Iteration of Differentiable Functions under m-Modal Maps with Aperiodic Kneading Sequences

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.

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 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 𝐼.

Let 𝐼⊂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 𝑐_𝑖, with 𝑖=1,2,…,𝑚, be the 𝑚 critical points of the map 𝑓 such that 𝑐₀<𝑐₁<⋅⋅⋅<𝑐_𝑚<𝑐_𝑚+1, where 𝑐₀ and 𝑐_𝑚+1 represent the boundary points of the interval 𝐼. In these conditions, consider the partition of the interval 𝐼 into disjoint subsets 𝐼=𝐼₁∪𝐼_𝐶1∪𝐼₂∪𝐼_𝐶2∪⋅⋅⋅∪𝐼_𝐶𝑚∪𝐼_𝑚+1, (2.1) where 𝐼_𝐶𝑖 is the set {𝑐_𝑖},𝑖=1,2,…,𝑚, and 𝐼_𝑖,𝑖=1,2,…,𝑚+1, are the intervals 𝐼₁=𝑐₀,𝑐₁,𝐼₂=𝑐₁,𝑐₂,…,𝐼_𝑚+1=𝑐_𝑚,𝑐_𝑚+1. (2.2)

A maximal interval on which 𝑓 is monotone is called a lap of 𝑓 and the number 𝑙=𝑙(𝑓) of distinct laps is called the lap number of 𝑓. Thus, each interval 𝐼_𝑖,𝑖=1,2,…,𝑚+1, is a lap.

According to [8], the limit 𝑠𝑓=lim_𝑛→∞𝑙𝑓^𝑛1/𝑛 (2.3) is a real number in the interval [1,𝑙(𝑓)] and is called the growth number of 𝑓.

Thetopological entropy of 𝑓 is ℎ_top𝑓=log𝑠𝑓=lim_𝑛→∞1𝑛𝑙𝑓^𝑛, (2.4) see [9].

Next, to each point 𝑥 in 𝐼_𝑖,𝑖=1,2,…,𝑚+1, we assign the symbol 𝑖,𝑖=1,2,…,𝑚+1, or 𝐶_𝑖,𝑖=1,2,…,𝑚, if 𝑥=𝑐_𝑖,𝑖=1,2,…,𝑚. This assignment is called the address of 𝑥, and it is denoted by ad(𝑥). The address of the point 𝑥, ad(𝑥), is thus given by ad(𝑥)=𝑖if𝑥∈𝐼_𝑖𝐶,𝑖=1,2,…,𝑚+1,_𝑖if𝑥∈𝐼_𝐶𝑖,𝑖=1,2,…,𝑚. (2.5)

As usual, we get a correspondence between orbits of points and symbolic sequences of the alphabet {1,𝐶₁,2,…,𝑚+1}, the itinerary of 𝑥 under 𝑓, which is defined by it(𝑥)∶=ad(𝑥)ad𝑓(𝑥)ad𝑓²(𝑥)⋯∈1,𝐶₁,2,…,𝑚+1^ℕ. (2.6)

The orbits, under 𝑓, of the critical points are of special importance, in particular, their itineraries. Following [8], for each critical point the kneading sequence is 𝒦_𝑖∶=it(𝑓(𝑐_𝑖)),𝑖=1,2,…,𝑚, and the collection of symbolic sequences 𝒦_𝑓∶=(𝒦₁,…,𝒦_𝑚) is called the kneading invariant of 𝑓.

A symbolic sequence (𝑖_𝑘)_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 space Σ, we define the usual shift map 𝜎∶Σ→Σ by 𝜎𝑖₁𝑖₂𝑖₃⋯=𝑖₂𝑖₃⋅⋅⋅. (2.7) Moreover, the following relation with 𝑓 and the itinerary map is satisfied 𝜎(it(𝑥))=it𝑓(𝑥). (2.8) Therefore, we obtain the symbolic system (Σ,𝜎) associated with the discrete dynamical system (𝐼,𝑓).

An admissible 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 𝒲_𝑘=𝒲_𝑘(𝑓). We define the cylinder set 𝐼_{𝑖0𝑖1⋅⋅⋅𝑖𝑘}⊂𝐼, 𝑖₀𝑖₁⋅⋅⋅𝑖_𝑘∈𝒲_𝑘+1, by 𝐼_{𝑖0𝑖1⋅⋅⋅𝑖𝑘}=𝑥∈𝐼∶𝑥∈𝐼_𝑖0,𝑓(𝑥)∈𝐼_𝑖1,…,𝑓^𝑘(𝑥)∈𝐼_𝑖𝑘=𝐼_𝑖0∩𝑓⁻¹𝐼_𝑖1∩⋯∩𝑓^−𝑘𝐼_𝑖𝑘. (2.9) In other words, 𝑥∈𝐼_{𝑖0𝑖1⋅⋅⋅𝑖𝑘} means that ad(𝑥)=𝑖₀, ad(𝑓(𝑥))=𝑖₁,…, ad(𝑓^𝑘(𝑥))=𝑖_𝑘.

