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

Authors: Wolf Bayer
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

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


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## Body

## 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 characterizedhypercyclic 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 operatorD 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(2.1)f(0):=f,f(n+1):=D(f(n))forn∈ℕ0={0,1,2,3,…}.Hence, we have for f∈𝒜,f(x)=∑i=0∞ai(xi/i!),ai∈ℂ, the relation(2.2)f(n)(x)=∑i=0∞an+ixii!holds. The sequence (ai)i∈ℕ0 of coefficients of the Taylor series ∑i=0∞ai(xi/i!) we call Taylor sequence.  Equipped with the supremum norm ∥f∥:=maxx∈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∈ℕ0 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].


## 3. Hyperbolic Functions

For the reason
of self-containedness, we inform on hyperbolic functions. Exponential functions
of the type(3.1)eα:E→ℂ,eα(x):=αex,α∈ℂare fixed points or fixed
elements of the dynamical system (2.1),
whereas the so-called
hyperbolic functions of ordern are periodic elements of system (2.1).Withn∈ℕ,i:=−1,ω:=e(2π/n)i Erdélyi et al. defined in [2] Hn:ℂ→ℂ(3.2)Hn(x):=1n∑ν=1neωνx=1n(eωx+eω2x+eω3x+⋯+eωnx)(3.3)=∑ν=0∞xνn(νn)!=1+xn(n)!+x2n(2n)!+x3n(3n)!+⋯.For the real part hn of Hn,
we know from [14] that  hn:=Re(Hn):ℝ→ℝ
(for real x), using the abbreviation αν:=(2π/n)ν,(3.4)hn(x):=1n∑ν=0n−1excosαν⋅cos(xsinαν)(3.5)=∑i=0∞xνn(νn)!=1+xn(n)!+x2n(2n)!+x3n(3n)!+⋯.The Taylor series (3.3) and (3.5)
reveal that hyperbolic functions coincide with their nth derivative, that is,(3.6)Hn(n)=Hn,hn(n)=hn.With  (3.1), (3.2), and (3.4), we find
that (3.7)H1=e1,H2=cosh,H4=12(cosh+cos)
and, see [14], (3.8)h3(x)=13(ex+2e−(1/2)xcos(32x)),h5(x)=15(ex+2excos(2π/5)cos(xsin2π5)+2excos(4π/5)cos(xsin4π5)),h6(x)=13(coshx+2cos(32x)cosh(12x)).We should note that we consider
the function Hn for complex x,
and hn for real x.
It is easy to prove.Proposition 3.1.
The statements (A) and (B) are
equivalent.
(A)
p∈𝒜 is a linear combination of Hn and Hn(1),Hn(2),…,Hn(n−1):(3.9)p=∑ν=0n−1ανHn(ν)=α0Hn+α1Hn(1)+⋯+αn−1Hn(n−1),αν∈ℂ.(B)
The sequence (p(n))n∈ℕ0 is a periodic orbit of the dynamical system
(2.1).Hence, the
orbit ofp in (3.9) move in circles planet-like in the
function space 𝒜.


