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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 251298, 19 pages
Research Article

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

Hans und Hilde Coppi Gymnasium, Römerweg 32, 10318 Berlin, Germany

Received 27 July 2008; Accepted 2 November 2008

Academic Editor: Marco Squassina

Copyright © 2008 Wolf Bayer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For analytic functions, we investigate the limit behavior of the sequence of their derivatives by means of Taylor series, the attractors are characterized by -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.

1. 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, Lücken-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 near to the definition of homocline orbits. In 1991, Godefroy and Shapiro [6] connected these two lines. Based on results of Rolewicz [7], they proved that common integral- and differential operators are hypercyclic. A widespread research activity followed. A quite good survey on the theory of hypercyclic operators has been given in 1999 by Grosse-Erdmann [8] and too in the conference report of the Congress of Mathematics in Zaragoza 2007, see [9].

In 1999, respectively, 2000 the author of the present article verified the chaos properties of Devaney and of Li and Yorke for the common differential operator, see [10, 11].

This paper continues the article of MacLane [1]. It gives more insight into the limit behavior of sequences of derivatives characterizing them by convergence properties of their Taylor coefficients. It gives predictions for their attractors, describing these by means of the concept Omega-limit sets, see Alligood et al. [12].

For our investigations, we choose the supremum norm, although in the topology of that norm the differential operator is discontinuous. By this, we can prove convergence properties very easily. Moreover, we focus our attention only on the cardinality of the Omega-limit sets.

2. Introduction

We investigate the dynamical system generated by the common differential operator which maps a function to its first derivative . Let its domain be the function space of all functions which are analytic on the complex unit disc . An analytic function means in complex analysis that the Taylor series of exists and is absolutely convergent for all . Thus, all derivatives of are contained in too. They are continuous and differentiable. For and , we consider the sequence of functionsHence, we have for the relationholds. The sequence of coefficients of the Taylor series we call Taylor sequence. Equipped with the supremum norm the set is a normed linear space, and in the topology of this norm, (2.1) is a regular dynamical system with the linear operator . For each function there is an orbit of this dynamical system (2.1).

Due to results in [6, 10], we conclude that the common differential operator is chaotic in the sense of Devaney [13], and from [11] in the sense of Li and Yorke [4].

3. Hyperbolic Functions

For the reason of self-containedness, we inform on hyperbolic functions. Exponential functions of the typeare fixed points or fixed elements of the dynamical system (2.1), whereas the so-called hyperbolic functions of order are periodic elements of system (2.1).

With Erdélyi et al. defined in [2] For the real part of , we know from [14] that (for real ), using the abbreviation ,The Taylor series (3.3) and (3.5) reveal that hyperbolic functions coincide with their th derivative, that is,With (3.1), (3.2), and (3.4), we find that and, see [14], We should note that we consider the function for complex , and for real . It is easy to prove.

Proposition 3.1. The statements (A) and (B) are equivalent.
(A) is a linear combination of and :(B)The sequence is a periodic orbit of the dynamical system (2.1).

Hence, the orbit of in (3.9) move in circles planet-like in the function space .

4. Preliminaries

Let be a normed linear space and a sequence in .

Definition 4.1 (Li/Yorke-property). One calls an aperiodic sequence or achaotic orbit if it is bounded but not asymptotically periodic, that is, for each periodic sequence 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 of an element . It contains all cluster points of the orbit . Thus, for the functions (3.1) and (3.7),

Proposition 4.2. The -operator is linear in the following sense. Let and defined in (3.1). Then

For Taylor sequences of type one introduces the concept of lacuna cluster.

Definition 4.3. Let the Taylor sequence of have the cluster point , and let be the index set defined by Then is called lacuna cluster of if the sequence is unbounded.

For 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 . Hence, the flutter function , introduced in [11], 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 its cluster sequence by identifying elements of convergent subsequences by their limit point. Note that is bounded, thus cluster points exist (Bolzano-Weierstraß).