Consider the sign function 𝜀∶∪^∞_𝑘=1𝒲_𝑘→{−1,0,1} defined by 𝜀𝑖₁⋅⋅⋅𝑖_𝑘=^𝑘_𝑗=1𝜀𝑗, (2.10) with 𝑖₁⋅⋅⋅𝑖_𝑘∈𝒲_𝑘, 𝜀(𝐶_𝑗)=0 and 𝜀(𝑗)=+1 or 𝜀(𝑗)=−1, accordingly 𝑓 restricted to 𝐼_𝑗 is increasing or decreasing.

The parity, with respect to the map 𝑓, of a given admissible word 𝑖₁,…,𝑖_𝑘∈𝒲_𝑘, is even if 𝜀(𝑖₁⋅⋅⋅𝑖_𝑘)=1 and odd if 𝜀(𝑖₁⋅⋅⋅𝑖_𝑘)=−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 (𝑖_𝑘)_𝑘≥1, (𝑗_𝑘)_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⋯𝑖_𝑟=𝑗₁⋯𝑗_𝑟 and 𝑖_𝑟+1≠𝑗_𝑟+1, we will say that (𝑖_𝑘)_𝑘≥1≺(𝑗_𝑘)_𝑘≥1 if and only if 𝑖_𝑟+1≺𝑗_𝑟+1 and 𝜀(𝑖₁⋯𝑖_𝑟)=1 or 𝑗_𝑟+1≺𝑖_𝑟+1 and 𝜀(𝑖₁⋯𝑖_𝑟)=−1.

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

Now, consider an m-modal map 𝑓 in the class 𝐶¹(𝐼,𝐼), for a certain interval 𝐼⊂ℝ, and the class of differentiable functions 𝒜=𝜑∈𝐶¹([0,1],𝐼)∶𝜑^′(0)=𝜑^′(1)=0,𝑐𝑝𝜑<∞, (2.11) where |𝑐𝑝(𝜑)| denotes the number of critical points of 𝜑.

Let 𝑇 be the operator 𝑇∶𝒜⟶𝒜,𝜑⟼𝑓∘𝜑. (2.12) 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(𝑇^𝑘𝜙)⊂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]): 𝒜_𝑐=(𝑥)[,𝒜𝜙∈𝒜∶𝜙isconstantin0,1]₀=],𝒜𝜙∈𝒜∶𝜙hasnocriticalpointsin0,1[_𝑗=𝜙∈𝒜∶𝜙has𝑗criticalpointsin]0,1[,𝑗=1,2,…. (2.13)

Let 𝜙∈𝒜 and let 𝜂(𝜙) be the number of nontrivial critical points of 𝜙 (inside [0,1]). In this case, if 𝜙∈𝒜_𝑗, 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∈ℕ0Σ^𝑗+1, where Σ^𝑗+1=Σ×Σ×⋅⋅⋅×Σ (𝑗+1 times), and we define the generalized address map for the space 𝒜 by ad∶𝒜⟶1,𝐶₁,2,…,𝑚+1^𝜂(𝜙)+2𝜙⟼ad𝜙∶=ad𝑑₀,ad𝑑₁,…,ad𝑑_𝜂(𝜙),ad𝑑_𝜂(𝜙)+1 (2.14) and the generalized itinerary map for the space 𝒜 by it∶𝒜⟶Σ𝜙⟼it𝜙=it𝑑₀,it𝑑₁,…,it𝑑_𝜂(𝜙),it𝑑_𝜂(𝜙)+1, (2.15) where 𝑑_𝑗∶=𝜙(𝑎_𝑗), 𝑗=0,1,…,𝜂(𝜙)+1, are the critical values of 𝜙 in the interval 𝐼 (with 𝑑₀=𝜙(0) and 𝑑_𝜂(𝜙)+1=𝜙(1)).

