Research Article | Open Access
Iteration of Differentiable Functions under m-Modal Maps with Aperiodic Kneading Sequences
We consider the dynamical system (, ), where is a class of differentiable functions defined on some interval and : → is the operator , where is a differentiable m-modal map. Using an algorithm, we obtained some numerical and symbolic results related to the frequencies of occurrence of critical values of the iterated functions when the kneading sequences of are aperiodic. Moreover, we analyze the evolution as well as the distribution of the aperiodic critical values of the iterated functions.
In this paper, we pursue the study of the properties of certain infinite dynamical systems that arise from iterated differentiable interval maps. Regarding infinite dimensional systems, important progresses were obtained in the study of boundary value problems for partial differentiable equations reducible to difference equations, as those studied in [1–3]. An approach, in that context, using symbolic dynamics was made in [4–6].
We consider the dynamical system , where is a class of differentiable real functions defined on some interval and is the operator , where is an m-modal map. The class is the class of differentiable functions defined on an interval (eventually could be defined on ) such that if with a finite number of critical points, satisfying the boundary conditions and , for a given interval . The nature of in terms of topological, metrical, or algebraic closure, for now, is not discussed since we are mainly interested in analyzing the qualitative changes of the elements in , under the iteration of , from a combinatorial point of view. In particular, we study the changes on the number of critical points, the relative localization of the critical points and critical values. The developed techniques allow us to study, in future work, the changes, under iteration of , of attributes of the elements in , such as the oscillations, the mean value, the measure, the mean derivative, the length of the graph, among others.
The dynamical system has infinite dimension, although induced by a one-dimensional discrete dynamical system . From the topological point of view, the dynamical system is formally contained in , since the constant functions, , belong trivially to and . Moreover, a monotone function (nontrivial) determine a signed interval, if is an increasing function and if is a decreasing function. Then the dynamics of intervals under iteration of an interval map is also formally contained in .
We determine the itineraries of a sufficiently large number of critical values of the iterated functions using the algorithm introduced in . We analyze the evolution and the distribution of these critical values whose frequencies are displayed in histograms. Moreover, we obtained a numerical result which relates the relative frequencies of critical values with the growth number of , . In this paper, we analyze the case of being an m-modal map with aperiodic kneading sequences.
2. Preliminaries on Symbolic Dynamics
2.1. Symbolic Dynamics for m-Modal Maps
In this section, we describe some preliminaries on symbolic dynamics, in particular, aspects concerning to the m-modal maps on an interval .
Let be an interval. A map is called m-modal if it is in and has critical points. Let , with , be the critical points of the map such that , where and represent the boundary points of the interval . In these conditions, consider the partition of the interval into disjoint subsets where is the set , and , are the intervals
A maximal interval on which is monotone is called a lap of and the number of distinct laps is called the lap number of . Thus, each interval , is a lap.
According to , the limit is a real number in the interval and is called the growth number of .
Thetopological entropy of is see .
Next, to each point in , we assign the symbol , or , if . This assignment is called the address of , and it is denoted by . The address of the point , , is thus given by
As usual, we get a correspondence between orbits of points and symbolic sequences of the alphabet , the itinerary of under , which is defined by
The orbits, under , of the critical points are of special importance, in particular, their itineraries. Following , for each critical point the kneading sequence is , and the collection of symbolic sequences is called the kneading invariant of .
A symbolic sequence in is called admissible, with respect to , if it occurs as an itinerary for some point in . The set of all admissible sequences in is denoted by .
In the sequence space , we define the usual shift map by Moreover, the following relation with and the itinerary map is satisfied Therefore, we obtain the symbolic system associated with the discrete dynamical system .
An admissible word is a finite sub-sequence occurring in an admissible sequence. The set of admissible words of size occurring in some sequence from is denoted by . We define the cylinder set , , by In other words, means that , , .
Consider the sign function defined by with , and or , accordingly restricted to is increasing or decreasing.
The parity, with respect to the map , of a given admissible word , is even if and odd if . From the order relation , inherited from the order of the intervals of the partition of the interval , we introduce an order relation between symbolic sequences as follows: given any distinct sequences , , admitting that they have a common initial subsequence, that is, there is a natural such that and , we will say that if and only if and or and .
2.2. Symbolic Dynamics for the Infinite Dynamical System
Now, consider an m-modal map in the class , for a certain interval , and the class of differentiable functions where denotes the number of critical points of .
