# Iteration of Differentiable Functions underm-Modal Maps with Aperiodic Kneading Sequences

Authors: Maria F. Correia; Carlos C. Ramos; Sandra Vinagre
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We consider the dynamical system (𝒜, T), where 𝒜 is a class of differentiable functions defined on some interval and T : 𝒜 → 𝒜 is the operator Tϕ:=f∘ϕ, where f 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 f are aperiodic. Moreover, we analyze the evolution as well as the distribution of the aperiodic critical values of the iterated functions.


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## Body

## 1. Introduction

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(𝒜,T), where 𝒜 is a class of differentiable real functions defined on some interval and T:𝒜→𝒜 is the operator Tϕ:=f∘ϕ, where f is an m-modal map. The class 𝒜 is the class of differentiable functions ϕ defined on an interval (eventually could be defined on ℝ) such that ϕ∈𝒜 if ϕ∈C1([0,1]) with a finite number of critical points, satisfying the boundary conditions ϕ′(0)=ϕ′(1)=0 and Im(ϕ)⊂I, for a given interval I. The nature of 𝒜 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 𝒜, under the iteration of T, 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 T, of attributes of the elements in 𝒜, such as the oscillations, the mean value, the measure, the mean derivative, the length of the graph, among others.The dynamical system(𝒜,T) has infinite dimension, although induced by a one-dimensional discrete dynamical system (I,f). From the topological point of view, the dynamical system (I,f) is formally contained in (𝒜,T), since the constant functions, ϕ≡c, belong trivially to 𝒜 and T(c)=f(c). Moreover, a monotone function ϕ∈𝒜 (nontrivial) determine a signed interval, ([ϕ(0),ϕ(1)],+) if ϕ is an increasing function and ([ϕ(1),ϕ(0)],-) if ϕ is a decreasing function. Then the dynamics of intervals under iteration of an interval map f is also formally contained in (𝒜,T).We determine the itineraries of a sufficiently large number of critical values of the iterated functions using the algorithm introduced in [7]. 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 f, s(f). In this paper, we analyze the case of f being an m-modal map with aperiodic kneading sequences.


## 2. Preliminaries on Symbolic Dynamics

### 2.1.Symbolic Dynamics form-Modal Maps

In this section, we describe some preliminaries on symbolic dynamics, in particular, aspects concerning to them-modal maps on an interval I.LetI⊂ℝ be an interval. A map f:I→I is called m-modal if it is in C1(I) and has m critical points. Let ci, with i=1,2,…,m, be the m critical points of the map f such that c0<c1<···<cm<cm+1, where c0 and cm+1 represent the boundary points of the interval I. In these conditions, consider the partition of the interval I into disjoint subsets
(2.1)I=I1∪IC1∪I2∪IC2∪⋅⋅⋅∪ICm∪Im+1,
where ICi is the set {ci},i=1,2,…,m, and Ii,i=1,2,…,m+1, are the intervals
(2.2)I1=[c0,c1[,I2=]c1,c2[,…,Im+1=]cm,cm+1].A maximal interval on whichf is monotone is called a lap off and the number l=l(f) of distinct laps is called the lap number off. Thus, each interval Ii,i=1,2,…,m+1, is a lap.According to [8], the limit
(2.3)s(f)=lim⁡n→∞l(fn)1/n
is a real number in the interval [1,l(f)] and is called the growth number off.Thetopologicalentropyoff is
(2.4)htop(f)=log⁡(s(f))=lim⁡n→∞⁡1nl(fn),
see [9].Next, to each pointx in Ii,i=1,2,…,m+1, we assign the symbol i,i=1,2,…,m+1, or Ci,i=1,2,…,m, if x=ci,i=1,2,…,m. This assignment is called the address ofx, and it is denoted by ad(x). The address of the point x, ad(x), is thus given by
(2.5)ad(x)={iifx∈Ii,i=1,2,…,m+1,Ciifx∈ICi,i=1,2,…,m.As usual, we get a correspondence between orbits of points and symbolic sequences of the alphabet{1,C1,2,…,m+1}, the itinerary ofx under f, which is defined by
(2.6)it(x):=ad(x)ad(f(x))ad(f2(x))⋯∈{1,C1,2,…,m+1}N.The orbits, underf, of the critical points are of special importance, in particular, their itineraries. Following [8], for each critical point the kneading sequence is 𝒦i:=it(f(ci)),i=1,2,…,m, and the collection of symbolic sequences 𝒦f:=(𝒦1,…,𝒦m) is called the kneading invariant off.A symbolic sequence(ik)k≥1 in {1,C1,2,…,m+1}ℕ0 is called admissible, with respect to f, if it occurs as an itinerary for some point x in I. The set of all admissible sequences in {1,C1,2,…,m+1}ℕ0 is denoted by Σ.In the sequence spaceΣ, we define the usual shift mapσ:Σ→Σ by
(2.7)σ(i1i2i3⋯)=i2i3⋅⋅⋅.
Moreover, the following relation with f and the itinerary map is satisfied
(2.8)σ(it(x))=it(f(x)).
Therefore, we obtain the symbolic system (Σ,σ) associated with the discrete dynamical system (I,f).Anadmissible word is a finite sub-sequence occurring in an admissible sequence. The set of admissible words of size k occurring in some sequence from Σ is denoted by 𝒲k=𝒲k(f). We define the cylinder set Ii0i1···ik⊂I, i0i1···ik∈𝒲k+1, by
(2.9)Ii0i1⋅⋅⋅ik={x∈I:x∈Ii0,f(x)∈Ii1,…,fk(x)∈Iik}=Ii0∩f-1(Ii1)∩⋯∩f-k(Iik).
In other words, x∈Ii0i1···ik means that ad(x)=i0, ad(f(x))=i1,…, ad(fk(x))=ik.Consider thesign functionɛ:∪k=1∞𝒲k→{-1,0,1} defined by
(2.10)ɛ(i1⋅⋅⋅ik)=∏j=1kɛ(j),
with i1···ik∈𝒲k, ɛ(Cj)=0 and ɛ(j)=+1 or ɛ(j)=-1, accordingly f restricted to Ij is increasing or decreasing.The parity, with respect to the mapf, of a given admissible word i1,…,ik∈𝒲k, is even if ɛ(i1···ik)=1 and odd if ɛ(i1···ik)=-1. From the order relation 1≺C1≺2≺···≺m+1, inherited from the order of the intervals of the partition of the interval I, we introduce an order relation between symbolic sequences as follows: given any distinct sequences (ik)k≥1, (jk)k≥1∈{1,C1,2,…,m+1}ℕ0, admitting that they have a common initial subsequence, that is, there is a natural r≥0 such that i1⋯ir=j1⋯jr and ir+1≠jr+1, we will say that (ik)k≥1≺(jk)k≥1 if and only if ir+1≺jr+1 and ɛ(i1⋯ir)=1 or jr+1≺ir+1 and ɛ(i1⋯ir)=-1.