Let 𝑖^(𝑗)=𝑖₁^(𝑗)𝑖₂^(𝑗)⋯be such that 𝑖^(𝑗)=it(𝑑_𝑗), 𝑗=0,1,2,…,𝜂(𝜙)+1. The generalized shift map is then defined by 𝜎𝑖⁽⁰⁾,…,𝑖^(𝑗),𝑖^(𝑗+1),…,𝑖^{(𝜂(𝜙)+1)}𝜎𝑖∶=⁽⁰⁾,…,𝜎𝑖^(𝑗),𝜎𝑖^(𝑗+1),…,𝜎𝑖^{(𝜂(𝜙)+1)}if𝑖₁^(𝑗)=𝑖₁^(𝑗+1),𝜎𝑖⁽⁰⁾,…,𝜎𝑖^(𝑗),𝒦_𝑖1(𝑗),…,𝒦_{𝑖1(𝑗+1)−1},𝜎𝑖^(𝑗+1),…,𝜎𝑖^{(𝜂(𝜙)+1)}if𝑖₁^(𝑗)≠𝑖₁^(𝑗+1),𝑖₁^(𝑗)≺𝑖₁^(𝑗+1),𝜎𝑖⁽⁰⁾,…,𝜎𝑖^(𝑗),𝒦_{𝑖1(𝑗)−1},…,𝒦_{𝑖1(𝑗+1)},𝜎𝑖^(𝑗+1),…,𝜎𝑖^{(𝜂(𝜙)+1)}if𝑖₁^(𝑗)≠𝑖₁^(𝑗+1),𝑖₁^(𝑗+1)≺𝑖₁^(𝑗), (2.16) where 𝒦_𝑖1(𝑗) is the kneading sequence corresponding to the critical point of 𝑓, localized between the critical values 𝑑_𝑗 and 𝑑_𝑗+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. Let 𝜙, 𝜙∈𝒜 with 𝜙≠𝜙 so that 𝑖𝑡(𝜙)=𝑖𝑡(𝜙), then 𝑖𝑡𝑇^𝑘𝜙=𝑖𝑡𝑇^𝑘𝜙,𝑘∈ℕ₀. (2.17)
Moreover, 𝜎∘𝑖𝑡=𝑖𝑡∘𝑇. (2.18)

The previous definition and results allow us, knowing only the itineraries of the critical values of an initial condition 𝜙₀ and the kneading invariant of 𝑓, 𝒦_𝑓, to obtain explicitly the itineraries of the critical values of 𝜙_𝑘=𝑇^𝑘𝜙₀, 𝑘∈ℕ, see the next example.

Example 2.2. Let us consider a 4-modal map 𝑓 characterized by the kneading invariant 𝒦_𝑓=555𝐶₁^∞,15𝐶₂^∞,53𝐶₃^∞,124𝐶₄^∞, (2.19) (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 𝒦_𝑓. Using the Theorem 2.1, we obtain the temporal evolution by 𝑇 of 𝜙₀: (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)^∞⟶⋯ (2.20)

Figure 1 

Graph of the map 𝑓 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 of 𝑓 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 of 𝜙_𝑛). Let 𝐾₁, 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 Algorithm 3.1 in Mathematica 6.0 for a bimodal 𝑓 with periodic kneading sequences.

Example 3.2. Let us consider a bimodal map 𝑓∶𝐼→𝐼 characterized by the kneading invariant 𝒦_𝑓=((332𝐶₁)^∞,(112𝐶₂)^∞) and the function 𝜙₀∈𝒜 such that it(𝜙₀)=((312)^∞,(32)^∞). Considering 𝐾₁=332𝐶₁,𝐾₂=112𝐶₂,𝑋₀={312,32},𝑛=14, (3.1) 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(𝜙_𝑘), for 𝑘=0,1,2,3,…,7,…,14, given by ()↓123,23231,332𝐶₁↓,32312,112𝐶₂,32𝐶₁↓3,23123,112𝐶₂,332𝐶₁,12𝐶₂1,332𝐶₁,112𝐶₂,2𝐶₁33,112𝐶₂↓⋮↓,32123,32𝐶₁3,112𝐶₂,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,𝐶₂112,112𝐶₂,32𝐶₁3,112𝐶₂,332𝐶₁,12𝐶₂1,332𝐶₁,112𝐶₂,2𝐶₁33,112𝐶₂,332𝐶₁,12𝐶₂1,332𝐶₁,112𝐶₂,32𝐶₁3,112𝐶₂,332𝐶₁,2𝐶₂11,332𝐶₁,112𝐶₂,32𝐶₁↓⋮↓3,23231,332𝐶₁,112𝐶₂,2𝐶₁33,112𝐶₂,332𝐶₁,…,2𝐶₁3,112𝐶₂,332𝐶₁,12𝐶₂1,332𝐶₁,32. (3.2)

