The Attractors of the Common Differential Operator Are Determined by Hyperbolic and Lacunary Functions

For analytic functions, we investigate the limit behavior of the sequence of their derivatives by means of Taylor series, the attractors are characterized by ω-limit sets. We describe four different classes of functions, with empty, finite, countable, and uncountable attractors. The paper reveals that Erdelyiés hyperbolic functions of higher order and lacunary functions play an important role for orderly or chaotic behavior. Examples are given for the sake of confirmation.


Historical Remarks
In 1952, MacLane 1 presented a strongly pioneering article, which studied sequences of derivatives for holomorphic functions and their limit behavior.He acted with sequences in a function space, generated by the common differential operator.When describing convergent and periodic behaviors, he found functions which Erdélyi et al. 2 have called hyperbolic functions of higher order.Besides he constructed a function whose limit behavior nowadays is called chaotic.
Lacunary functions, that is, L ücken-functions have been studied already by Hadamard 1892, he proved his Lacuna-theorem, see 3 .
Li and Yorke 4 introduced the idea of chaos in the theory of dynamical systems 1975; they described periodic and chaotic behaviors of orbits in finite-dimensional systems.In 1978, Marotto 5 introduced snap-back repellers, the so-called homocline orbits, to enrich dynamics by a sufficient criterion for chaos.In 1989, Devaney gave a topological characterization of chaos by introducing sensitivy, transitivy, and the notion dense periodical points.
Parallel to these, in operator theory, a lot of investigations concerning iterated linear operators appeared.In 1986, Beauzamy characterized hypercyclic operators by a property very

Introduction
We investigate the dynamical system generated by the common differential operator D which maps a function f to its first derivative f .Let its domain be the function space A of all functions which are analytic on the complex unit disc E : {x ∈ C : |x| ≤ 1}.An analytic function f means in complex analysis that the Taylor series of f exists and is absolutely convergent for all x ∈ E. Thus, all derivatives of f are contained in A too.They are continuous and differentiable.For f ∈ A and D : A → A, D f f , we consider the sequence of functions f 0 : f, f n 1 : D f n for n ∈ N 0 {0, 1, 2, 3, . ..}.

2.1
Hence, we have for f ∈ A, f x ∞ i 0 a i x i /i! , a i ∈ C, the relation holds.The sequence a i i∈N 0 of coefficients of the Taylor series ∞ i 0 a i x i /i! we call Taylor sequence.Equipped with the supremum norm f : max x∈E {|f x |} the set A, • is a normed linear space, and in the topology of this norm, 2.1 is a regular dynamical system with the linear operator D. For each function f ∈ A, there is an orbit f n n∈N 0 of this dynamical system 2.1 .
Due to results in 6, 10 , we conclude that the common differential operator D is chaotic in the sense of Devaney 13 , and from 11 in the sense of Li and Yorke 4 .

Hyperbolic functions
For the reason of self-containedness, we inform on hyperbolic functions.Exponential functions of the type are fixed points or fixed elements of the dynamical system 2.1 , whereas the so-called hyperbolic functions of order n are periodic elements of system 2.1 .
For the real part h n of H n , we know from 14 that h n : Re H n : R → R for real x , using the abbreviation α ν : 2π/n ν, The Taylor series 3.3 and 3.5 reveal that hyperbolic functions coincide with their nth derivative, that is,

H n n H n , h
n n h n .

3.8
We should note that we consider the function H n for complex x, and h n for real x.It is easy to prove.: B The sequence p n n∈N 0 is a periodic orbit of the dynamical system 2.1 .
Hence, the orbit of p in 3.9 move in circles planet-like in the function space A.