## 4. Preliminaries

Let(X,∥⋅∥) be a normed linear space and (xn)n∈ℕ0 a sequence in X.Definition 4.1 (Li/Yorke-property).
One calls(xn)n∈ℕ0 an aperiodic sequence or achaotic orbit if it is bounded
but not asymptotically periodic, that is, for each periodic sequence (yn)n∈ℕ0⊂X,
one has
(4.1)0<lim supn→∞∥xn−yn∥<∞.
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 setω(f) of an element f∈𝒜.
It contains all cluster points of the orbit (f(n))n∈ℕ0.
Thus, for the functions (3.1) and (3.7),
(4.2)ω(eα)={eα},ω(cosh)={cosh,sinh},ω(sin)={sin,cos,−sin,−cos}.Proposition 4.2.
Theω-operator is linear in the following sense.  Let f∈𝒜 and eα defined in (3.1). Then(4.3)ω(αf)=αω(f),ω(f+eα)=eα+ω(f).For Taylor sequences(ai)i∈ℕ0 of type(4.4)(…,α,α,α,…,α,α,α,α,…,α,α,α,α,α,…),
one
introduces the concept of lacuna cluster.Definition 4.3.
Let the Taylor sequence(ai)i∈ℕ0 of f∈𝒜 have the cluster point α∈ℂ,
and let I⊂ℕ0 be the index set defined by I:={ni:ani≠α}. Then α is  called  lacuna cluster
of (ai)i∈ℕ0, if the sequence (ni+1−ni)ni∈I is unbounded.Forα=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], (4.5)Φ(x):=1+x1!+x44!+x99!+x1616!+x2525!+x3636!+⋯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(ai)i∈ℕ0 its cluster sequence(a¯i)i∈ℕ0 by identifying elements of convergent
subsequences by their limit point. Note that (ai)i∈ℕ0 is bounded, thus cluster points exist
(Bolzano-Weierstraß).Definition 4.4.
Let{α0,α1,α2,…} be the set of cluster points of the sequence (ai)i∈ℕ0.
One constructs inductively a mapping ai→ai¯∈{α0,α1,α2,…}.
(1)
Due toα0 is cluster point, there is a subsequence (ai)i∈I0,I0⊂ℕ0, converging to α0.
For i∈I0, we define ai¯:=α0.(2)
Ifℕ0∖I0 is a finite set, one defines
for i∈ℕ0∖I0ai¯:=α0.
Otherwise there is a cluster point α1≠α0 and a subset (ai)i∈I1,I1⊂ℕ0∖I0, converging to α1.
For i∈I1,
one defines
ai¯:=α1.(3)
Continuing inductively fork=0,1,2,3,…. If ℕ0∖⋃j=0kIj is a finite set, one
defines
ai¯:=αk, otherwise there is a cluster point αk+1 different from αj for j=0,1,2,…,k, and a subset (ai)i∈Ik+1,Ik+1⊂ℕ0∖⋃j=0kIj, converging to αk+1.
For i∈Ik+1,
one defines
ai¯:=αk+1.Note that the
set of indices{Ik:k=0,1,2,…} are pairwise disjoint and their union is ℕ0.The cluster sequence(a¯i)i∈ℕ0 reveals the asymptotical
behavior of an orbit (f(n))n∈ℕ0,
Property (C) will be very useful.Proposition 4.5.
Forf∈𝒜,f(x)=∑i=0∞ai(xi/i!) and f¯(x):=∑i=0∞a¯i(xi/i!),
one has
(4.6)(A)|ai−a¯i|→0fori→∞.(B)∥f(n)−f¯(n)∥→0forn→∞.(C)ω(f)=ω(f¯).