### 2.2. Symbolic Dynamics for the Infinite Dynamical System(𝒜,T)

Now, consider anm-modal map f in the class C1(I,I), for a certain interval I⊂ℝ, and the class of differentiable functions
(2.11)A={φ∈C1([0,1],I):φ′(0)=φ′(1)=0,|cp(φ)|<∞},
where |cp(φ)| denotes the number of critical points of φ.LetT be the operator
(2.12)T:A⟶A,φ⟼f∘φ.
Note that this operator is well defined since (f∘φ)′(0)=(f∘φ)′(1)=0. Moreover, if ϕ∈𝒜 and Im(ϕ)⊂I, then Im(Tkϕ)⊂I for every k∈ℕ. Therefore, we obtain a discrete dynamical system (𝒜,T) in the sense that we have a set 𝒜 (eventually with additional structure, a topology or a metric, for now not specified) and a self-map T, which characterizes the discrete time evolution.In order to introduce a symbolic description for the discrete dynamical system(𝒜,T), let us consider the decomposition of 𝒜 into the following classes (see [10–12]):
(2.13)Ac={ϕ∈A:ϕ(x)isconstantin[0,1]},A0={ϕ∈A:ϕhasnocriticalpointsin]0,1[},Aj={ϕ∈A:ϕhasjcriticalpointsin]0,1[},j=1,2,….Letϕ∈𝒜 and let η(ϕ) be the number of nontrivial critical points of ϕ (inside [0,1]). In this case, if ϕ∈𝒜j, j∈ℕ0, then η(ϕ)=j and the total number of critical points is η(ϕ)+2=j+2. We are interested in the symbolic description of the dynamical evolution of a function ϕ under iteration of f, which has essentially a topological meaning; therefore, the important aspect is to distinguish and codify the critical points and the critical values of ϕ. Given ϕ∈𝒜, we identify its critical points and collect the addresses and itineraries of the corresponding critical values. The generalized symbolic space is Σ_:=∪j∈ℕ0Σj+1, where Σj+1=Σ×Σ×···×Σ (j+1 times), and we define the generalized address map for the space 𝒜 by
(2.14)ad_:A⟶{1,C1,2,…,m+1}η(ϕ)+2ϕ⟼ad_(ϕ):=(ad(d0),ad(d1),…,ad(dη(ϕ)),ad(dη(ϕ)+1))
and the generalized itinerary map for the space 𝒜 by
(2.15)it_:A⟶Σ_ϕ⟼it_(ϕ)=(it(d0),it(d1),…,it(dη(ϕ)),it(dη(ϕ)+1)),
where dj:=ϕ(aj), j=0,1,…,η(ϕ)+1, are the critical values of ϕ in the interval I (with d0=ϕ(0) and dη(ϕ)+1=ϕ(1)).Leti(j)=i1(j)i2(j)⋯be such that i(j)=it(dj), j=0,1,2,…,η(ϕ)+1. The generalized shift map is then defined by
(2.16)σ_(i(0),…,i(j),i(j+1),…,i(η(ϕ)+1))i(η(ϕ)+1):={(σ(i(0)),…,σ(i(j)),σ(i(j+1)),…,σ(i(η(ϕ)+1)))ifi1(j)=i1(j+1),(σ(i(0)),…,σ(i(j)),Ki1(j),…,Ki1(j+1)-1,σ(i(j+1)),…,σ(i(η(ϕ)+1)))ifi1(j)≠i1(j+1),i1(j)≺i1(j+1),(σ(i(0)),…,σ(i(j)),Ki1(j)-1,…,Ki1(j+1),σ(i(j+1)),…,σ(i(η(ϕ)+1)))ifi1(j)≠i1(j+1),i1(j+1)≺i1(j),
where 𝒦i1(j) is the kneading sequence corresponding to the critical point of f, localized between the critical values dj and dj+1, with i1(j)∈{1,…,m+1}, j=0,1,2,…,η(ϕ). We obtain a symbolic system (Σ_,σ_) associated to (𝒜,T). Similarly to the finite-dimensional discrete dynamical systems, it is verified the following result.Theorem 2.1.
Letϕ, ϕ~∈𝒜 with ϕ≠ϕ~ so that it_(ϕ)=it_(ϕ~), then
(2.17)it_(Tkϕ)=it_(Tkϕ~),k∈N0.
Moreover,(2.18)σ_∘it_=it_∘T.The previous definition and results allow us, knowing only the itineraries of the critical values of an initial conditionϕ0 and the kneading invariant of f, 𝒦f, to obtain explicitly the itineraries of the critical values of ϕk=Tkϕ0, k∈ℕ, see the next example.Example 2.2.
Let us consider a4-modal map f characterized by the kneading invariant
(2.19)Kf=((555C1)∞,(15C2)∞,(53C3)∞,(124C4)∞),
(with the alphabet {1,C1,2,C2,3,C3,4,C4,5}), see Figure 1. Now, we consider the generalized itinerary of the function ϕ0∈𝒜 given by it_(ϕ0)=((15)∞,(253)∞,(413)∞), with respect to the kneading invariant 𝒦f. Using the Theorem 2.1, we obtain the temporal evolution by T of ϕ0:
(2.20)((15)∞,(253)∞,(413)∞)⟶((51)∞,K1,(532)∞,K2,K3,(134)∞)=((51)∞,(555C1)∞,(532)∞,(15C2)∞,(53C1)∞,(134)∞)⟶((15)∞,(55C15)∞,(325)∞,K4,K3,K2,K1,(5C21)∞,(3C15)∞,(341)∞)=((15)∞,(55C15)∞,(325)∞,(124C4)∞,(53C3)∞,(15C2)∞,(555C1)∞,(5C21)∞,(3C15)∞,(341)∞)⟶((51)∞,K1,K2,K3,K4,(5C155)∞,K4,K3,(253)∞,K2,K1,(24C41)∞,K1,K2,K3,K4,(3C35)∞,K4,K3,K2,K1,(5C21)∞,K1,K2,K3,K4,(55C15)∞,(C215)∞,(C353)∞,(413)∞)=((51)∞,(555C1)∞,(15C2)∞,(53C3)∞,(124C4)∞,(5C155)∞,(124C4)∞,(53C3)∞,(253)∞,(15C2)∞,(24C41)∞,(555C1)∞,(15C2)∞,(53C3)∞,(124C4)∞,(3C35)∞,(124C4)∞,(53C3)∞,(15C2)∞,(555C1)∞,(5C21)∞,(555C1)∞,(15C2)∞,(53C3)∞,(124C4)∞,(55C15)∞,(C215)∞,(C353)∞,(413)∞)⟶⋯Figure 1
Graph of the mapf characterized by the kneading invariant 𝒦f=((555C1)∞,(15C2)∞,(53C3)∞,(124C4)∞).