Let be the operator Note that this operator is well defined since . Moreover, if and , then for every . Therefore, we obtain a discrete dynamical system in the sense that we have a set (eventually with additional structure, a topology or a metric, for now not specified) and a self-map , which characterizes the discrete time evolution.
Let and let be the number of nontrivial critical points of (inside ). In this case, if , , then and the total number of critical points is . We are interested in the symbolic description of the dynamical evolution of a function under iteration of , which has essentially a topological meaning; therefore, the important aspect is to distinguish and codify the critical points and the critical values of . Given , we identify its critical points and collect the addresses and itineraries of the corresponding critical values. The generalized symbolic space is , where ( times), and we define the generalized address map for the space by and the generalized itinerary map for the space by where , , are the critical values of in the interval (with and ).
Let be such that , . The generalized shift map is then defined by where is the kneading sequence corresponding to the critical point of , localized between the critical values and , with , . We obtain a symbolic system associated to . Similarly to the finite-dimensional discrete dynamical systems, it is verified the following result.
Theorem 2.1. Let , with so that , then
The previous definition and results allow us, knowing only the itineraries of the critical values of an initial condition and the kneading invariant of , , to obtain explicitly the itineraries of the critical values of , , see the next example.
Example 2.2. Let us consider a -modal map characterized by the kneading invariant (with the alphabet , see Figure 1. Now, we consider the generalized itinerary of the function given by , with respect to the kneading invariant . Using the Theorem 2.1, we obtain the temporal evolution by of :
3. The Evolution and Distribution of the Aperiodic Critical Values of the Iterated Functions
In our previous work, we analyze the evolution and distribution of the periodic critical values of the iterated functions when the kneading sequences of are periodic. Moreover, we developed the following algorithm based only on the kneading invariant of . This algorithm computes symbolically all the itineraries of a sufficiently large number of critical values of iterated functions in few minutes. Using this algorithm, we overcome many difficulties in the study related with itineraries of critical values when we use numerical methods.
Algorithm 3.1 (to compute the itineraries of all critical values of ). Let , , , and be the inputs and let and be the outputs.
Step 0. Set , and .
Step 1. If , compute and append the sequence to the vector . Set and go to the Step 2.
Step 2. If , then design by the first symbol of and by the first symbol of , and go to the Step 3. Otherwise, go to the Step 7.
Step 3. If (resp. or and or (resp. , then append to the vector and the , set and return to the Step 2. Otherwise, go to the Step 4.
Step 4. If (resp. or and , or (resp. , then append to the vector and the , set , and return to the Step 2. Otherwise, go to the Step 5.
Step 5. If and then, append and to the vector and the , set , and return to the Step 2. Otherwise, go to the Step 6.
Step 6. If and , then append and the to the vector and the , set , and return to the Step 2.
Step 7. Append the vector to the set and set . If , the algorithm ends; otherwise, set and return to the Step 1.
In the following example, we illustrate an implementation of the Algorithm 3.1 in Mathematica 6.0 for a bimodal with periodic kneading sequences.
Example 3.2. Let us consider a bimodal map characterized by the kneading invariant and the function such that . Considering we obtain a large number of critical values of , in this case critical values for . Moreover, we obtain the itineraries , for , given by
To apply the algorithm when the keading sequences of are periodic, we only consider the periodic part of each kneading sequence. However, there is no difficulty in applying the algorithm when the kneading sequences of are aperiodic. In this case, the length of the kneading sequences is not finite. Then, we make explicit at least the first symbols of each kneading sequence to compute the itineraries of critical values of , for some fixed .
In this section, we analyze the evolution and distribution of the periodic critical values of the iterated functions when the kneading sequences of are aperiodic. As we present in the following results, the new distinct critical values obtained at each iteration depend on the kneading sequences of . Then, the new critical values will also be aperiodic points which belong to some critical orbits. The number of different critical values will always grow and never stabilize. The reason is that the itinerary of each critical value will never repeat again under iteration. We also analyze the type of growth of these critical values.
The next result illustrates the set of the itineraries of all critical values of an iterated function . As previously, denotes the number of the nontrivial critical points of and , denote the critical values of . Let be the number of the critical values of .