To apply the algorithm when the keading sequences of 𝑓 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 𝜙_𝑝=𝑇^𝑝𝜙₀, 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 of 𝑓 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 function 𝜙_𝑘. As previously, 𝜂(𝜙₀) denotes the number of the nontrivial critical points of 𝜙₀ and 𝑑_𝑗,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 map 𝑓 restricted to a subinterval 𝐼_𝑖 is monotone, 𝑓 is invertible if restricted to 𝐼_𝑖,𝑖=1,…,𝑚. Denote by 𝑓_𝑖⁻¹ the inverse map 𝑓⁻¹_|𝐼𝑖∶𝑓(𝐼_𝑖)→𝐼_𝑖. Let 𝑐_𝑖,𝑖=1,…,𝑚, be the critical points of the map 𝑓, and let 𝑈_𝑖⊂𝐼 be the maximal interval containing 𝑐_𝑖, consisting of points whose orbits tend to the stable periodic orbit of 𝑐_𝑖, see [13].

Let Λ_{𝑖1𝑖2⋯𝑖𝑝𝐶𝑖}⊂𝐼 be given by Λ_{𝑖1𝑖2⋯𝑖𝑝𝐶𝑖}∶=𝑓_𝑖⁻¹₁∘𝑓_𝑖⁻¹₂∘⋅⋅⋅∘𝑓_𝑖⁻¹_𝑝(𝑈_𝑖),𝑖=1,…,𝑚,  𝑖₁𝑖₂⋅⋅⋅𝑖_𝑝∈𝒲_𝑝, 𝑝∈ℕ. Therefore, the basin of attraction for 𝑓 is given by Λ=⋃^∞_𝑝=1⋃^𝑚_𝑖=1⋃_{𝑖1𝑖2⋅⋅⋅𝑖𝑝∈𝒲𝑝}Λ_{𝑖1𝑖2⋅⋅⋅𝑖𝑝𝐶𝑖}.

Proposition 3.3. Let 𝑓 be an m-modal map, 𝜙₀∈𝒜 and 𝜙_𝑘=𝑇^𝑘𝜙₀,𝑘∈ℕ₀. If the kneading sequences of 𝑓, 𝒦_𝑖, are aperiodic, then the itineraries of the critical values of 𝜙_𝑘, 𝑘∈ℕ₀, will belong to the set ^𝑚_𝑖=1𝜎^𝑞𝒦_𝑖,𝑞=0,1,2,…,𝑘−1∪^{𝜂(𝜙0)+1}_𝑗=0𝜎^𝑙it𝑑_𝑗,𝑙=0,1,2,…,𝑘−1, (3.3) where 𝑑_𝑗, 𝑗=0,1,2,…,𝜂(𝜙₀)+1, are the critical values of 𝜙₀. Moreover, 𝜐(𝜙_𝑘) will always grow and never stabilize.

Proof. Let 𝑐_𝑖,𝑖=1,2,…,𝑚, be the critical points of the map 𝑓. In [12], we proved that all critical points of 𝜙_𝑘=𝑇^𝑘𝜙₀,𝑘∈ℕ₀ arise in the following two ways:(i)a critical point of 𝜙_𝑘−1 will be a critical point of 𝜙_𝑘;(ii)every critical point of 𝜙_𝑘, which is not a critical point of 𝜙_𝑘−1, is a point 𝑦 in which 𝜙_𝑘−1(𝑦)=𝑐_𝑖 for some 𝑖=1,2,…,𝑚.
Therefore, from (i), the critical values of 𝜙_𝑘 correspond to the image under 𝑓 of the critical values of 𝜙_𝑘−1.
From (ii), the new critical values of 𝜙_𝑘 are always 𝑓(𝑐_𝑖),𝑖=1,2,…,𝑚. Then the critical values of 𝜙_𝑘 are either points in the set {𝑓^𝑘(𝑑_𝑗),𝑘∈ℕ} or points in the set ⋃^𝑚_𝑖=1{𝑓^𝑘(𝑐_𝑖),𝑘∈ℕ}. Recalling that it(𝑓^𝑘(𝑥))=𝜎^𝑘(it(𝑥)),𝑘∈ℕ₀, for all 𝑥∈𝐼, the itineraries of the critical values of 𝜙_𝑘 are in the set ^𝑚_𝑖=1𝜎^𝑞𝒦_𝑖,𝑞=0,1,2,…,𝑘−1∪^{𝜂(𝜙0)+1}_𝑗=0𝜎^𝑙it𝑑_𝑗,𝑙=0,1,2,…,𝑘−1. (3.4) Since we are considering aperiodic points 𝑐_𝑖, 𝑖=1,2,…,𝑚, the set ^𝑚_𝑖=1𝑓^𝑘𝑐_𝑖,𝑘∈ℕ₀ (3.5) is infinite.
For a fixed 𝑘∈ℕ₀, we have that the number of distinct critical values of a certain 𝜙_𝑘 is 𝜐(𝜙_𝑘)≤𝜐(𝜙_𝑘+1). We have that 𝜐(𝜙_𝑘) grows with 𝑘.