## 2.1.Symbolic Dynamics form-Modal Maps

In this section, we describe some preliminaries on symbolic dynamics, in particular, aspects concerning to them-modal maps on an interval I.LetI⊂ℝ be an interval. A map f:I→I is called m-modal if it is in C1(I) and has m critical points. Let ci, with i=1,2,…,m, be the m critical points of the map f such that c0<c1<···<cm<cm+1, where c0 and cm+1 represent the boundary points of the interval I. In these conditions, consider the partition of the interval I into disjoint subsets
(2.1)I=I1∪IC1∪I2∪IC2∪⋅⋅⋅∪ICm∪Im+1,
where ICi is the set {ci},i=1,2,…,m, and Ii,i=1,2,…,m+1, are the intervals
(2.2)I1=[c0,c1[,I2=]c1,c2[,…,Im+1=]cm,cm+1].A maximal interval on whichf is monotone is called a lap off and the number l=l(f) of distinct laps is called the lap number off. Thus, each interval Ii,i=1,2,…,m+1, is a lap.According to [8], the limit
(2.3)s(f)=lim⁡n→∞l(fn)1/n
is a real number in the interval [1,l(f)] and is called the growth number off.Thetopologicalentropyoff is
(2.4)htop(f)=log⁡(s(f))=lim⁡n→∞⁡1nl(fn),
see [9].Next, to each pointx in Ii,i=1,2,…,m+1, we assign the symbol i,i=1,2,…,m+1, or Ci,i=1,2,…,m, if x=ci,i=1,2,…,m. This assignment is called the address ofx, and it is denoted by ad(x). The address of the point x, ad(x), is thus given by
(2.5)ad(x)={iifx∈Ii,i=1,2,…,m+1,Ciifx∈ICi,i=1,2,…,m.As usual, we get a correspondence between orbits of points and symbolic sequences of the alphabet{1,C1,2,…,m+1}, the itinerary ofx under f, which is defined by
(2.6)it(x):=ad(x)ad(f(x))ad(f2(x))⋯∈{1,C1,2,…,m+1}N.The orbits, underf, of the critical points are of special importance, in particular, their itineraries. Following [8], for each critical point the kneading sequence is 𝒦i:=it(f(ci)),i=1,2,…,m, and the collection of symbolic sequences 𝒦f:=(𝒦1,…,𝒦m) is called the kneading invariant off.A symbolic sequence(ik)k≥1 in {1,C1,2,…,m+1}ℕ0 is called admissible, with respect to f, if it occurs as an itinerary for some point x in I. The set of all admissible sequences in {1,C1,2,…,m+1}ℕ0 is denoted by Σ.In the sequence spaceΣ, we define the usual shift mapσ:Σ→Σ by
(2.7)σ(i1i2i3⋯)=i2i3⋅⋅⋅.
Moreover, the following relation with f and the itinerary map is satisfied
(2.8)σ(it(x))=it(f(x)).
Therefore, we obtain the symbolic system (Σ,σ) associated with the discrete dynamical system (I,f).Anadmissible word is a finite sub-sequence occurring in an admissible sequence. The set of admissible words of size k occurring in some sequence from Σ is denoted by 𝒲k=𝒲k(f). We define the cylinder set Ii0i1···ik⊂I, i0i1···ik∈𝒲k+1, by
(2.9)Ii0i1⋅⋅⋅ik={x∈I:x∈Ii0,f(x)∈Ii1,…,fk(x)∈Iik}=Ii0∩f-1(Ii1)∩⋯∩f-k(Iik).
In other words, x∈Ii0i1···ik means that ad(x)=i0, ad(f(x))=i1,…, ad(fk(x))=ik.Consider thesign functionɛ:∪k=1∞𝒲k→{-1,0,1} defined by
(2.10)ɛ(i1⋅⋅⋅ik)=∏j=1kɛ(j),
with i1···ik∈𝒲k, ɛ(Cj)=0 and ɛ(j)=+1 or ɛ(j)=-1, accordingly f restricted to Ij is increasing or decreasing.The parity, with respect to the mapf, of a given admissible word i1,…,ik∈𝒲k, is even if ɛ(i1···ik)=1 and odd if ɛ(i1···ik)=-1. From the order relation 1≺C1≺2≺···≺m+1, inherited from the order of the intervals of the partition of the interval I, we introduce an order relation between symbolic sequences as follows: given any distinct sequences (ik)k≥1, (jk)k≥1∈{1,C1,2,…,m+1}ℕ0, admitting that they have a common initial subsequence, that is, there is a natural r≥0 such that i1⋯ir=j1⋯jr and ir+1≠jr+1, we will say that (ik)k≥1≺(jk)k≥1 if and only if ir+1≺jr+1 and ɛ(i1⋯ir)=1 or jr+1≺ir+1 and ɛ(i1⋯ir)=-1.