Since the map restricted to a subinterval is monotone, is invertible if restricted to . Denote by the inverse map . Let ,, be the critical points of the map , and let be the maximal interval containing , consisting of points whose orbits tend to the stable periodic orbit of , see .
Let be given by , , . Therefore, the basin of attraction for is given by .
Proposition 3.3. Let be an m-modal map, and . If the kneading sequences of , , are aperiodic, then the itineraries of the critical values of , , will belong to the set where , , are the critical values of . Moreover, will always grow and never stabilize.
Proof. Let , be the critical points of the map . In , we proved that all critical points of arise in the following two ways:(i)a critical point of will be a critical point of ;(ii)every critical point of , which is not a critical point of , is a point in which for some .
Therefore, from (i), the critical values of correspond to the image under of the critical values of .
From (ii), the new critical values of are always . Then the critical values of are either points in the set or points in the set . Recalling that , for all , the itineraries of the critical values of are in the set Since we are considering aperiodic points , , the set is infinite.
For a fixed , we have that the number of distinct critical values of a certain is . We have that grows with .
Let be an aperiodic sequence in and let denote the number of times which occurs as itinerary of a critical value of . The following result illustrates the frequencies of occurence of the critical values whose itineraries are obtained of , .
Proposition 3.4. Let be an m -modal map, (with , and . If the kneading sequences of , , are aperiodic then for each , we have for with , .
Proof. As defined previously, is the number of times, which occurs as itinerary of a critical value of , for some fixed and some fixed .
If the itinerary of some critical values of is , then the itinerary of these critical values is for , which correspond to the new critical points of . These new critical points are derived of the )th-preimages of , that is, .
Similarly, if the itinerary of some critical values of is , then these critical values correspond to new critical points of that are obtained of the )th-preimages, under of , that is, .
Since , then . In this case, the number of the admissible -preimages is less or equal to the number of the admissible -preimages since the number of laps of is less than the number of laps of . Therefore, for with , .
In order to illustrate the previous results, consider the following example.
Example 3.5. Consider the bimodal map characterized by the kneading invariant
which analytical expression is given approximately by
and the function such that
According to the Proposition 3.3, we have that the itineraries of the critical values of belong to the set Regarding that and , by the Proposition 3.4, we have Indeed, we have
Let us consider a bimodal map with aperiodic kneading sequences and growth number . Let be a function and , . Recall that is the number of times that , , occurs as itinerary of the critical values of . We define the relative frequency of the critical values of by where is the itinerary of the -th critical value of . These values range between and and verify the following equality:
We denote by the set of the values , , and we construct histograms using these values.
4. Numerical Results
Next, we present a numerical result that relates , the relative frequencies of the critical values of given by , , for each .
Let be the basin of attraction for and such that . If is the kneading sequence of , then
In a simple and concise way, we can explain the numerical result in (4.1), relating it with the growth rate of the critical points of . There is a relationship between the growth rate of the critical points of and the growth number of , . Since the growth rate of the critical points of is related with the growth of the preimages under of the critical points , , we have that the growth of the number of each distinct critical value, that results of the kneading sequence , is related with growth of the critical points correspondent to the preimages under of the critical points , for each .
An estimation of the growth number of , , is given by Usually, a practical way to compute the growth number of , , when the kneading sequences of are periodic is through the spectral radius of the transition matrix associated to the kneading invariant of , see . Using the algorithm, we spend less time computing the growth number of than computing the transition matrix associated to the kneading invariant of . Here, we compute numerically the total number of critical points of the iterated functions calculating the total number of the itineraries of their critical values.
In the following example, we illustrate the previous result.
Example 4.1. Consider the bimodal map characterized by the kneading invariant
whose analytical expression is given approximately by
Let be a function such that
whose analytical expression is given by .
An estimation of the growth number of , , is given by
The number is easily computed symbolically using the algorithm. The itineraries of the critical values correspondent to the new critical points of are given by , ,, and are presented in the following set ordered by increasing , which is suitable to present graphically
The respective values ,, are The relative frequencies of the critical values of whose itineraries are , with , , , are given in the set These values (dependent only on the kneading invariant of ) are independent of the initial functions and are presented in the histogram of Figure 2. Note that in this histogram, the distribution of the critical values is not symmetric and has two peaks at and . In this example, the kneading sequences and are not symmetric, and we have
In general, if then
In this particular case, according to the values , we have
Moreover, we have the following: