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

We consider the dynamical system A, T , whereA is a class of differentiable functions defined on some interval and T :A → A is the operator Tφ : f ◦φ, where f is a differentiablem-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.


Introduction and Preliminaries
One-dimensional maps of the interval have had an important role for the understanding of many properties and phenomena found in nonlinear dynamics and complex systems.Part of its success is due to symbolic dynamics, which allowed the simplification of the dynamical description to its essential features, usually concerning topological dynamics.Some subsets of the state space are identified, represented symbolically and treated as equivalent.Then, instead of orbits, we deal with sequences of symbols.Important results on classification and computation of invariants, such topological entropy, was obtained for one-dimensional iterated maps, using symbolic dynamics.Many efforts have been made to extend the techniques and results obtained for iterated maps of attributes, as referred above, of a function φ ∈ A change under iteration of T in a combinatorial and topological perspective.In the particular case of f being a m-modal map the combinatorial description is particularly interesting.
Now, in what follows 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 map if is in C 1 (I) and has m critical points.
Let c i , with i = 1, 2, . . ., m, be the m critical points of the m-modal map f such that c 0 < c 1 < • • • < c m < c m+1 , where c 0 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 where I C j is the set {c j } , j = 1, 2, . . ., m, and I j , j = 1, 2, . . ., m + 1, are given by Each interval I j , j = 1, 2, . . ., m + 1, is a maximal interval of monotonicity of f .Next, to each point x in I j , j = 1, 2, . . ., m + 1, we assign the symbol j, j = 1, 2, . . ., m + 1, or C j , j = 1, 2, . . ., m, if the point x is the critical point c j , j = 1, 2, . . ., m, of f .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 As usual, we get a correspondence between orbits of points and symbolic sequences of the alphabet {1, C 1 , 2, . . ., m + 1}, the itinerary of x under f is defined by The orbits, under f , of the critical points are of special importance, in particular, their itineraries.Following Milnor and Thurston in [3], for each critical point, the kneading sequence is given by K i := it (f (c i )) , i = 1, 2, . . .m, and the collection of symbolic sequences , . . ., m + 1} N 0 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, C 1 , 2, . . ., m + 1} N 0 is denoted by Σ.
In the sequence space Σ, we define the usual shift map σ : Σ → Σ by Moreover, the following relation with f and the itinerary map is satisfied 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 We define the cylinder set In other words, x ∈ I i 0 i 1 ...i k means that ad (x) = i 0 , ad (f (x)) = i 1 , . . ., ad f k (x) = i k .
Consider the sign function ε : The parity, with respect to the map f , of a given admissible word i 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, C 1 , 2, . . ., m + 1} N 0 , admitting that they have a common initial subsequence, i.e., there is a natural r ≥ 0 such that i 1 . . .i r = j 1 . . .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(i 1 . . .i r ) = 1 or j r+1 ≺ i r+1 and ε(i 1 . . .i r ) = −1.

Iteration on a Class of Differentiable Functions
Now, consider a m-modal map f in the class C 1 (I), for a certain interval I, and the class of differentiable functions Let T be the operator Note that this operator is well defined since (f 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 gives the discrete time evolution.Let us consider the following results about the localization of the critical values and critical points of the iterates of φ 0 in A under the map f .Proposition 1.Let f be a m-modal map and let φ ∈ A and J ⊂ [0, 1] be an interval and x ∈ J.If φ |J (x) is a maximal (resp.minimal) value in Proof.First, by the definition of ..i k for any admissible word i 0 . . .i k .Next, since f |I i 0 is decreasing it reverses the order, therefore maximal (resp.minimal) points for φ |J in some J, subinterval of [0, 1], correspond to minimal (resp.maximal) points for f • φ |J .The same reasoning, noting that f |I j 0 is increasing and preserves the order, leads to the claimed result.
Let cp(φ) denote the set of the critical points of φ ∈ A, cv(φ) denote the set of the critical values of φ ∈ A and s(φ) denote the set of points A consequence of this result is the following: ).This last result means that the maxima and the minima of the iterates φ k = f k (φ 0 ) of some initial function φ 0 ∈ A arise from the orbits of the maxima and minima of φ 0 and also from the appearance of points where φ 0 , φ 1 , . . ., φ k , attained the critical points of f .The preimages, under f , of the critical points c j , j = 1, 2, . . ., m, that is, points y ∈ I such that f k (y) = c j , for some natural k and some j = 1, 2, . . ., m, can be labeled by admissible words in W k .For example, the admissible word i 1 i 2 . . .i k together with the symbol C j (for a fixed j = 1, . . ., m), i 1 i 2 . . .i k C j , represents the k-preimage of c j , that is, ad (x) = i 1 , ad (f (x)) = i 2 , . . ., ad f k−1 (x) = i k and f k (x) = c j .An admissible word gives the prefix of the itinerary of a preimage y.If we consider the graphics of the constant functions equal to the preimages of the critical points of f we obtain a grid with each line is labeled by an admissible word.If one of such line, labeled by a certain admissible word i 1 i 2 . . .i k C j , intersects φ 0 in a point x ∈ [0, 1], then T k (φ 0 ) will have a new critical point in x with critical value localized in the cylinder I i 1 i 2 ...i k .Note that x is not necessarily a critical point for T k−1 (φ 0 ).In order to illustrate the results given above consider the Example 1.