## 2.2. Symbolic Dynamics for the Infinite Dynamical System(𝒜,T)

Now, consider anm-modal map f in the class C1(I,I), for a certain interval I⊂ℝ, and the class of differentiable functions
(2.11)A={φ∈C1([0,1],I):φ′(0)=φ′(1)=0,|cp(φ)|<∞},
where |cp(φ)| denotes the number of critical points of φ.LetT be the operator
(2.12)T:A⟶A,φ⟼f∘φ.
Note that this operator is well defined since (f∘φ)′(0)=(f∘φ)′(1)=0. Moreover, if ϕ∈𝒜 and Im(ϕ)⊂I, then Im(Tkϕ)⊂I for every k∈ℕ. Therefore, we obtain a discrete dynamical system (𝒜,T) in the sense that we have a set 𝒜 (eventually with additional structure, a topology or a metric, for now not specified) and a self-map T, which characterizes the discrete time evolution.In order to introduce a symbolic description for the discrete dynamical system(𝒜,T), let us consider the decomposition of 𝒜 into the following classes (see [10–12]):
(2.13)Ac={ϕ∈A:ϕ(x)isconstantin[0,1]},A0={ϕ∈A:ϕhasnocriticalpointsin]0,1[},Aj={ϕ∈A:ϕhasjcriticalpointsin]0,1[},j=1,2,….Letϕ∈𝒜 and let η(ϕ) be the number of nontrivial critical points of ϕ (inside [0,1]). In this case, if ϕ∈𝒜j, j∈ℕ0, then η(ϕ)=j and the total number of critical points is η(ϕ)+2=j+2. We are interested in the symbolic description of the dynamical evolution of a function ϕ under iteration of f, which has essentially a topological meaning; therefore, the important aspect is to distinguish and codify the critical points and the critical values of ϕ. Given ϕ∈𝒜, we identify its critical points and collect the addresses and itineraries of the corresponding critical values. The generalized symbolic space is Σ_:=∪j∈ℕ0Σj+1, where Σj+1=Σ×Σ×···×Σ (j+1 times), and we define the generalized address map for the space 𝒜 by
(2.14)ad_:A⟶{1,C1,2,…,m+1}η(ϕ)+2ϕ⟼ad_(ϕ):=(ad(d0),ad(d1),…,ad(dη(ϕ)),ad(dη(ϕ)+1))
and the generalized itinerary map for the space 𝒜 by
(2.15)it_:A⟶Σ_ϕ⟼it_(ϕ)=(it(d0),it(d1),…,it(dη(ϕ)),it(dη(ϕ)+1)),
where dj:=ϕ(aj), j=0,1,…,η(ϕ)+1, are the critical values of ϕ in the interval I (with d0=ϕ(0) and dη(ϕ)+1=ϕ(1)).Leti(j)=i1(j)i2(j)⋯be such that i(j)=it(dj), j=0,1,2,…,η(ϕ)+1. The generalized shift map is then defined by
(2.16)σ_(i(0),…,i(j),i(j+1),…,i(η(ϕ)+1))i(η(ϕ)+1):={(σ(i(0)),…,σ(i(j)),σ(i(j+1)),…,σ(i(η(ϕ)+1)))ifi1(j)=i1(j+1),(σ(i(0)),…,σ(i(j)),Ki1(j),…,Ki1(j+1)-1,σ(i(j+1)),…,σ(i(η(ϕ)+1)))ifi1(j)≠i1(j+1),i1(j)≺i1(j+1),(σ(i(0)),…,σ(i(j)),Ki1(j)-1,…,Ki1(j+1),σ(i(j+1)),…,σ(i(η(ϕ)+1)))ifi1(j)≠i1(j+1),i1(j+1)≺i1(j),
where 𝒦i1(j) is the kneading sequence corresponding to the critical point of f, localized between the critical values dj and dj+1, with i1(j)∈{1,…,m+1}, j=0,1,2,…,η(ϕ). We obtain a symbolic system (Σ_,σ_) associated to (𝒜,T). Similarly to the finite-dimensional discrete dynamical systems, it is verified the following result.Theorem 2.1.
Letϕ, ϕ~∈𝒜 with ϕ≠ϕ~ so that it_(ϕ)=it_(ϕ~), then
(2.17)it_(Tkϕ)=it_(Tkϕ~),k∈N0.
Moreover,(2.18)σ_∘it_=it_∘T.The previous definition and results allow us, knowing only the itineraries of the critical values of an initial conditionϕ0 and the kneading invariant of f, 𝒦f, to obtain explicitly the itineraries of the critical values of ϕk=Tkϕ0, k∈ℕ, see the next example.Example 2.2.
Let us consider a4-modal map f characterized by the kneading invariant
(2.19)Kf=((555C1)∞,(15C2)∞,(53C3)∞,(124C4)∞),
(with the alphabet {1,C1,2,C2,3,C3,4,C4,5}), see Figure 1. Now, we consider the generalized itinerary of the function ϕ0∈𝒜 given by it_(ϕ0)=((15)∞,(253)∞,(413)∞), with respect to the kneading invariant 𝒦f. Using the Theorem 2.1, we obtain the temporal evolution by T of ϕ0:
(2.20)((15)∞,(253)∞,(413)∞)⟶((51)∞,K1,(532)∞,K2,K3,(134)∞)=((51)∞,(555C1)∞,(532)∞,(15C2)∞,(53C1)∞,(134)∞)⟶((15)∞,(55C15)∞,(325)∞,K4,K3,K2,K1,(5C21)∞,(3C15)∞,(341)∞)=((15)∞,(55C15)∞,(325)∞,(124C4)∞,(53C3)∞,(15C2)∞,(555C1)∞,(5C21)∞,(3C15)∞,(341)∞)⟶((51)∞,K1,K2,K3,K4,(5C155)∞,K4,K3,(253)∞,K2,K1,(24C41)∞,K1,K2,K3,K4,(3C35)∞,K4,K3,K2,K1,(5C21)∞,K1,K2,K3,K4,(55C15)∞,(C215)∞,(C353)∞,(413)∞)=((51)∞,(555C1)∞,(15C2)∞,(53C3)∞,(124C4)∞,(5C155)∞,(124C4)∞,(53C3)∞,(253)∞,(15C2)∞,(24C41)∞,(555C1)∞,(15C2)∞,(53C3)∞,(124C4)∞,(3C35)∞,(124C4)∞,(53C3)∞,(15C2)∞,(555C1)∞,(5C21)∞,(555C1)∞,(15C2)∞,(53C3)∞,(124C4)∞,(55C15)∞,(C215)∞,(C353)∞,(413)∞)⟶⋯Figure 1
Graph of the mapf characterized by the kneading invariant 𝒦f=((555C1)∞,(15C2)∞,(53C3)∞,(124C4)∞).


