval , so the equation induces the infinite-dimensional dynamical system on the space of functions . In the latter case, not only are the long-term behaviours of solutions critically dependent on the limit behaviour of the sequence (as in the discrete case) but also on the internal structure of as . Assuming to be continuous, we consider the iterations of as the semigroup generated by on the space of continuous maps, and introduce the notion of a limit semigroup for in a wider map space in order to investigate asymptotic properties of . We construct a limit semigroup in the space of upper semicontinuous maps. This enables us to describe both of the aforementioned aspects of our interest around the iterations of.
1. Introduction
The study of continuous-time difference equations pave the way for the discovery of new kinds of complexity. These equations appear as models of systems whose future depends not only on the current state but also on part of the past history. Even the simplest nonlinear continuous-time difference equation
gives a wonderful example of how very complex behaviors can be described with the help of very simple models (for example, see [13, 14] and references therein). Over last decades, the focus of our research has been on this equation. As a result, the qualitative theory of equation (1) was completed and published not so long ago in [8] (the key ideas were put forward in [7, 11]).
Each solution of (1) is determined by its values on and can be written as
where . Typical solutions of (1), even solutions of any smoothness index, behave in bizarre ways, for example as in Figure 1.
To describe the long-term properties of the solutions and explain how and why they occur, it is necessary to examine both the asymptotic properties of the iterative sequence and the internal structure of the maps by themselves for large . As a rule, the structure of becomes more and more intricate with the increase of . The complexities in the behaviour of trajectories of the one-dimensional dynamical system are transformed into a very complicated (even chaotic) structure of as . But, at the same time, the sequence is in a certain way (explained below) asymptotically periodic for almost all .
With the aim to present in a uniform manner the findings in this direction, we considered the maps as the semigroup generated by the map in the space of continuous maps of into itself (with being a closed bounded interval of the real line). If is finite, then the sequence is periodic and hence the structure of is simple and repetitive as increases. To study the general case where is infinite, we developed the notion of a limit semigroup for in a wider map space. We succeeded in constructing a limit semigroup in the space of upper semicontinuous maps (naturally, this space is the first thing that comes to mind because semicontinuous maps are pointwise limits of sequences of continuous maps). All this enabled us to achieve our aim as well as to discover some new insights.
The results just discussed was published only in Russian and, moreover, in their entirety only in [8]. Therefore, we thought it appropriate to cover them all in this article (with some unimportant simplifications and omissions for brevity).
2. Case of Finite Semigroup
In order to indicate conditions for to be infinite, we introduce the set of maps
Clearly, the set consists of continuous maps such that for each one there exists an interval with the properties:
If then . If in addition is smooth, then either or .
Theorem 1 The semigroup is finite if and only if
Proof. The `if' part is obvious since Consider the `only if' part. If is finite, then one can find such that This gives for all Putting we get and consequently,
The dynamics of maps generating finite semigroups is very simple (see Figure 2). Let , , and be the sets of fixed, periodic, and unstable points of , respectively.
Theorem 2 If is finite, then is empty, is connected and hence .
Proof. As bing continuous, whatever , , and , there exists such that , if . As is finite, for a certain . Therefore for all , if , i.e., all points are stable under and .
If , then the set is connected: otherwise one can find such that ; therefore the function maintains its sign in and hence one of the points is unstable under , which gives a contradiction. Since implies , all the sets are connected. Hence, if form a cycle of , then i.e. for As is known [1], this equality cannot hold for Consequently, coincides with and is therefore connected.
From Theorem 2 it follows that the set of continuous maps generating finite semigroups is exhausted by the maps with the property: for there exists its own interval such that
Two simplest maps that satisfy (3) are shown in Figure 2. In the case where the semigroup is finite, its structure is easily derived from (3) by elementary calculations.
Theorem 3 Let Then is finite semigroup that has:
type if
type if and
type if and
and is a group if and only if is the identity map.
The map of Figure 2(a) generates a semigroup of type (1,1), and the map of Figure 2(b) generates a semigroup of type (1,2). Theorem 3 defines completely the finite semigroup . For example, if has type then consists of the maps
3. Limit Semigroup Concept
The above section shows that the semigroup is typically infinite. In this case, we will try to describe its asymptotic properties using another, simpler semigroup.
Definition 1 Let the space be embedded in a wider space of maps , endowed with the metric If there exists a periodic or almost periodic semigroup , , such that
then we say that is the limit semigroup for in
Definition 2 The semigroup is called periodic if there exists a positive integer such that with referring to as the period of ;
The semigroup is called almost periodic if there are positive integers such that for any one can find satisfying with referring to as the almost-periods of .