## 5. Finite Attractors

Like the oracle
of Delphi in ancient Greece informed people about their future, our theorems
will show that the Taylor sequence(ai)i∈ℕ0 predicts the asymptotical behavior of an orbit (f(n))n∈ℕ0 for n→∞.
The following theorem deals with empty and finite attractors, it reveals the
role of Erdelyi's hyperbolic functions Hn for the attractors ω(f) of the differential operator.Theorem 5.1.
Letf∈𝒜,f(x)=∑i=0∞ai(xi/i!) for x∈E.
(A)
If(ai)i∈ℕ0 is unbounded and contains no lacuna cluster,  then (f(n))n∈ℕ0 is unbounded too and ω(f) is empty.(B)
If(ai)i∈ℕ0 is convergent to α∈ℂ,  then (f(n))n∈ℕ0 converges to eα and ω(f)={eα}.(C)
If(ai)i∈ℕ0 is asymptotically periodic to {β0,β1,…,βn−1}⊂ℂ,  then (f(n))n∈ℕ0 is asymptotically periodic too, and(5.1)ω(f)={p,p(1),p(2),…,p(n−1)},wherep:=∑k=0n−1βkHn(n−k).We
give some examples as follows.(1) The functiontan+1/cos∈𝒜 possess for |x|<π/2 with the Bernoulli numbers Bν and the Euler numbers Eν the Taylor series(5.2)tan(x)+1cos(x)=∑ν=1∞4ν(4ν−1)2νBνx2ν−1(2ν−1)!+1+∑ν=1∞Eνx2ν(2ν)!=1+x+x22!+2x33!+5x44!+16x55!+61x66!+272x77!+1385x88!+⋯.Its Taylor sequence {1,1,1,2,5,16,61,272,1385,…} is unbounded without any  cluster point, hence ω(tan+1/cos)=∅.(2)  For each
polynomialq∈𝒜, we have ω(q)={e0}={0}.(3) Letf:E→ℂ defined by (5.3)f(x):={1,ifx=0,sinx(1+1x),else. Then (5.4)f(x)=1+x−x23⋅2!−x33!+x45⋅4!+x55!−x67⋅6!−x77!+x89⋅8!+x99!+⋯.Its Taylor sequence (1,1,−1/3,−1,1/5,1,−1/7,−1,1/9,1,−1/11,…) is asymptotically periodic to the periodic
sequence {0,1,0,−1,0,1,0,−1,0,1,0,−1,…}. Using (5.1), (3.7), and  sin=H4(3)−H4(1),
its attractor becomes ω(f)={sin,cos,−sin,−cos}.