## 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 off are periodic. Moreover, we developed the following algorithm based only on the kneading invariant of f. 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 ofϕn).
LetK1, K2, X0, and n be the inputs and let L and Y be the outputs.
Step 0. Set L={}, Z=X0 and i=1.
Step 1. If Y={}, compute |Z| and append the sequence σ(Z[1]) to the vector Y. Set j=1 and go to the Step 2.
Step 2. If j≤|Z|-1, then design by y the first symbol of Z[j] and by z the first symbol of Z[j+1], and go to the Step 3. Otherwise, go to the Step 7.
Step 3. If y=1  (resp. y=C2 or y=2) and z=C2 or z=2   (resp. z=1), then append K1 to the vector Y and the σ(Z[j+1]), set j=j+1 and return to the Step 2. Otherwise, go to the Step 4.
Step 4. If y=3   (resp. y=C1 or y=2) and z=C1, or z=2   (resp. z=3), then append K2 to the vector Y and the σ(Z[j+1]), set j=j+1, and return to the Step 2. Otherwise, go to the Step 5.
Step 5. If y=1 and z=3 then, append K1 and K2 to the vector Y and the σ(Z[j+1]), set j=j+1, and return to the Step 2. Otherwise, go to the Step 6.
Step 6. If y=3 and z=1, then append K2 and the K1 to the vector Y and the σ(Z[j+1]), set j=j+1, and return to the Step 2.
Step 7. Append the vector Y to the set L and set Z=Y. If i=n, the algorithm ends; otherwise, set i=i+1 and return to the Step 1.In the following example, we illustrate an implementation of the Algorithm3.1 in Mathematica6.0 for a bimodal f with periodic kneading sequences.Example 3.2.
Let us consider a bimodal mapf:I→I characterized by the kneading invariant 𝒦f=((332C1)∞,(112C2)∞) and the function ϕ0∈𝒜 such that it_(ϕ0)=((312)∞,(32)∞). Considering
(3.1)K1=332C1,K2=112C2,X0={312,32},n=14,
we obtain a large number of critical values of ϕ14=T14ϕ0, in this case 181369 critical values for ϕ14. Moreover, we obtain the itineraries it_(ϕk), for k=0,1,2,3,…,7,…,14, given by
(3.2)(123,23)↓(231,332C1,32)↓(312,112C2,32C13,23)↓(123,112C2,332C1,12C21,332C1,112C2,2C133,112C2,32)↓⋮↓(123,32C13,112C2,C2112,112C2,32C13,112C2,332C1,12C21,332C1,112C2,2C133,112C2,332C1,12C21,332C1,112C2,32C13,112C2,332C1,2C211,332C1,112C2,32C13,112C2,332C1,12C21,332C1,C1332,332C1,12C21,332C1,112C2,32C13,112C2,332C1,2C211,332C1,112C2,32C13,112C2,332C1,12C21,332C1,112C2,2C133,112C2,332C1,12C21,332C1,112C2,332C1,12C21,332C1,112C2,2C133,112C2,332C1,12C21,332C1,112C2,32C13,112C2,332C1,2C211,332C1,112C2,32C13,112C2,332C1,12C21,332C1,C1332,332C1,12C21,332C1,112C2,32C13,112C2,332C1,2C211,332C1,112C2,32C13,112C2,332C1,12C21,332C1,112C2,2C133,112C2,332C1,12C21,332C1,112C2,32C13,112C2,C2112,112C2,32C13,112C2,332C1,12C21,332C1,112C2,2C133,112C2,332C1,12C21,332C1,112C2,32C13,112C2,332C1,12C21,332C1,112C2,2C133,112C2,332C1,12C21,332C1,112C2,32C13,112C2,C2112,112C2,32C13,112C2,332C1,12C21,332C1,112C2,2C133,112C2,332C1,12C21,332C1,112C2,32C13,112C2,332C1,2C211,12332C1,112C2,32C13,23)↓⋮↓(231,332C1,112C2,2C133,112C2,332C1,…,2C13,112C2,332C1,12C21,332C1,32).To apply the algorithm when the keading sequences off 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 f are aperiodic. In this case, the length of the kneading sequences is not finite. Then, we make explicit at least the first p symbols of each kneading sequence to compute the itineraries of critical values of ϕp=Tpϕ0, for some fixed p∈ℕ.In this section, we analyze the evolution and distribution of the periodic critical values of the iterated functions when the kneading sequences off are aperiodic. As we present in the following results, the new distinct critical values obtained at each iteration depend on the kneading sequences of f. 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 functionϕk. As previously, η(ϕ0) denotes the number of the nontrivial critical points of ϕ0 and dj,j=0,1,2,…,η(ϕ0)+1, denote the critical values of ϕ0. Let υ(ϕ0) be the number of the critical values of ϕ0.Since the mapf restricted to a subinterval Ii is monotone, f is invertible if restricted to Ii,i=1,…,m. Denote by fi-1 the inverse map f|Ii-1:f(Ii)→Ii. Let ci,i=1,…,m, be the critical points of the map f, and let Ui⊂I be the maximal interval containing ci, consisting of points whose orbits tend to the stable periodic orbit of ci, see [13].LetΛi1i2⋯ipCi⊂I be given by Λi1i2⋯ipCi:=fi1-1∘fi2-1∘···∘fip-1(Ui),i=1,…,m,  i1i2···ip∈𝒲p, p∈ℕ. Therefore, the basin of attraction for f is given by Λ=⋃p=1∞‍⋃i=1m‍⋃i1i2···ip∈𝒲p‍Λi1i2···ipCi.Proposition 3.3.
Letf be an m-modal map, ϕ0∈𝒜 and ϕk=Tkϕ0,k∈ℕ0. If the kneading sequences of f, 𝒦i, are aperiodic, then the itineraries of the critical values of ϕk, k∈ℕ0, will belong to the set
(3.3)⋃i=1m‍{σq(Ki),q=0,1,2,…,k-1}∪⋃j=0η(ϕ0)+1‍{σl(it(dj)),l=0,1,2,…,k-1},