Given , the semigroup have just one limit semigroup or none at all. Taking for this or that map space and using the corresponding limit semigroup (if exists), one can describe the asymptotic properties of in less or more detail.
In this connection, the question arises whether can have a limit semigroup in ``its own'' space . From the above section it follows that every finite has a limit semigroup in and this is its largest subgroup. Indeed, let be finite. Then the map
generates the semigroup , which consists of the one element if and of the two elements and otherwise. Hence is a periodic group with period 1 or 2; moreover, is the largest subgroup of . By virtue of Theorem 3, as Consequently: If is finite, then its largest subgroup is the limit semigroup for in
There is a representation for , in which does not appear explicitly. Let be for the -limit set of the trajectory of the point under . The map can be written as
The map in the form (6) can also exist for certain continuous maps that generate infinite semigroup. These are maps with the property: for there is its own interval such that
An illustration is in Figure 3: although is finite and is infinite, (6) yields .
If (8) is not met, then no longer has a limit semigroup in and we need a wider map space. We choose the space of upper semicontinuous maps endowed with the metric
where is for the graph of and is for the Hausdorff distance between two sets. The space is compact in the topology induced by , and is embedded in .
Let denote the operation of passing to the limit in . The meaning of the convergence in becomes clear from the equivalence of the relations
where is for the operation of passing to the topological limit. Thus, the convergence of maps in reduced to the existence of the topological limit of their graphs, which are treated as sets in the plane. It is therefore important to understand how the convergence of map graphs is related to the local behaviour of these maps. The answer is given by the following lemma.
Lemma 1 Let be a sequence of maps from with being closed bounded intervals, and let stand for the -neighborhood of The limit exists if and only if the limit exists for any and any less than a certain Moreover, if so, then
Proof. The 'if' part is proved by verifying that with and being for the upper and lower topological limits, respectively. Take, say, the latter inclusion. If then , , i.e., for any there is a sequence of points such that and as . Let us consider the sequence Since , one can find such that and as Therefore as , and hence as claimed.
The 'only if' part follows from the definition of topological limit. If exists, then also exists for every given and sufficiently small . Hence, the set is well defined and, by what was just shown, (10) holds.
The composition of is understood as the map which obviously belongs to Let and From the topological sense of the convergence in , it follows that, if exists, then and also exist, moreover,
The equality is usually not satisfied.
4. Resolvent Map
Now the goal is to construct a special map so as to generates a limit semigroup for (of course, must be identical to , if is finite). We call the resolvent map for and accept that
So far, this is only a formal notation for , since the existence of the limit in (12) is not obvious. If exists, then it belongs to the space , where is the subset of , consisting of closed intervals (including degenerated ones).
Let . Then exists for any connected (see [6]) and, by Lemma 1, the sequences and converge in to the same limit. Therefore ; an example is in Figure 4. If similar arguments lead to the formula and for finite .
Where for any the resolvent map may not exist. The simplest example occurs when some interval collapses to a point, say, under . Then when the limit on the right-hand side exists. If the trajectory of is neither asymptotically periodic nor asymptotically almost-periodic, then the sequence diverges and cannot exist. An example is delivered by the maps drawn in Figure 5. For each of and , almost all points of [a,b] are not asymptotically periodic or asymptotically almost-periodic. Thereby, is non-existent for almost all z* (because collapses to under ) and exists for all regardless of whether has the periodicity property or not, namely, . Trajectories of almost all points of are dense in under both and . This is what ensures the existence of , but for this argument does not ``work'' whenever does not have periodicity properties under .
Thus, in order for to exists, must be subject to further restrictions. Before stating the theorem on the resolvent map, we need few more notions.
Definition 3 The set
is called the domain of influence of the point under the map .
The set is nothing but the upper topological limit of the sets as and . Consequently, shows how far from the trajectory of the point the trajectories of its nearby points go. For example, in Figure 4, the fixed point is attracting and the fixed point is repelling. Therefore, is the one-point set (moreover, for all that are attracted to ), and is the interval .
Let stand for the -limit set of the trajectory of under . Clearly, contains . More precisely: if , then and has no interior points, but if , then at least one of its half-neighborhoods ``expands'' under , therefore the interior of is nonempty and .
The notion of the domain of influence of a point produces the notion of the domain of influence of a set:
which, of course, is equivalent to the formula .