## 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.
Letf∈𝒜,f(x)=∑i=0∞ai(xi/i!) for x∈E.  Then the statements (A) and (B) are equivalent as follows.
(A)
The Taylor
sequence(ai)i∈ℕ0 is aperiodic.(B)
The sequence
of derivatives(f(n))n∈ℕ0 is a chaotic orbit of the system (2.1).Figure1 presents the sequence (∥Φ(n)∥)n∈ℕ0 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.Figure 1
The sequence(∥Φ(n)∥)n∈ℕ0.The next theorems reveal the part of lacunary
functions and exponential functionseα for chaotic orbits of the differential
operator and its attractor.Theorem 6.2.
Letf∈𝒜,f(x)=∑i=0∞ai(xi/i!) for x∈E and (ai)i∈ℕ0 aperiodic. Then
(A)
if(ai)i∈ℕ0 possesses  only a finite number of cluster
points,  then the
cluster sequence (a¯i)i∈ℕ0 contains at least one lacuna cluster.(B)
Ifα∈ℂ is a lacuna cluster of the cluster sequence (a¯i)i∈ℕ0,  then for the
exponential function eα,
one has
eα∈ω(f).We introduce
abbreviations splitting the exponential functione1 into a Taylor  polynomial Tn and its remainder Rn:(6.1)Tn(x):=∑i=0nxii!,Rn(x):=∑i=n+1∞xii!,qn(x):=xnn!.Thus, e1=Tn+Rn for each n∈ℕ.
We will use it for constructing a stairway βTn+γRn between exponential functions eβ and eγ in the function space 𝒜.Theorem 6.3.
Letf∈𝒜,f(x)=∑i=0∞ai(xi/i!) for x∈E and the Taylor sequence (ai)i∈ℕ0 be aperiodic. Then
(A)
for each lacuna clusterβ in the cluster sequence (a¯i)i∈ℕ0, infinitely  often followed
by a lacuna cluster γ≠β,
one has with Un:=βTn+γRn,(6.2){eβ,eγ}∪{Un:n∈ℕ0}⊂ω(f).(B)
For each tupel(b0,b1,…,bk−1)⊂ℂk in the cluster sequence (a¯i)i∈ℕ0,  which appears infinitely
often between the lacuna clusters β and γ,
one has
with Sn:=βTn−1+∑j=0k−1bjqn+j+γRn+k−1(6.3){eβ,eγ}∪{Sn:n∈ℕ0}⊂ω(f).(C)
If the cluster sequence(a¯i)i∈ℕ0 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 functionΛ∈𝒜, we choose ai(=a¯i) according to the rule (6.4)ai:={0,iffork∈ℕ0:k2(3k+3)≤i≤k2(3k+5),1,ifk2(3k+5)<i<k+12(3k+4),2,otherwise. Then (6.5)(ai)i∈ℕ0=(0,1,2,0,0,1,1,2,2,0,0,0,1,1,1,2,2,2,0,0,0,0,1,1,1,1,2,…),Λ(x):=x1!+2x22!+x55!+x66!+2x77!+2x88!+x1212!+x1313!+x1414!+2x1515!+⋯.We see three lacuna clusters 0,1,2.
The attractor of Λ becomes(6.6)ω(Λ)={e0,e1,e2}∪{Rn:n∈ℕ0}∪{Tn+2Rn:n∈ℕ0}∪{2Tn:n∈ℕ0}.Figure 2 presents the sequence (∥Λ(n)∥)n∈ℕ0 graphically, Figure 3 shows schematically the
orbit (Λ(n))n∈ℕ0 and its attractor ω(Λ) in the function space 𝒜.
In both figures, we see the stairways up from e0 to e1 and from e1 to e2,
and the stairway down from e2 to e0.Figure 2
The sequence(∥Λ(n)∥)n∈ℕ0  of the lacunary function Λ.Figure 3
Orbit andω-limit set of the lacunary function Λ.Example for Statement (B)
Is given by the flutter functionΦ defined in (4.5), with k=1,b0=1,β=γ=0. It leads to the attractor of Φ: (6.7)ω(Φ)={e0}∪{qn:n∈ℕ0}.Figure 1 shows the stairway {qn:n∈ℕ0}.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:(6.8)(an)n∈ℕ0:=(1,12,1,1,12,12,13,13,1,1,1,12,12,12,13,13,13,14,14,14,1,1,1,1,12,12,12,12,13,13,13,13,14,14,14,14,15,15,15,15,1,1,1,1,1,…).Mathematical research on lacunary
functions deals usually with unbounded coefficients. In addition to Theorem5.1(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(6.9)f(x):=1+x1!+1!x44!+3!x99!+5!x1616!+7!x2525!+9!x3636!+⋯=1+x+∑i=2∞(2i−3)!xi2(i2)!,we show e0∈ω(f).
The k2th derivative of f is(6.10)f(k2)(x)=∑i=k∞(2i−3)!xi2−k2(i2−k2)!=(2k−3)!+∑i=k+1∞(2i−3)!xi2−k2(i2−k2)!.Because 0∈E, we have ∥f(k2)∥≥(2k−3)!, which means that the orbit is unbounded.We consider its successorf(k2+1) and ∥f(k2+1)∥:(6.11)f(k2+1)(x)=∑i=k+1∞(2i−3)!xi2−k2−1(i2−k2−1)!=(2k−1)!x2k(2k)!+∑i=k+2∞(2i−3)!xi2−k2−1(i2−k2−1)!=x2k2k+∑i=k+2∞xi2−k2−1∏j=2i−2i2−k2−1j,∥f(k2+1)∥=12k+∑i=k+2∞1∏j=2i−2i2−k2−1j<12k+∑i=k+2∞1(2i−2)(i−1)2−k2.Hence, ∥f(k2+1)∥→0 with k→∞.
We conclude the exponential function e0∈ω(f).