Preliminaries
Let X, • be a normed linear space and x n n∈N 0 a sequence in X.
Definition 4.1 Li/Yorke-property .One calls x n n∈N 0 an aperiodic sequence or a chaotic orbit if it is bounded but not asymptotically periodic, that is, for each periodic sequence y n n∈N 0 ⊂ X,one has Hence, an aperiodic sequence has at least two cluster points.
According to Alligood et al. 12 , one defines attractors of an orbit by the Omega-limit set ω f of an element f ∈ A. It contains all cluster points of the orbit f n n∈N 0 .Thus, for the functions 3.1 and 3.For α 0, this definition coincides with the classical definition of lacunary functions used by Hadamard, Polya, and so on.Thus, an analytic function is lacunary function, if its Taylor sequence has the lacuna cluster 0. Hence, the flutter function Φ : E → C, introduced in 11 , x 4 4!
is lacunary function, its Taylor sequence has the lacuna cluster 0. Lacunary functions have been already discussed by Weierstraß and Hadamard; Polya 1939 proved that functions of this type possess no extension to any point on their periphery, see 3 .In recent time, lacunary functions with unbounded Taylor sequence play a role in complex analysis again.
Next, we introduce for each sequence a i i∈N 0 its cluster sequence a i i∈N 0 by identifying elements of convergent subsequences by their limit point.Note that a i i∈N 0 is bounded, thus cluster points exist Bolzano-Weierstraß .Definition 4.4.Let {α 0 , α 1 , α 2 , . ..} be the set of cluster points of the sequence a i i∈N 0 .One constructs inductively a mapping a i → a i ∈ {α 0 , α 1 , α 2 , . ..}.
1 Due to α 0 is cluster point, there is a subsequence a i i∈I 0 , I 0 ⊂ N 0 , converging to α 0 .
For i ∈ I 0 , we define a i : α 0 .
If N 0 \ k j 0 I j is a finite set, one defines a i : α k , otherwise there is a cluster point α k 1 different from α j for j 0, 1, 2, . . ., k, and a subset Note that the set of indices {I k : k 0, 1, 2, . ..} are pairwise disjoint and their union is N 0 .
The cluster sequence a i i∈N 0 reveals the asymptotical behavior of an orbit f n n∈N 0 , Property C will be very useful.

Finite attractors
Like the oracle of Delphi in ancient Greece informed people about their future, our theorems will show that the Taylor sequence a i i∈N 0 predicts the asymptotical behavior of an orbit International Journal of Mathematics and Mathematical Sciences The following theorem deals with empty and finite attractors, it reveals the role of Erdelyi's hyperbolic functions H n for the attractors ω f of the differential operator.
A If a i i∈N 0 is unbounded and contains no lacuna cluster, then f n n∈N 0 is unbounded too and ω f is empty.

5.1
We give some examples as follows.
1 The function tan 1/cos ∈ A possess for |x| < π/2 with the Bernoulli numbers B ν and the Euler numbers E ν the Taylor series

Countable attractors
We now consider chaotic orbits of the differential operator.The next theorem shows that these are characterized by aperiodic Taylor sequences.

Then the statements (A) and (B) are equivalent as follows.
A The Taylor sequence a i i∈N 0 is aperiodic.
B The sequence of derivatives f n n∈N 0 is a chaotic orbit of the system 2.1 .
Figure 1 presents the sequence Φ n n∈N 0 of the flutter function Φ, see 4.5 , graphically.Imagine a chicken that wants to escape the kitchen.It flutters up to a window one meter high, it bumps against the window and crashes down to the bottom.Then it starts the same procedure again, but it has lost energy, so it needs a longer way to flutter up again.There is no periodicity, the time difference between "downs" and "ups" increases.This fluttering upward and crashing down may be seen in Figure 1.
The next theorems reveal the part of lacunary functions and exponential functions e α for chaotic orbits of the differential operator and its attractor.
A if a i i∈N 0 possesses only a finite number of cluster points, then the cluster sequence a i i∈N 0 contains at least one lacuna cluster.
B If α ∈ C is a lacuna cluster of the cluster sequence a i i∈N 0 , then for the exponential function e α, one has e α ∈ ω f .
We introduce abbreviations splitting the exponential function e 1 into a Taylor polynomial T n and its remainder R n : Thus, e 1 T n R n for each n ∈ N. We will use it for constructing a stairway βT n γR n between exponential functions e β and e γ in the function space A.
x ∈ E and the Taylor sequence a i i∈N 0 be aperiodic.Then A for each lacuna cluster β in the cluster sequence a i i∈N 0 , infinitely often followed by a lacuna cluster γ / β, one has with U n : βT n γR n , the cluster sequence a i i∈N 0 , which appears infinitely often between the lacuna clusters β and γ, one has with S n : C If the cluster sequence a i i∈N 0 contains arbitrary many lacuna clusters but only a finite number of nonlacuna clusters, then the attractor ω f is a countably infinite set.