Definition 4 A non-degenerated interval is called a -interval of , if the trajectories of all its points have the same -limit set that is different from a cycle or the closure of an almost periodic trajectory.
The simplest variant of -intervals is an interval that collapses (under ) into a point whose trajectory is neither asymptotically periodic nor asymptotically almost periodic, as in Figure 5(b).
Definition 5 An ordered set of intervals is said to be a cycle of intervals of the map with period if these intervals are cyclically permutated by and have mutually no common interior points.
Having introduced these definitions, we can pass to the resolvent map.
Theorem 4 If the map has no -intervals, then the resolvent map exists and it can be written in the form
Proof. The full proof is too long to include here, but the idea is that the theorem will be confirmed (in view of Lemma 1) if we will show that for any there is such that exists for and
We consider only the simplest case where is an attracting cycle. If so, then: for ; with being the period of the attracting cycle, and there exists such that for if . Consequently,
As is a subsequence of , we obtain (15) from the equalities
In case is the closure of an almost-periodic trajectory, the proof is similar but more technically involved, it is presented in [11]. A proof that covers all cases can be found in [8], so far only in Russian.
Theorem 4 will hold if
is a cycle or the closure of an almost-periodic trajectory for all . It is known [10, 12] that (i) certainly holds if is closed (then is a cycle whatever ) and fails if has a cycle with a non-power-of-two period. Furthermore, (i) is no longer guaranteed if has cycles of all periods but no others.
Here is another sufficient condition for Theorem 4
is a cycle or a cycle of intervals for all .
If (ii) is modified as follows
(iii) is a cycle or a cycle of intervals for all and their periods are uniformly bounded,
then takes the very simple form
where is the least common multiple of the periods of all
So, the resolvent map may or may not exist, and when it does, it is usually arranged complexly. The main properties of are listed below.
- The map thought of as a function form into is upper semicontinuous. Its value is a non-degenerated closed interval for and a one-point set (singleton) for
- The map thought of as a function form into , is single-valued and continuous over , and is multi-valued over
Definition 6 The set of values that takes on for is called its spectrum of jumps and denoted by .
Two simple illustrations are given by the maps of Figure 4 and Figure 5(b): here consists of the interval and consists of the intervals and .
- The spectrum of jumps is composed from the connected component of domains of influence of the unstable points of . Any two elements of either do not have common interior points, or one of them contains the other.
- In typical situations, as well as the set of values of on are finite (in particular, this is the case for structurally stable ).
- In many cases, the graph of is locally self-similar and, furthermore, fractal: is a plane curve whose fractal dimension is greater than 1.
The first four properties are easy to obtain from Theorem 4, and the last one needs some comments. Let denote the box-counting dimension (one of the commonly used versions of fractal dimension). It is clear that
usually holds (at least when is finite). The set is known to be often fractal. Therefore, if then the fractal dimension of is greater than 1 and, moreover, it can even be equal to 2 (the latter is always the case where contains an interval).
More details and arguments about can be found in [8], but here the properties of are illustrated in Figure 6, which displays the resolvent map for the logistics parabola
The dynamical properties of (17) are well known, so we use them without explanations.
If , then has an attracting period-4 cycle, which attracts all points of [1] except for the set , which consists of the repelling period-1 and period-2 periodic points of and their inverse images, and is therefore countable and nowhere dense. The resolvent map , thought of as a function , is piecewise constant over and takes on four values that correspond to the points of the attracting cycle, but over , it becomes interval-valued with the spectrum of jumps consisting of three intervals (see Figure 6(a)).
Where and has an attractive period-3 cycle and repelling cycles of any period, the properties of are nearly the same as in the previous case (see Figure 6(c)). The only difference is that , over which is interval-valued, is already uncountable but again nowhere dense (now is a Cantor-like set, since it is the closure of the repelling periodic points of and their inverse images). In addition, has positive fractal dimension and, hence, the graph of is a fractal.
Now a very different situation where with being the parameter value at which has only cycles with periods (see Figure 6(b)). These cycles are repelling and their inverse images are dense on . Hence, is a countable dense set and all points of are attracted to the Cantor-like set consisting of the condensation points of . It follows that is singleton-valued over and its values run through the continual set ; whereas is interval-valued over and its spectrum of jumps is countable (the infimum of the lengths of intervals from equals ).