## 7. Uncountable Attractors

Finally, we demonstrate that not
only lacunary functions may have chaotic orbits. We use the Cantor
sequence(7.1)(ci)i∈ℕ0=(11,12,22,13,23,33,14,24,34,44,15,25,35,45,55,16,26,36,46,56,66,17,…)to define an analytic function C∈𝒜.
It has countably infinitely many cluster points, but no lacuna
cluster:(7.2)C(x):=∑i=0∞cixii!=1+12x+x22!+13x33!+23x44!+x55!+14x66!+12x77!+34x88!+⋯.With the abbreviation(7.3)sn:=n2(n+1),n∈ℕ0,the elements of the Cantor
sequence (ci)i∈ℕ0 maybe given by(7.4)ci:=i+1−snn+1forsn≤i<sn+1.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(7.5)1=∥T0∥,2=∥T1∥,2.5=∥T2∥,2.6¯=∥T3∥,2.7083¯=∥T4∥,….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 {eα:α∈[0;1]}. After that the orbit jumps to a Taylor
polynomial Tm (the squirrel jumps to a branch), and to
another one, lower one Tm−1,
and then to Tm−2,…,T0.
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 e1.Figure 4
Graph of the Cantor functionC for real arguments.Figure 5
Sequence(∥C(n)∥)n∈ℕ  of the Cantor function C.Figure 6
Orbit andω-limit set of the Cantor function C.We describe the properties of the orbit ofC in a theorem, using Taylor polynomials Tn,
remainders Rn (6.1), exponential functions (3.1), and sn (7.3).Theorem 7.1.
For the orbit(C(n))n∈ℕ0 of the Cantor function C,
one has
(A)
∥C(sn)∥→0 for n→∞;(B)
form with0≤m<n:∥C(sn−m−1)−Tm∥→0 for n→∞;(C)
for eachα∈[0;1], there is a subsequence (C(nj))nj∈I,I⊂ℕ,  of the orbit(C(n))n∈ℕ with ∥C(nj)−eα∥→0 for j→∞;(D)
lim supn→∞∥C(n)∥=e1,lim infn→∞∥C(n)∥=0;(E)
ω(C)={Tn:n∈ℕ0}∪{eα:α∈[0;1]},
containing uncountably many different elements.Properties (A), (B), and (D) can be seen in
Figure5, property (E) in Figure 6.


## 8. Proofs

### 8.1. Proof of Theorem5.1

Property (A)
Using (2.2),(8.1)∥f(n)∥=maxx∈E|∑i=0∞an+ixii!|=maxx∈E|an+∑i=1∞an+ixii!|≥|an|because 0∈E.
Thus, (|an|)n∈ℕ0 is an unbounded minorizing sequence.Property (B)
It follows from
(C) withn=1 and β0=α.Property (C)
By assumption
the cluster sequence(a¯i)i∈ℕ0 of the Taylor sequence becomes a¯i=βimodn. Consider p∈𝒜 defined by(8.2)p(x):=∑i=0∞a¯ixii!=∑i=0∞βimodnxii!=∑k=0n−1βk∑i=0∞xin+k(in+k)!=∑k=0n−1βkHn(n−k).Proposition 3.1 implies that the
orbit (p(n))n∈ℕ0 is a periodic orbit. Using (4.6), the orbit (f(n))n∈ℕ0 is asymptotically periodic to
its attractor ω(f)={p,p(1),p(2),…,p(n−1)}.


### 8.2. Proof of Theorem6.1

Using the Li/Yorke-property (4.1).(A)⇒ (B) Let (bi)i∈ℕ0⊂ℂ be a periodic sequence. Then Theorem 5.1(C)
implies that the sequence (p(n))n∈ℕ,
defined by p(x)=∑i=0∞bi(xi/i!), is a periodic orbit of system (2.1).Letδ:=lim supi→∞|ai−bi|, assuming δ>0. Using (2.2),(8.3)∥f(n)−p(n)∥=maxx∈E|∑i=0∞an+ixii!−∑i=0∞bn+ixii!|=maxx∈E|∑i=0∞(an+i−bn+i)xii!|≥|an−bn|because 0∈E.
For infinitely many n, we have |an−bn|>δ/2.
Thus, (8.4)lim supn→∞∥f(n)−p(n)∥≥δ2>0, where (f(n))n∈ℕ and (p(n))n∈ℕ are bounded, and hence (∥f(n)−p(n)∥)n∈ℕ0 is bounded too.(B)⇒ (A)Let (p(n))n∈ℕ0 be a periodic orbit of system (2.1). Because (∥f(n)∥)n∈ℕ0 and (∥p(n)∥)n∈ℕ0 are bounded, (∥f(n)−p(n)∥)n∈ℕ0 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 Theorem6.2