where dj, j=0,1,2,…,η(ϕ0)+1, are the critical values of ϕ0. Moreover, υ(ϕk) will always grow and never stabilize.Proof.
Letci,i=1,2,…,m, be the critical points of the map f. In [12], we proved that all critical points of ϕk=Tkϕ0,k∈ℕ0 arise in the following two ways:(i)
a critical point ofϕk-1 will be a critical point of ϕk;(ii)
every critical point ofϕk, which is not a critical point of ϕk-1, is a point y in which ϕk-1(y)=ci for some i=1,2,…,m.
Therefore, from (i), the critical values ofϕk correspond to the image under f of the critical values of ϕk-1.
From (ii), the new critical values ofϕk are always f(ci),i=1,2,…,m. Then the critical values of ϕk are either points in the set {fk(dj),k∈ℕ} or points in the set ⋃i=1m‍{fk(ci),k∈ℕ}. Recalling that it(fk(x))=σk(it(x)),k∈ℕ0, for all x∈I, the itineraries of the critical values of ϕk are in the set
(3.4)⋃i=1m‍{σq(Ki),q=0,1,2,…,k-1}∪⋃j=0η(ϕ0)+1‍{σl(it(dj)),l=0,1,2,…,k-1}.
Since we are considering aperiodic points ci, i=1,2,…,m, the set
(3.5)⋃i=1m‍{fk(ci),k∈N0}
is infinite.
For a fixedk∈ℕ0, we have that the number of distinct critical values of a certain ϕk is υ(ϕk)≤υ(ϕk+1). We have that υ(ϕk) grows with k.Leti1i2⋯ be an aperiodic sequence in Σ and let Ni1i2⋯(ϕ) denote the number of times which i1i2⋯ occurs as itinerary of a critical value of ϕ∈𝒜. The following result illustrates the frequencies of occurence of the critical values whose itineraries are obtained of 𝒦i, i=1,2,…,m.Proposition 3.4.
Letf be an m -modal map, ϕ0∈𝒜   (with ϕ0∉𝒜c), and ϕk=Tkϕ0,k∈ℕ0. If the kneading sequences of f, 𝒦i, are aperiodic then for each k∈ℕ0, we have
(3.6)Nσq(Ki)(ϕk)≤Nσp(Ki)(ϕk),
for q>p with p,  q=0,1,2,…,k-1,i=1,2,…,m.Proof.
As defined previously,Nσq(𝒦i)(ϕk) is the number of times, which σq(𝒦i) occurs as itinerary of a critical value of ϕk, for some fixed i∈{1,2,…,m} and some fixed q∈{0,1,2,…,k-1}.
If the itinerary of some critical values ofϕk is σq(𝒦i), then the itinerary of these critical values is 𝒦i for ϕk-q, which correspond to the new critical points of ϕk-q. These new critical points are derived of the (k-q-1)th-preimages of ci, that is,  ϕk-q-1(y)=ci.
Similarly, if the itinerary of some critical values ofϕk is σp(𝒦i), then these critical values correspond to new critical points of ϕk-p that are obtained of the (k-p-1)th-preimages, under f of ci, that is, ϕk-q-1(y)=ci.
Sinceq>p, then k-q-1<k-p-1. In this case, the number of the admissible k-1-q-preimages is less or equal to the number of the admissible k-1-p-preimages since the number of laps of fk-q is less than the number of laps of fk-p. Therefore,
(3.7)Nσq(Ki)(ϕk)≤Nσp(Ki)(ϕk),
for q>p with p,  q=0,1,2,…,k-1,i=1,2,…,m.In order to illustrate the previous results, consider the following example.Example 3.5.
Consider the bimodal mapf characterized by the kneading invariant
(3.8)Kf=(332311321321…,113113323323…)
which analytical expression is given approximately by
(3.9)f(x)=3.9333x3-0.00665413x2-2.9333x+0.00665413
and the function ϕ0∈𝒜 such that
(3.10)it_(ϕ0)=(1∞,3∞,1∞,3∞).
According to the Proposition3.3, we have that the itineraries of the critical values of ϕ10=T10ϕ0 belong to the set
(3.11){σq(332311321321…),q=0,1,…,9}∪{σq(113113323323…),q=0,1,…,9}∪{σl(1∞),l=0,1,…,9}∪{σl(3∞),l=0,1,…,9}={332311321321…,32311321321…,2311321321…,311321321…,11321321…,1321321…,321321…,21321…,1321…,113113323323…,13113323323…,3113323323…,113323323…,13323323…,3323323…,323323…,23323…,3323…,323…,1∞,3∞}.
Regarding that 𝒦1=332311321321… and 𝒦2=113113323323…, by the Proposition 3.4, we have
(3.12)Nσ9(K1)(ϕ10)≤Nσ8(K1)(ϕ10)≤Nσ7(K1)(ϕ10)≤⋅⋅⋅≤Nσ(K1)(ϕ10)≤NK1(ϕ10),Nσ9(K2)(ϕ10)≤Nσ8(K2)(ϕ10)≤Nσ7(K2)(ϕ10)≤⋅⋅⋅≤Nσ(K2)(ϕ10)≤NK2(ϕ10).
Indeed, we have
(3.13)Nσ9(K1)(ϕ10)=4,Nσ8(K1)(ϕ10)=12,Nσ7(K1)(ϕ10)=36,Nσ6(K1)(ϕ10)=100,Nσ5(K1)(ϕ10)=268,Nσ4(K1)(ϕ10)=716,Nσ3(K1)(ϕ10)=1908,Nσ2(K1)(ϕ10)=5060,Nσ(K1)(ϕ10)=13412,NK1(ϕ10)=35548,Nσ9(K2)(ϕ10)=4,Nσ8(K2)(ϕ10)=12,Nσ7(K2)(ϕ10)=36,Nσ6(K2)(ϕ10)=92,Nσ5(K2)(ϕ10)=244,Nσ4(K2)(ϕ10)=644,Nσ3(K2)(ϕ10)=1708,Nσ2(K2)(ϕ10)=4524,Nσ(K2)(ϕ10)=11996,NK2(ϕ10)=31780.Let us consider a bimodal mapf with aperiodic kneading sequences and growth number s(f). Let ϕ0∈𝒜 be a function and ϕk=Tkϕ0, k∈ℕ0. Recall that Ni(j)(ϕ) is the number of times that i(j), j=0,1,2,…,υ(ϕk), occurs as itinerary of the critical values of ϕ∈𝒜. We define the relative frequency of the critical values of ϕk by
(3.14)Hi(j)(ϕk):=Ni(j)(ϕk)η(ϕk)+2,j=0,1,2,…,υ(ϕk),
where i(j) is the itinerary of the j-th critical value of ϕk. These values Hi(j) range between 0 and 1 and verify the following equality:
(3.15)Hi(0)+Hi(1)+⋅⋅⋅+Hi(υ(ϕk))=1.We denote byH the set of the values Hi(j)(ϕk), j=0,1,2,…,υ(ϕk), and we construct histograms using these values.