Let 𝑖₁𝑖₂⋯ be an aperiodic sequence in Σ and let 𝑁_{𝑖1𝑖2⋯}(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 𝒦_𝑖, 𝑖=1,2,…,𝑚.

Proposition 3.4. Let 𝑓 be an m -modal map, 𝜙₀∈𝒜   (with 𝜙₀∉𝒜_𝑐), and 𝜙_𝑘=𝑇^𝑘𝜙₀,𝑘∈ℕ₀. If the kneading sequences of 𝑓, 𝒦_𝑖, are aperiodic then for each 𝑘∈ℕ₀, we have 𝑁_{𝜎𝑞(𝒦𝑖)}𝜙_𝑘≤𝑁_{𝜎𝑝(𝒦𝑖)}𝜙_𝑘, (3.6) for 𝑞>𝑝 with 𝑝,  𝑞=0,1,2,…,𝑘−1,𝑖=1,2,…,𝑚.

Proof. As defined previously, 𝑁_{𝜎𝑞(𝒦𝑖)}(𝜙_𝑘) is the number of times, which 𝜎^𝑞(𝒦_𝑖) occurs as itinerary of a critical value of 𝜙_𝑘, for some fixed 𝑖∈{1,2,…,𝑚} and some fixed 𝑞∈{0,1,2,…,𝑘−1}.
If the itinerary of some critical values of 𝜙_𝑘 is 𝜎^𝑞(𝒦_𝑖), then the itinerary of these critical values is 𝒦_𝑖 for 𝜙_𝑘−𝑞, which correspond to the new critical points of 𝜙_𝑘−𝑞. These new critical points are derived of the (𝑘−𝑞−1)th-preimages of 𝑐_𝑖, that is,  𝜙_{𝑘−𝑞−1}(𝑦)=𝑐_𝑖.
Similarly, if the itinerary of some critical values of 𝜙_𝑘 is 𝜎^𝑝(𝒦_𝑖), then these critical values correspond to new critical points of 𝜙_𝑘−𝑝 that are obtained of the (𝑘−𝑝−1)th-preimages, under 𝑓 of 𝑐_𝑖, that is, 𝜙_{𝑘−𝑞−1}(𝑦)=𝑐_𝑖.
Since 𝑞>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 𝑓^𝑘−𝑞 is less than the number of laps of 𝑓^𝑘−𝑝. Therefore, 𝑁_{𝜎𝑞(𝒦𝑖)}𝜙_𝑘≤𝑁_{𝜎𝑝(𝒦𝑖)}𝜙_𝑘, (3.7) 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 map 𝑓 characterized by the kneading invariant 𝒦_𝑓=(332311321321…,113113323323…) (3.8) which analytical expression is given approximately by 𝑓(𝑥)=3.9333𝑥³−0.00665413𝑥²−2.9333𝑥+0.00665413 (3.9) and the function 𝜙₀∈𝒜 such that it𝜙₀=1^∞,3^∞,1^∞,3^∞. (3.10)
According to the Proposition 3.3, we have that the itineraries of the critical values of 𝜙₁₀=𝑇¹⁰𝜙₀ belong to the set 𝜎^𝑞(332311321321…),𝑞=0,1,…,9∪𝜎^𝑞()∪𝜎113113323323…,𝑞=0,1,…,9_𝑙1^∞,𝑙=0,1,…,9∪𝜎^𝑙3^∞={,𝑙=0,1,…,9332311321321…,32311321321…,2311321321…,311321321…,11321321…,1321321…,321321…,21321…,1321…,113113323323…,13113323323…,3113323323…,113323323…,13323323…,3323323…,323323…,23323…,3323…,323…,1^∞,3^∞. (3.11) Regarding that 𝒦₁=332311321321… and 𝒦₂=113113323323…, by the Proposition 3.4, we have 𝑁_𝜎9(𝒦1)𝜙₁₀≤𝑁_𝜎8(𝒦1)𝜙₁₀≤𝑁_𝜎7(𝒦1)𝜙₁₀≤⋅⋅⋅≤𝑁_𝜎(𝒦1)𝜙₁₀≤𝑁_𝒦1𝜙₁₀,𝑁_𝜎9(𝒦2)𝜙₁₀≤𝑁_𝜎8(𝒦2)𝜙₁₀≤𝑁_𝜎7(𝒦2)𝜙₁₀≤⋅⋅⋅≤𝑁_𝜎(𝒦2)𝜙₁₀≤𝑁_𝒦2𝜙₁₀. (3.12) Indeed, we have 𝑁_𝜎9(𝒦1)𝜙₁₀=4,𝑁_𝜎8(𝒦1)𝜙₁₀=12,𝑁_𝜎7(𝒦1)𝜙₁₀𝑁=36,_𝜎6(𝒦1)𝜙₁₀=100,𝑁_𝜎5(𝒦1)𝜙₁₀=268,𝑁_𝜎4(𝒦1)𝜙₁₀𝑁=716,_𝜎3(𝒦1)𝜙₁₀=1908,𝑁_𝜎2(𝒦1)𝜙₁₀=5060,𝑁_𝜎(𝒦1)𝜙₁₀𝑁=13412,_𝒦1𝜙₁₀𝑁=35548,_𝜎9(𝒦2)𝜙₁₀=4,𝑁_𝜎8(𝒦2)𝜙₁₀=12,𝑁_𝜎7(𝒦2)𝜙₁₀𝑁=36,_𝜎6(𝒦2)𝜙₁₀=92,𝑁_𝜎5(𝒦2)𝜙₁₀=244,𝑁_𝜎4(𝒦2)𝜙₁₀N=644,_𝜎3(K2)𝜙₁₀=1708,N_𝜎2(K2)𝜙₁₀=4524,N_𝜎(K2)𝜙₁₀𝑁=11996,_𝒦2𝜙₁₀=31780. (3.13)