Example 1. Let us consider the bimodal map
given by f (x) = 4x 3 − 3x, with the critical points c 1 = −1/2 and c 2 = 1/2.The partition of the interval I is given by

Symbolic Dynamics for the Infinite Dynamical System (A, T )
In order to introduce a symbolic dynamics description for the discrete dynamical system (A, T ), let us consider the decomposition of A into the following classes, as in [2]: Let φ ∈ A and let η (φ) be the number of non-trivial critical points of φ (inside [0, 1]).In this case if φ ∈ A j , j ∈ N 0 , 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 .Moreover, we are interested that this symbolic description has essentially a topological meaning, therefore the important point is to distinguish and codify the critical points and values of φ.Given φ ∈ A, we identify its critical points and collect the addresses and itineraries of the corresponding critical values.Our generalized symbolic space will be Σ := ∪ j∈N 0 Σ j+1 , where Σ j+1 = Σ × Σ × • • • × Σ (j + 1 times).We now define the generalized address, itinerary and shift maps for the space A, where d i = φ(a i ), i = 0, 1, . . .η (φ) + 1, are the critical values of φ in the interval [−1, 1] (with d 0 = φ(0) and d η(φ)+1 = φ(1)).As an example, consider f given approximately by the analytical expression f (x) = −2.15x 3 +1.15xand φ given approximately by the analytical expression φ (x) = 0.261 cos (πx) + 0.711 (see , for some j ∈ {0, 1, . . ., η (φ)} and by (1) we obtain and finally, in the last case, if i 1 , for some j ∈ {0, 1, . . ., η (φ)} and by (1) we obtain For k = 0, we obtain it(φ) = it( φ) (by assumption).We assume the result is true for k = n, i.e., assume it( which is the desired result for k = n + 1.
Next, we give the following examples to illustrate the previous definitions and results.
respectively, see the Figure 6.
We can verify that In the Figure 7, we show the graphs of φ, T φ, T 2 φ, T 3 φ, T 4 φ and T 5 φ and we compute their itineraries by the equality It is visually clear the accumulation of new critical points, in the last picture.We can make our symbolic analysis without an explicit function φ ∈ A, especially if the itineraries involved are periodic sequences, as we can see in the next example.
(which is in fact the only necessary information).Now, consider an address ((121) ∞ , (21) ∞ ).Both sequences are admissible, therefore, this address represents a class of functions belonging to A which have two critical points necessarily 0 and 1 (the boundary points) and the corresponding critical values have itineraries given by it(φ(0)) = (121) ∞ and it(φ(1)) = (21) ∞ (periodic points with respect to f ).Using the Theorem 1 we know that the time evolution under T of any function φ ∈ A which has address ((121) ∞ , (21) ∞ ), is given by and the first claim follows.The critical values of f • φ are naturally the images under f • φ of the critical points of f • φ.Moreover, f (cv(φ)) = f • φ(cp(φ)) and the result follows.
Figures 1, 2 and 3 illustrate the Propositions 1 and 2. In the Figure 4. a), each horizontal line is a k-preimage under f of each critical point of f , c 1 and c 2 , in particular, the line indicated is the first preimage of c 2 belonging to I 2 .Thus, for any initial function which intersect the indicated line in a point, we get a new critical point at the second iterate and not before that.In the Figure 4. b), we consider φ 0 (x) = a 0 + a 1 cos(2πx), with a 0 = −0.2 and a 1 = 0.06.Both φ 0 and φ 1 = T (φ 0 ) have one critical point inside [0, 1].However the second iterate φ 2 = T 2 (φ 0 ) = T (T (φ 0 )) has two new critical points precisely in the points where φ 0 intersects the line labeled by 2C 2 .

Figure 1 :
Figure 1: a) Graph of the restriction of a function φ, in two subintervals J 1 , J 2 , with a maximum (resp.minimum) in the region I 31 .b) Graph of the restriction of f • φ, in the same subintervals J 1 , J 2 , which has a maximum (resp.minimum) in the region I 1 .

Figure 2 :
Figure 2: a) Graph of the restriction of a function φ, two subintervals J 1 , J 2 , with a maximum (resp.minimum) in the region I 22 (resp.I 23 ).b) Graph of the restriction of f • φ, in the same subintervals J 1 , J 2 , which has a minimum (resp.maximum) in the region I 2 (resp.I 3 ).

Figure 3 :
Figure 3: a) Graph of the restriction of a function φ, in subintervals J 1 , J 2 , which intersect the horizontal line corresponding to a critical point of f at some points x 1 , x ′ 1 ∈ J 1 , x 2 , x ′ 2 ∈ J 2 and have a maximum (resp.minimum) in the region I 31 (resp.I 21 ).b) Graph of the restriction of the function f • φ, in the same subintervals J 1 , J 2 , with new minimal values at x 1 , x ′ 1 ∈ J 1 , x 2 , x ′ 2 ∈ J 2 and a maximum in the region I 1 .Note that in b) the vertical scale is changed.
Now, we consider, in A, the function given approximately by the analytical expression φ(x) = 0.05 cos(2πx) − 0.92.

Example 4 .
Consider a bimodal map f with kneading invariant