## 4. Numerical Results

Next, we present a numerical result that relatesHσq(𝒦i)(ϕk), the relative frequencies of the critical values of ϕk given by σq(𝒦i), q=0,1,2,…,k-1, for each i=1,2,…,m.LetΛ⊂I be the basin of attraction for f and ϕ0∈𝒜 such that Im(ϕ0)⊄Λ. If 𝒦i is the kneading sequence of f, then
(4.1)Hσq(Ki)(ϕk)Hσq+n(Ki)(ϕk)⟶sn(f),sincek⟶∞,q=0,1,2,…,k-1,n=1,2,…,k-2,i=1,2,…,m.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 ϕk. There is a relationship between the growth rate of the critical points of ϕk and the growth number of f, s(f). Since the growth rate of the critical points of ϕk is related with the growth of the preimages under f of the critical points ci, i=1,2,…,m, we have that the growth of the number of each distinct critical value, that results of the kneading sequence 𝒦i, is related with growth of the critical points correspondent to the preimages under f of the critical points ci, for each i∈{1,2,…,m}.An estimation of the growth number off, s(f), is given by
(4.2)s(f)≈η(ϕk)+2η(ϕk-1)+2,ask⟶∞.
Usually, a practical way to compute the growth number of f, s(f), when the kneading sequences of f are periodic is through the spectral radius of the transition matrix associated to the kneading invariant of f, see [14]. Using the algorithm, we spend less time computing the growth number of f than computing the transition matrix associated to the kneading invariant of f. 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 mapf:I→I characterized by the kneading invariant
(4.3)Kf=(331233113212…,112223213311…),
whose analytical expression is given approximately by
(4.4)f(x)=3.91322x3+0.0232761x2-2.91322x-0.0232761.
Let ϕ0∈𝒜 be a function such that
(4.5)it_(ϕ0)=(1∞,3∞),
whose analytical expression is given by ϕ0(x)=-cos(πx).
An estimation of the growth number off, s(f), is given by
(4.6)s(f)≈η(ϕk)+2η(ϕk-1)+2=17072465482=2.60719…,ask⟶∞.
The numberη(ϕk) is easily computed symbolically using the algorithm. The itineraries of the critical values correspondent to the new critical points of ϕ12=T12ϕ0 are given by σq(𝒦i), q=0,1,2,…,11,i=1,2, and are presented in the following set ordered by increasing k, which is suitable to present graphically
(4.7){K2,σ(K2),σ2(K2),σ3(K2),…,σ11(K2),K1,σ(K1),σ2(K1),σ3(K1),…,σ11(K1)}.
The respective valuesNσq(𝒦i)(ϕ12),q=0,1,2,…,11,i=1,2, are
(4.8){53765,20621,7911,3035,1163,447,171,65,25,9,3,1,51477,19749,7573,2905,1113,427,163,63,23,9,3,1}.
The relative frequencies of the critical values of ϕ12 whose itineraries are σq(𝒦i),