Let us consider a bimodal map 𝑓 with aperiodic kneading sequences and growth number 𝑠(𝑓). Let 𝜙₀∈𝒜 be a function and 𝜙_𝑘=𝑇^𝑘𝜙₀, 𝑘∈ℕ₀. Recall that 𝑁_𝑖(𝑗)(𝜙) is the number of times that 𝑖^(𝑗), 𝑗=0,1,2,…,𝜐(𝜙_𝑘), occurs as itinerary of the critical values of 𝜙∈𝒜. We define the relative frequency of the critical values of 𝜙_𝑘 by 𝐻_𝑖(𝑗)𝜙_𝑘𝑁∶=^𝑖(𝑗)𝜙_𝑘𝜂𝜙_𝑘𝜙+2,𝑗=0,1,2,…,𝜐_𝑘, (3.14) where 𝑖^(𝑗) is the itinerary of the 𝑗-th critical value of 𝜙_𝑘. These values 𝐻_𝑖(𝑗) range between 0 and 1 and verify the following equality: 𝐻_𝑖(0)+𝐻_𝑖(1)+⋅⋅⋅+𝐻_{𝑖𝑘(𝜐(𝜙))}=1. (3.15)

We denote by 𝐻 the set of the values 𝐻_𝑖(𝑗)(𝜙_𝑘), 𝑗=0,1,2,…,𝜐(𝜙_𝑘), and we construct histograms using these values.

4. Numerical Results

Next, we present a numerical result that relates 𝐻_{𝜎𝑞(𝒦𝑖)}(𝜙_𝑘), the relative frequencies of the critical values of 𝜙_𝑘 given by 𝜎^𝑞(𝒦_𝑖), 𝑞=0,1,2,…,𝑘−1, for each 𝑖=1,2,…,𝑚.

Let Λ⊂𝐼 be the basin of attraction for 𝑓 and 𝜙₀∈𝒜 such that Im(𝜙₀)⊄Λ. If 𝒦_𝑖 is the kneading sequence of 𝑓, then 𝐻_{𝜎𝑞(𝒦𝑖)}𝜙_𝑘𝐻_{𝜎𝑞+𝑛(𝒦𝑖)}𝜙_𝑘⟶𝑠^𝑛𝑓,since𝑘⟶∞,𝑞=0,1,2,…,𝑘−1,𝑛=1,2,…,𝑘−2,𝑖=1,2,…,𝑚. (4.1)

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 𝜙_𝑘. There is a relationship between the growth rate of the critical points of 𝜙_𝑘 and the growth number of 𝑓, 𝑠(𝑓). Since the growth rate of the critical points of 𝜙_𝑘 is related with the growth of the preimages under 𝑓 of the critical points 𝑐_𝑖, 𝑖=1,2,…,𝑚, we have that the growth of the number of each distinct critical value, that results of the kneading sequence 𝒦_𝑖, is related with growth of the critical points correspondent to the preimages under 𝑓 of the critical points 𝑐_𝑖, for each 𝑖∈{1,2,…,𝑚}.