Definition 4.4. Let be the set of cluster points of the sequence . One constructs inductively a mapping .
(1)Due to is cluster point, there is a subsequence converging to . For we define (2)If is a finite set, one defines for . Otherwise there is a cluster point and a subset converging to . For one defines (3)Continuing inductively for If is a finite set, one defines otherwise there is a cluster point different from for and a subset converging to . For one defines

Note that the set of indices are pairwise disjoint and their union is .

The cluster sequence reveals the asymptotical behavior of an orbit , Property (C) will be very useful.

Proposition 4.5. For and one has

5. Finite Attractors

Like the oracle of Delphi in ancient Greece informed people about their future, our theorems will show that the Taylor sequence predicts the asymptotical behavior of an orbit for . The following theorem deals with empty and finite attractors, it reveals the role of Erdelyi's hyperbolic functions for the attractors of the differential operator.

Theorem 5.1. Let for .
(A) If is unbounded and contains no lacuna cluster, then is unbounded too and is empty.(B)If is convergent to , then converges to and (C)If is asymptotically periodic to , then is asymptotically periodic too, and

We give some examples as follows.

(1) The function possess for with the Bernoulli numbers and the Euler numbers the Taylor seriesIts Taylor sequence is unbounded without any cluster point, hence .

(2) For each polynomial we have .

(3) Let defined by Then Its Taylor sequence is asymptotically periodic to the periodic sequence Using (5.1), (3.7), and , its attractor becomes .

6. Countable Attractors

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

Theorem 6.1. Let for . Then the statements (A) and (B) are equivalent as follows.
(A)The Taylor sequence is aperiodic.(B)The sequence of derivatives is a chaotic orbit of the system (2.1).

Figure 1 presents the sequence of the flutter function , 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.

Figure 1: The sequence .

The next theorems reveal the part of lacunary functions and exponential functions for chaotic orbits of the differential operator and its attractor.

Theorem 6.2. Let for and aperiodic. Then
(A)if possesses only a finite number of cluster points, then the cluster sequence contains at least one lacuna cluster.(B)If is a lacuna cluster of the cluster sequence , then for the exponential function one has

We introduce abbreviations splitting the exponential function into a Taylor polynomial and its remainder :Thus, for each . We will use it for constructing a stairway between exponential functions and in the function space .

Theorem 6.3. Let for and the Taylor sequence be aperiodic. Then
(A)for each lacuna cluster in the cluster sequence infinitely often followed by a lacuna cluster , one has with ,(B)For each tupel in the cluster sequence , which appears infinitely often between the lacuna clusters and , one has with (C)If the cluster sequence contains arbitrary many lacuna clusters but only a finite number of nonlacuna clusters, then the attractor is a countably infinite set.

Example for Statement (A)
To define the function we choose according to the rule Then We see three lacuna clusters . The attractor of becomesFigure 2 presents the sequence graphically, Figure 3 shows schematically the orbit and its attractor in the function space . In both figures, we see the stairways up from to and from to , and the stairway down from to .

Figure 2: The sequence of the lacunary function .
Figure 3: Orbit and -limit set of the lacunary function .

Example for Statement (B)
Is given by the flutter function defined in (4.5), with It leads to the attractor of : Figure 1 shows the stairway .

Example for Statement (C)
Is given by the sequence for the construction of a Taylor sequence with infinitely many lacuna clusters:

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 -limit set is nonempty. For , given bywe show . The th derivative of isBecause we have which means that the orbit is unbounded.

We consider its successor and :Hence, with . We conclude the exponential function .

7. Uncountable Attractors

Finally, we demonstrate that not only lacunary functions may have chaotic orbits. We use the Cantor sequenceto define an analytic function . It has countably infinitely many cluster points, but no lacuna cluster:With the abbreviationthe elements of the Cantor sequence maybe given byIn Figure 4, we see the graph of for real values. Figure 5 shows the sequence of the orbit of . It is bounded from below by and from above by the Euler number . It increases apparently linear in some subintervals, followed by a descent at the valuesFigure 6 shows and a subset of the orbit schematically. Like a squirrel runs up a tree, the orbit runs up along the stick After that the orbit jumps to a Taylor polynomial (the squirrel jumps to a branch), and to another one, lower one , and then to . 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 .