Graphics shown in Figure 6 have local self-similarity in the vicinity of each ``largest'' vertical segment in case (a), and each vertical segment in cases (b) and (c). Moreover, in the last case, there are an uncountable number of vertical segments, which leads to the fractality of the graph.
5. Limit Semigroup in the Space of Upper Semicontinuous Maps
The existence of given by (12) does not in itself guarantee that generates the limit semigroup for . The last is automatically true if
(the former of these equalities entails the latter, but not the other way around).
The condition of Theorem 4, generally speaking, do not imply (18). A confirming example comes from Figure 7. For depicted there, exists, but and do not commute: , , , and hence . Besides, . Here, either of the equalities in (18) fails because has an extremum at the unstable point .
There are maps for which only the former equality in (18) breaks down. One example are
(the drawing is left to the reader). Here, has the semi-attracting fixed point , which leads to that the domain of influence of a point does not coincide with the domain of influence of its -limit set, namely: and . Therefore, and for In order for the last equality in (18) to be also violated, we ``improve'' by replacing it with
which has much the same dynamical property as . More precisely: and for As a result, , but whereas . Indeed, and for .
There are smooth maps for which (18) is not satisfied, for example,
As the above examples demonstrate, violating (18) occurs when the set
is non-empty. It is not difficult to realize that this condition is necessary and sufficient.
Theorem 5 Let satisfy the condition of Theorem 4. The equalities (18) take place if and only if
Proof. The 'only if' part is evident. The 'if' part follows from and with being the greatest common divisor of and , because these formulas are valid only provided that
The condition does not seem very good for practical use, but it is much easier to check it where satisfies (iii) and, hence, is given by (16) with some . It suffices to analyse just and not all the iterations of .
Based on Theorem (5), from now on we will assume that
(v)the map has no -intervals,
(vv),
and refer to (v),(vv) as the limit semigroup conditions or LSG-conditions for short.
These conditions cover most of continuous intervals maps, in particular, structurally stable maps. For instance, the logistic parabola (17) is structurally stable (and, hence, satisfies LSG-conditions) on a parameter set containing an open subset; this subset consists of those for which there is an attracting cycle with multiplier less than 1 in absolute value. Of course, LSG-conditions can be met by maps that are not structurally stable. An example is again given by the logistic parabola. If , then has the attracting fixed point whose multiplier is equal to . In this case, and (see Figure 8), and thus the LSG-conditions are met.
Once LSG-conditions are valid, exists and generates in the semigroup that consists of just the maps~ Then from (11) and (18) it follows that
with the limit being uniform in .
Theorem 6 Let LSG-conditions be met.
- The semigroup is periodic or almost periodic.
- The semigroup is the limit semigroup for in the space .
Proof. We consider the simple case where satisfies (iii). Then is written in the form (16) with some . Therefore and . Hence is periodic, as claimed in item 1. Item 2 can be considered true if we succeed in proving that
It is clear that with some , . Since for any there exists such that if . Hence,
which gives (20), and item 2 follows. If (iii) is not satisfied, the proof becomes more complicated and needs some additional background. It can be found in [8].
If is finite (in which case satisfies (iii) with or ), then its limit semigroup is periodic with period 1 or 2. At the same time, is typically not periodic. It is periodic only if is the identity map, i.e., if either or with being the midpoint of . Indeed, in both the cases, and therefore is actually periodic with period 1 (in the former case) or 2 (in the latter one); moreover, and is the identity map, i.e., coincides with , which means that is the limit semigroup for itself in as well as in .
There is another approach to the study of transformation groups, which based on the notion of Ellis enveloping semigroup [2, 3, 4]. For , the enveloping semigroup is defined as the closure of the set in the space of maps of into , equipped with the pointwise convergence topology. The asymptotic properties of are described by the following subset of :
Where is a finite semigroup, it can be regarded in a certain sense as an analogue of the limit semigroup for in . Indeed, is finite if and only if
Then is a periodic group of period , namely,
This statement is a direct consequence of the following well-known fact: the trajectories of all are asymptotically periodic under with their asymptotic periods being uniformly bounded, if and only if (21) holds (see [9]). In this case, it is easy to verify that and for . Hence, is identical to the resolvent map over . In addition, is continuous just if (or, equivalently, is connected) and then over . Otherwise, is a first Baire class function. Thus, the limit semigroup approach, compared with the enveloping semigroup approach, can give more information about the asymptotic properties of and allows one to study a wider class of maps .