An estimation of the growth number of 𝑓, 𝑠(𝑓), is given by 𝑠𝑓≈𝜂𝜙_𝑘+2𝜂𝜙_𝑘−1+2,as𝑘⟶∞. (4.2) 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 map 𝑓∶𝐼→𝐼 characterized by the kneading invariant 𝒦_𝑓=(331233113212…,112223213311…), (4.3) whose analytical expression is given approximately by 𝑓(𝑥)=3.91322𝑥³+0.0232761𝑥²−2.91322𝑥−0.0232761. (4.4) Let 𝜙₀∈𝒜 be a function such that it𝜙₀=1^∞,3^∞, (4.5) whose analytical expression is given by 𝜙₀(𝑥)=−cos(𝜋𝑥).
An estimation of the growth number of 𝑓, 𝑠(𝑓), is given by 𝑠𝑓≈𝜂𝜙_𝑘+2𝜂𝜙_𝑘−1=[phone]=2.60719…,as𝑘⟶∞. (4.6)
The number 𝜂(𝜙_𝑘) 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 𝜎^𝑞(𝒦_𝑖), 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 𝒦₂,𝜎𝒦₂,𝜎²𝒦₂,𝜎³𝒦₂,…,𝜎¹¹𝒦₂,𝒦₁,𝜎𝒦₁,𝜎²𝒦₁,𝜎³𝒦₁,…,𝜎¹¹𝒦₁. (4.7)
The respective values 𝑁_{𝜎𝑞(𝒦𝑖)}(𝜙₁₂),𝑞=0,1,2,…,11,𝑖=1,2, are {}.53765,20621,7911,3035,1163,447,171,65,25,9,3,1,51477,19749,7573,2905,1113,427,163,63,23,9,3,1 (4.8) The relative frequencies of the critical values of 𝜙₁₂ whose itineraries are 𝜎^𝑞(𝒦_𝑖), 𝐻_{𝜎𝑞(𝒦𝑖)}𝜙₁₂=𝑁_{𝜎𝑞(𝒦𝑖)}𝜙₁₂𝜂𝜙₁₂,+2 (4.9) with 𝜂(𝜙₁₂)=170444, 𝑞=0,1,…,11, 𝑖=1,2, are given in the set {}.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… (4.10) 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 𝐻_{𝜎𝑞(𝒦1)}𝜙_𝑘≤𝐻_{𝜎𝑞(𝒦2)}𝜙_𝑘,as𝑘⟶∞,𝑞=0,1,…,11. (4.11)
In general, if 𝐻_𝒦1𝜙_𝑘≤𝐻_𝒦2𝜙_𝑘, (4.12) then 𝐻_{𝜎𝑞(𝒦1)}𝜙_𝑘≤𝐻_{𝜎𝑞(𝒦2)}𝜙_𝑘,as𝑘⟶∞. (4.13)
In this particular case, according to the values 𝐻_{𝜎𝑞(𝒦𝑖)}(𝜙₁₂), we have 𝐻_𝒦1≤𝐻_𝒦2≤𝐻_𝜎(𝒦1)≤𝐻_𝜎(𝒦2)≤⋅⋅⋅≤𝐻_{𝜎11(𝒦1)}≤𝐻_{𝜎11(𝒦2)}. (4.14)