(4.9)Hσq(Ki)(ϕ12)=Nσq(Ki)(ϕ12)η(ϕ12)+2,
with η(ϕ12)=170444, q=0,1,…,11, i=1,2, are given in the set
(4.10)H={0.314931…,0.120788…,0.046339…,0.0177776…,0.00681232…,0.00261832…,0.00100164…,0.00038074…,0.000146439…,0.000052717,…,0.0000175726…,0.301529…,0.115681…,0.0443592…,0.0170162…,0.00651945…,0.00250117…,0.00095478…,0.000369025…,0.000134724…,0.0000527179…,0.0000175726…}.
These values (dependent only on the kneading invariant of f) 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 𝒦1 and 𝒦2. In this example, the kneading sequences 𝒦1 and 𝒦2 are not symmetric, and we have
(4.11)Hσq(K1)(ϕk)≤Hσq(K2)(ϕk),ask⟶∞,q=0,1,…,11.
In general, if(4.12)HK1(ϕk)≤HK2(ϕk),
then
(4.13)Hσq(K1)(ϕk)≤Hσq(K2)(ϕk),ask⟶∞.
In this particular case, according to the valuesHσq(𝒦i)(ϕ12), we have
(4.14)HK1≤HK2≤Hσ(K1)≤Hσ(K2)≤⋅⋅⋅≤Hσ11(K1)≤Hσ11(K2).
Moreover, we have the following:(4.15)HK1(ϕk)Hσ(K1)(ϕk)=2.60656…,Hσ(K1)(ϕk)Hσ2(K1)(ϕk)=2.60782…,Hσ2(K1)(ϕk)Hσ3(K1)(ϕk)=2.60688,…,Hσ3(K1)(ϕk)Hσ4(K1)(ϕk)=2.61006…,Hσ4(K1)(ϕk)Hσ5(K1)(ϕk)=2.60656…,Hσ5(K1)(ϕk)Hσ6(K1)(ϕk)=2.61963,…,HK1(ϕk)Hσ2(K1)(ϕk)=6.7944…,Hσ(K1)(ϕk)Hσ3(K1)(ϕk)=6.79828…,Hσ2(K1)(ϕk)Hσ4(K1)(ϕk)=6.80413,…,Hσ3(K1)(ϕk)Hσ5(K1)(ϕk)=6.80328…,Hσ4(K1)(ϕk)Hσ6(K1)(ϕk)=6.82822…,Hσ5(K1)(ϕk)Hσ7(K1)(ϕk)=6.77778,…,HK1(ϕk)Hσ3(K1)(ϕk)=17.7201…,Hσ(K1)(ϕk)Hσ4(K1)(ϕk)=17.7439…,Hσ2(K1)(ϕk)Hσ5(K1)(ϕk)=17.7354,…,HK1(ϕk)Hσ4(K1)(ϕk)=46.2507…,Hσ(K1)(ϕk)Hσ5(K1)(ϕk)=46.2506…,Hσ2(K1)(ϕk)Hσ6(K1)(ϕk)=46.4601,…,HK2(ϕk)Hσ(K2)(ϕk)=2.60729…,Hσ(K2)(ϕk)Hσ2(K2)(ϕk)=2.60662…,Hσ2(K2)(ϕk)Hσ3(K2)(ϕk)=2.60659,…,Hσ3(K2)(ϕk)Hσ4(K2)(ϕk)=2.60963…,Hσ4(K2)(ϕk)Hσ5(K2)(ϕk)=2.60179…,Hσ5(K2)(ϕk)Hσ6(K2)(ϕk)=2.61404,…,HK2(ϕk)Hσ2(K2)(ϕk)=6.79623…,Hσ(K2)(ϕk)Hσ3(K2)(ϕk)=6.7944…,Hσ2(K2)(ϕk)Hσ4(K2)(ϕk)=6.80224,…,Hσ3(K2)(ϕk)Hσ5(K2)(ϕk)=6.78971…,Hσ4(K2)(ϕk)Hσ6(K2)(ϕk)=6.80117…,Hσ5(K2)(ϕk)Hσ7(K2)(ϕk)=6.87692,…,HK2(ϕk)Hσ3(K2)(ϕk)=17.715…,Hσ(K2)(ϕk)Hσ4(K2)(ϕk)=17.7309…,Hσ2(K2)(ϕk)Hσ5(K2)(ϕk)=17.698,…,HK2(ϕk)Hσ4(K2)(ϕk)=46.2296…,Hσ(K2)(ϕk)Hσ5(K2)(ϕk)=46.132…,Hσ2(K2)(ϕk)Hσ6(K2)(ϕk)=46.2632…,
where s(f)=2.60719…, (s(f))2=6.79744…, (s(f))3=17.7222…, and (s(f))4=46.2052….
In general, we have(4.16)Hσq(Ki)(ϕk)Hσq+n(Ki)(ϕk)⟶sn(f),ask⟶∞,q=0,1,2,…,k-1,n=1,2,…,k-2,i=1,2.Figure 2
Histogram of the relative frequencies of the distinct critical values ofϕ12=T12ϕ0, whose itineraries are given by σq(𝒦1), σq(𝒦2), q=0,1,2,3,…. Note that the order of these itineraries is 𝒦2, σ(𝒦2), σ2(𝒦2), σ3(𝒦2), …, 𝒦1, σ(𝒦1), σ2(𝒦1), σ3(𝒦1), ….



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*Source: 796180-2012-08-08.xml*