6. Limit Sequence
LSG-conditions (v),(vv) are always satisfied by structurally stable maps, and then the limit semigroup exists and consists of the maps
When is not structurally stable, LSG-conditions may not be met. If this is due to the failure of only (vv), then exists, but its associated maps (23) can no longer form a semigroup. Nevertheless, it may happen that the maps (23) constitute a periodic or almost periodic sequence and still retain the principal property of limit semigroups
If so, the maps (23) continue to describe the asymptotic properties of . In particular, such a situation occurs where satisfies (iii). Then (v) holds and the sequence (23) is periodic. Consequently, (24) holds true regardless of whether (vv) holds or not.
As an example, we again take the logistic parabola , where the parameter value is chosen so that the critical point is mapped onto the repelling fixed point in three iterations, i.e., (see Figure 9(b)). In this case, the domain of influence of each is an interval. Namely, setting and we get that and for , where absorbs all , each after finitely many iterations, and can be split into two adjoint intervals: and that passe into each other under (see Figure 9(a)). It follows that , which is structurally unstable, satisfies (iii), and hence exists (see Figure 9(b)) and has the form (16) with , namely,
But the condition (vv): is violated because of the critical point . Indeed, and . Therefore the critical point belongs to (the same goes for its inverse images). The result is that
But even so, the ``limit'' property (24) holds for (this can be checked directly or by using ). Since the sequence is periodic with period .
Based on these facts, we arrive at the following generalization of the limit semigroup notion. To the iterative sequence , whose elements form the semigroup~, we assign the sequence (23) and call it the limit sequence in if it is periodic or almost periodic and has the ``limit'' property (24).
As the proof of Theorem 6 shows, the sequence (23) has the property (24) only if is uniform in The uniformity of the limit is ensured by LSG-conditions (v),(vv). But they are not necessary, in particular, if satisfies (iii), then the uniformity follows from the periodicity of (23) (here (v) is met automatically and (vv) is optional). Thus, generally speaking, (24) needs verification.
Interestingly, the ``alternative'' sequence cannot be regarded as a limit sequence, of course except where
References
- Yu. S. Bogdanov, On the functional equation , Reports of the Academy of Sciences of the BSSR 5 (1961), pp. 235–237 (in Russian).
- I.U. Bronstein, Extensions of Minimal Transformation Groups, Springer, Dordrecht, 1979.[CrossRef]
- R. Ellis, A semigroup associated with a transformation group, Trans. Amer. Math. Soc. 94 (1960), pp. 272–281.[CrossRef]
- R. Ellis, Lectures on Topological Dynamics, W. A. Benjamin Inc., New York, 1969.
- V. V. Fedorenko, Simple dynamics of the interval maps, Preprint 186, Centre de recerca matematica Instut d’estudis Catalans, Barcelona (1993).
- V. V. Fedorenko, Topological limit of trajectories of intervals of simplest one-dimensional dynamical systems, Ukrainian Math. J. 54 (2002), pp. 527–532.[CrossRef]
- E. Yu. Romanenko, Attractors of continuous difference equations, Computers and Math. with Appl. 36 (1998), pp. 377–390.[CrossRef]
- O. Yu. Romanenko, Difference Equations With Continuous Argument, Kyiv: Institute of Math- ematics (National Academy of Sciences, Ukraine), 2014 (in Russian).
- A. N. Sharkovsky, On cycles and the structure of a continuous map, Ukrainskyi Matematychnyi Zhurnal 1965, pp. 104–111 (in Russian).
- A. N. Sharkovsky, Behaviour of a map in the vicinity of an attracting set, Ukrainskyi Matem- atychnyi Zhurnal 18 (1966), pp. 60–83 (in Russian).
- A. N. Sharkovsky, Yu. L. Maistrenko, and E. Yu. Romanenko, Difference Equations and Their Applications, Dordrecht: Kluwer Academic Publishers, 1993.[CrossRef]
- A. N. Sharkovsky, Attractors of trajectories and their basins, Kyiv: Naukova Dumka, 2013 (in Russian).
- A. N. Sharkovsky and E. Yu. Romanenko, Turbulence: Ideal, Encyclopedia of Nonlinear Sci- ence, Alwyn Scott, ed., Routledge, New York and London, 2005, pp. 955–957.
- A. N. Sharkovsky, and E. Yu. Romanenko,Ideal turbulence: fractal and stochastic attractors of trajectories in idealised models of mathematical physics, Kyiv: Institute of Mathematics (National Academy of Sciences, Ukraine), 2020 (in Ukrainian, in Russian).
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