Moreover, we have the following: 𝐻_𝒦1𝜙_𝑘𝐻_𝜎(𝒦1)𝜙_𝑘𝐻=2.60656…,^𝜎(𝒦1)𝜙_𝑘𝐻_𝜎2(𝒦1)𝜙_𝑘𝐻=2.60782…,^𝜎2(𝒦1)𝜙_𝑘𝐻_𝜎3(𝒦1)𝜙_𝑘𝐻=2.60688,…,_𝜎3(𝒦1)𝜙_𝑘𝐻_𝜎4(𝒦1)𝜙_𝑘𝐻=2.61006…,^𝜎4(𝒦1)𝜙_𝑘𝐻_𝜎5(𝒦1)𝜙_𝑘𝐻=2.60656…,^𝜎5(𝒦1)𝜙_𝑘𝐻_𝜎6(𝒦1)𝜙_𝑘𝐻=2.61963,…,_𝒦1𝜙_𝑘𝐻_𝜎2(𝒦1)𝜙_𝑘𝐻=6.7944…,^𝜎(𝒦1)𝜙_𝑘𝐻_𝜎3(𝒦1)𝜙_𝑘𝐻=6.79828…,^𝜎2(𝒦1)𝜙_𝑘𝐻_𝜎4(𝒦1)𝜙_𝑘𝐻=6.80413,…,_𝜎3(𝒦1)𝜙_𝑘𝐻_𝜎5(𝒦1)𝜙_𝑘𝐻=6.80328…,^𝜎4(𝒦1)𝜙_𝑘𝐻_𝜎6(𝒦1)𝜙_𝑘𝐻=6.82822…,^𝜎5(𝒦1)𝜙_𝑘𝐻_𝜎7(𝒦1)𝜙_𝑘𝐻=6.77778,…,_𝒦1𝜙_𝑘𝐻_𝜎3(𝒦1)𝜙_𝑘𝐻=17.7201…,^𝜎(𝒦1)𝜙_𝑘𝐻_𝜎4(𝒦1)𝜙_𝑘𝐻=17.7439…,^𝜎2(𝒦1)𝜙_𝑘𝐻_𝜎5(𝒦1)𝜙_𝑘𝐻=17.7354,…,_𝒦1𝜙_𝑘𝐻_𝜎4(𝒦1)𝜙_𝑘𝐻=46.2507…,^𝜎(𝒦1)𝜙_𝑘𝐻_𝜎5(𝒦1)𝜙_𝑘𝐻=46.2506…,^𝜎2(𝒦1)𝜙_𝑘𝐻_𝜎6(𝒦1)𝜙_𝑘𝐻=46.4601,…,_𝒦2𝜙_𝑘𝐻_𝜎(𝒦2)𝜙_𝑘𝐻=2.60729…,^𝜎(𝒦2)𝜙_𝑘𝐻_𝜎2(𝒦2)𝜙_𝑘𝐻=2.60662…,^𝜎2(𝒦2)𝜙_𝑘𝐻_𝜎3(𝒦2)𝜙_𝑘𝐻=2.60659,…,_𝜎3(𝒦2)𝜙_𝑘𝐻_𝜎4(𝒦2)𝜙_𝑘𝐻=2.60963…,^𝜎4(𝒦2)𝜙_𝑘𝐻_𝜎5(𝒦2)𝜙_𝑘𝐻=2.60179…,^𝜎5(𝒦2)𝜙_𝑘𝐻_𝜎6(𝒦2)𝜙_𝑘𝐻=2.61404,…,_𝒦2𝜙_𝑘𝐻_𝜎2(𝒦2)𝜙_𝑘𝐻=6.79623…,^𝜎(𝒦2)𝜙_𝑘𝐻_𝜎3(𝒦2)𝜙_𝑘𝐻=6.7944…,^𝜎2(𝒦2)𝜙_𝑘𝐻_𝜎4(𝒦2)𝜙_𝑘𝐻=6.80224,…,_𝜎3(𝒦2)𝜙_𝑘𝐻_𝜎5(𝒦2)𝜙_𝑘𝐻=6.78971…,^𝜎4(𝒦2)𝜙_𝑘𝐻_𝜎6(𝒦2)𝜙_𝑘𝐻=6.80117…,^𝜎5(𝒦2)𝜙_𝑘𝐻_𝜎7(𝒦2)𝜙_𝑘𝐻=6.87692,…,_𝒦2𝜙_𝑘𝐻_𝜎3(𝒦2)𝜙_𝑘𝐻=17.715…,^𝜎(𝒦2)𝜙_𝑘𝐻_𝜎4(𝒦2)𝜙_𝑘𝐻=17.7309…,^𝜎2(𝒦2)𝜙_𝑘𝐻_𝜎5(𝒦2)𝜙_𝑘𝐻=17.698,…,_𝒦2𝜙_𝑘𝐻_𝜎4(𝒦2)𝜙_𝑘𝐻=46.2296…,^𝜎(𝒦2)𝜙_𝑘𝐻_𝜎5(𝒦2)𝜙_𝑘𝐻=46.132…,^𝜎2(𝒦2)𝜙_𝑘𝐻_𝜎6(𝒦2)𝜙_𝑘=46.2632…, (4.15) 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 𝐻_{𝜎𝑞(𝒦𝑖)}𝜙_𝑘𝐻_{𝜎𝑞+𝑛(𝒦𝑖)}𝜙_𝑘⟶𝑠^𝑛𝑓,as𝑘⟶∞,𝑞=0,1,2,…,𝑘−1,𝑛=1,2,…,𝑘−2,𝑖=1,2. (4.16)

Figure 2 

Histogram of the relative frequencies of the distinct critical values of 𝜙12=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), ….

Acknowledgment

M. F. Correia was supported by Calouste Gulbenkian Foundation. This work has been partially supported by the research center CIMA-UE, FCT-Portugal funding program.