Figure 4: Graph of the Cantor function for real arguments.
Figure 5: Sequence of the Cantor function .
Figure 6: Orbit and -limit set of the Cantor function .

We describe the properties of the orbit of in a theorem, using Taylor polynomials , remainders (6.1), exponential functions (3.1), and (7.3).

Theorem 7.1. For the orbit of the Cantor function one has
(A) for ; (B) for with for ; (C)for each , there is a subsequence , of the orbit with for ; (D)(E), containing uncountably many different elements.

Properties (A), (B), and (D) can be seen in Figure 5, property (E) in Figure 6.

8. Proofs

8.1. Proof of Theorem 5.1

Property (A)
Using (2.2),because . Thus, is an unbounded minorizing sequence.

Property (B)
It follows from (C) with and .

Property (C)
By assumption the cluster sequence of the Taylor sequence becomes Consider defined byProposition 3.1 implies that the orbit is a periodic orbit. Using (4.6), the orbit is asymptotically periodic to its attractor .

8.2. Proof of Theorem 6.1

Using the Li/Yorke-property (4.1).

(A) (B) Let be a periodic sequence. Then Theorem 5.1(C) implies that the sequence , defined by is a periodic orbit of system (2.1).

Let assuming Using (2.2),because . For infinitely many we have . Thus, where and are bounded, and hence is bounded too.

(B) (A)Let be a periodic orbit of system (2.1). Because and are bounded, is bounded too, hence we have the right part of relation (4.1). The left part follows indirectly with Theorem 5.1(C).

8.3. Proof of Theorem 6.2

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

Statement (B)
(1) For , let be the set of indices, where a block of zeros in the cluster sequence starts. Then for Using (2.2) and (4.6), we have for The Taylor sequence is bounded by . Thus, (8.5) guarantees that it is the case for its cluster sequence too. The latter estimateHence, for and .
(2) For the function is lacunary function. Thus, its Taylor sequence has the lacuna cluster . Using (4.3) and the case above imply

8.4. Proof of Theorem 6.3

Statement (A)
Let . For we construct a subsequence , of the orbit converging to .
As assumed, the cluster sequence of the Taylor sequence is of typeThe last in a row of 's defines an index , whereNote that is an infinite set. Using (2.2), we find for the derivative As assumed, the sequence is bounded, thus is bounded by a real number , thus,Hence, for and .

Statement (B)
Let . For we construct a subsequence , of the orbit converging to . At index starts a row of 's, at index the tupel, and at a row of 's, which has its end at index We define the set bywhere contains infinitely many elements. Because is a lacuna cluster, we haveFor , we consider the th derivative , using (2.2) and the abbreviation ,To it has the distanceThe sequence is bounded by , thus the latter term Because of (8.14), the sum converges to for . Thus,

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

8.5. Proof of Theorem 7.1

From (7.4), we find for

Statement (A)
Using (8.19) and , we have for sufficiently large

Statement (B)
Let . For , see (6.1) and (7.5), using (2.2) and (8.19), we haveThis leads to

Statement (C)
Statement (A) implies the statement is valid for . From statement (B), we conclude for each . Because of and Theorem 5.1, we find . Thus, the statement is true for .
Let and Due to the fact that is dense in , we find a rational number , , and
For we choose such thatWe define and
This leads to andFurthermore, we haveUsing (2.2), (8.19), (8.23), (8.24), (8.25), we deduceThus, we have proved statement (C). Statements (D) and (E) follow by using (A), (B), and (C).


The author would like to thank Rudolf Gorenflo (Free University of Berlin) for discussions.


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