Example for statement (A)
To define the function Λ ∈ A, we choose a i a i according to the rule 2, otherwise.

6.5
We see three lacuna clusters 0, 1, 2. The attractor of Λ becomes Figure 2 presents the sequence Λ n n∈N 0 graphically, Figure 3 shows schematically the orbit Λ n n∈N 0 and its attractor ω Λ in the function space A. In both figures, we see the stairways up from e 0 to e 1 and from e 1 to e 2 , and the stairway down from e 2 to e 0 .

Example for statement (B)
Is given by the flutter function Φ defined in 4.5 , with k 1, b 0 1, β γ 0. It leads to the attractor of Φ: Figure 1 shows the stairway {q n : n ∈ N 0 }.

Example for statement (C)
Is given by the sequence 1/n n∈N for the construction of a Taylor sequence with infinitely many lacuna clusters: International Journal of Mathematics and Mathematical Sciences , 1, 1, 1, 1, 1, . . . .

6.8
Mathematical research on lacunary functions deals usually with unbounded coefficients.In addition to Theorem 5.1 A , we give an example of a lacunary function with unbounded Taylor sequence, whose ω-limit set is nonempty.For f : E → C, given by we show e 0 ∈ ω f .The k 2 th derivative of f is Because 0 ∈ E, we have f k 2 ≥ 2k − 3 !, which means that the orbit is unbounded.We consider its successor f k 2 1 and f k 2 1 :

Uncountable attractors
Finally, we demonstrate that not only lacunary functions may have chaotic orbits.We use the Cantor sequence  to define an analytic function C ∈ A. It has countably infinitely many cluster points, but no lacuna cluster: x 4 4!
x 5 5! 1 4 x 6 6! 1 2 With the abbreviation the elements of the Cantor sequence c i i∈N 0 maybe given by In Figure 4, we see the graph of C for real values.Figure 5 shows the sequence C n n∈N of the orbit of C. It is bounded from below by 0 and from above by the Euler number e.It increases apparently linear in some subintervals, followed by a descent at the values Figure 6 shows ω C and a subset of the orbit schematically.Like a squirrel runs up a tree, the orbit runs up along the stick {e α : α ∈ 0; 1 }.After that the orbit jumps to a Taylor polynomial T m the squirrel jumps to a branch , and to another one, lower one T m−1 , and then to T m−2 , . . ., T 0 .Then it starts again to run upward along the stick, with one step more than before, at each circulation it reaches a higher level.It climbs up nearer and nearer to the top of the stick e 1 .
We describe the properties of the orbit of C in a theorem, using Taylor polynomials T n , remainders R n 6.1 , exponential functions 3.1 , and s n 7.3 .
Properties A , B , and D can be seen in Figure 5, property E in Figure 6.

Proof of Theorem 5.1
Property (A) because 0 ∈ E. Thus, |a n | n∈N 0 is an unbounded minorizing sequence.

Property (B)
It follows from C with n 1 and β 0 α.

Property (C)
By assumption the cluster sequence a i i∈N 0 of the Taylor sequence becomes a i β i mod n .Consider p ∈ A defined by

8.2
Proposition 3.1 implies that the orbit p n n∈N 0 is a periodic orbit.Using 4.6 , the orbit f n n∈N 0 is asymptotically periodic to its attractor ω f p, p 1 , p 2 , . . ., p n−1 .

Proof of Theorem 6.1
Using the Li/Yorke-property 4.1 .A ⇒ B Let b i i∈N 0 ⊂ C be a periodic sequence.Then Theorem 5.1 C implies that the sequence p n n∈N , defined by p x

Statement (A)
It can be proved directly from elementary combinatorics.

Statement (B)
1 For α 0, let I ⊂ N 0 be the set of indices, where a block of zeros in the cluster sequence a i i∈N 0 starts.Then for n ∈ I, Using 2.2 and 4.6 , we have for n ∈ I, The Taylor sequence is bounded by a ∈ R .Thus, 8.5 guarantees that it is the case for its cluster sequence too.The latter estimate Hence, f n → e 0 for n → ∞ and e 0 ∈ ω f ω f . 2 For α / 0, the function g : f − e α is lacunary function.Thus, its Taylor sequence has the lacuna cluster 0. Using 4.3 and the case α 0 above imply e 0 ∈ ω g ω f − e α −e α ω f ⇐⇒ e α e 0 e α ∈ ω f .8.8

Statement (A)
Let m ∈ N.For U m βT m γR m , we construct a subsequence f n n∈I , I ⊂ N 0 , of the orbit f n n∈N 0 , converging to U m .

8.11
As assumed, the sequence a i i∈N 0 is bounded, thus |a i − γ| i∈N 0 is bounded by a real number c ∈ R , thus,

Statement (B)
Let m ∈ N.For S m , we construct a subsequence f n n∈I , I ⊂ N 0 , of the orbit f n n∈N 0 , converging to S m .At index n starts a row of β's, at index n m the tupel, and at n m k a row of γ's, which has its end at index n m k M n − 1.We define the set I by

8.13
where I contains infinitely many elements.Because γ is a lacuna cluster, we have 8.14

International Journal of Mathematics and Mathematical Sciences
For n ∈ I, we consider the nth derivative f n , using 2.2 and the abbreviation M : M n ,

8.15
To S m , it has the distance

8.16
The sequence |a i − γ| i∈N is bounded by c ∈ R , thus the latter term 8.17 Because of 8.14 , the sum converges to 0 for n → ∞.Thus,

Statement (C)
If there is only one lacuna cluster in the cluster sequence, then statement B implies with β γ that the attractor is countably infinite.If there are infinitely many lacuna clusters α 1 , α 2 , α 3 , . . . in the cluster sequence, then statement A implies for each couple α j , α j 1 countably infinitely many elements of the attractor.Using countable × countable = countable, we conclude C .

Statement (B)
Let n, m ∈ N 0 , m < n.For T m , see 6.1 and 7.5 , using 2.2 and 8.19 , we have c s n −m−1 i x i i! .

8.21
This leads to

Statement (C)
Statement A implies the statement is valid for α 0. From statement B , we conclude T m ∈ ω C for each m ∈ N 0 .Because of lim m → ∞ T m e 1 and Theorem 5.1, we find e 1 ∈ ω C .Thus, the statement is true for α 1.
Let 0 < α < 1 and ε > 0. Due to the fact that Q is dense in R, we find a rational number p/q ∈ Q, p < q, and |p/q − α| < ε/3e.

8.26
Thus, we have proved statement C .Statements D and E follow by using A , B , and C .

Figure 4 :
Figure 4: Graph of the Cantor function C for real arguments.

Figure 5 :
Figure 5: Sequence C n n∈N of the Cantor function C.

Proposition 4.2. The ω-operator is linear in the following sense
. Let f ∈ A and e α defined in 3.1 .Let the Taylor sequence a i i∈N 0 of f ∈ A have the cluster point α ∈ C, and let I ⊂ N 0 be the index set defined by I : {n i : a n i / α}.Then α is called lacuna cluster of a i i∈N 0 , if the sequence n i 1 − n i n i ∈I is unbounded.

3
International Journal of Mathematics and Mathematical Sciences because 0 ∈ E. For infinitely many n, we have |a n − b n | > δ/2.Thus, International Journal of Mathematics and Mathematical SciencesFor ε > 0, we choose k ∈ N such that We define m : kp − 1 and n : kq − 1.This leads to n − m k q − p ≥ k and