Statement (A)
It can be
proved directly from elementary combinatorics.Statement (B)
(1) Forα=0,  let I⊂ℕ0 be the set of indices, where a block of zeros
in the cluster sequence (a¯i)i∈ℕ0 starts. Then for n∈I,(8.5)a¯i=0forn≤i<n+kn,kn∈ℕ,kn→∞forn→∞.Using (2.2) and (4.6), we have for n∈I,(8.6)∥f¯(n)∥=maxx∈E|∑i=0∞a¯n+ixii!|=maxx∈E|∑i=kn∞a¯n+ixii!|≤∑i=kn∞|a¯n+i|1i!.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(8.7)≤a∑i=kn∞1i!→0forn→∞.Hence, f¯(n)→e0 for n→∞ and e0∈ω(f¯)=ω(f).
(2) Forα≠0, the function g:=f−eα is lacunary function. Thus, its Taylor
sequence has the  lacuna cluster 0.
Using (4.3) and the case α=0 above imply(8.8)e0∈ω(g)=ω(f−eα)=−eα+ω(f)⟺eα+e0=eα∈ω(f).


### 8.4. Proof of Theorem6.3

Statement (A)
Letm∈ℕ.
For Um=βTm+γRm, we construct a subsequence
(f¯n)n∈I,I⊂ℕ0,
of the orbit (f¯n)n∈ℕ0, converging to Um.
As assumed, the cluster sequence(a¯i)i∈ℕ0 of the Taylor sequence is of
type(8.9)(a¯i)i∈ℕ0=(…,β,γ,…,β,β,γ,γ,…,β,β,β,γ,γ,γ,…).The last β in a row of β's defines an index n∈I,
where(8.10)I:={n∈ℕ:a¯i=βforn−m≤i≤nanda¯i=γforn<i≤n+kwithk≥m}.Note that I is an infinite set. Using (2.2), we find for the
derivative f¯(n−m),(8.11)∥f¯(n−m)−Um∥=maxx∈E|∑i=0∞a¯n−m+ixii!−β∑i=0mxii!−γ∑i=m+1∞xii!|=maxx∈E|∑i=n+k+1∞(a¯n−m+i−γ)xii!|≤∑i=n+k+1∞|a¯n−m+i−γ|1i!. As assumed, the sequence (ai)i∈ℕ0 is bounded, thus (|a¯i−γ|)i∈ℕ0 is bounded by a real number c∈ℝ+,
thus,(8.12)∥f¯(n−m)−Um∥≤c∑i=n+k+1∞1i!→0forn→∞,n∈I.Hence, f¯(n−m)→Um for n→∞,n∈I and Um∈ω(f¯)=ω(f).Statement (B)
Letm∈ℕ.
For Sm, we construct a subsequence (f¯(n))n∈I,I⊂ℕ0,
of the orbit (f¯(n))n∈ℕ0, converging to Sm.
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+Mn−1. We define the set I by(8.13)I:={n∈ℕ0:a¯n+i=βfor0≤i<m,a¯n+i=bi−mform≤i<m+k,a¯n+i=γform+k≤i<m+k+Mn,a¯n+i≠γfori=m+k+Mn},where I contains infinitely many elements. Because γ is a lacuna cluster, we have(8.14)Mn→∞forn→∞.For n∈I,
we consider the nth derivative f¯(n),
using (2.2) and  the abbreviation M:=Mn,(8.15)f¯(n)=∑i=0∞a¯n+iqi=β∑i=0m−1qi+∑i=mm+k−1bi−mqi+γ∑i=m+km+k+M−1qi+∑i=m+k+M∞a¯n+iqi=βTm−1+∑j=0k−1bjqm+j+γ(Rm+k−1−Rm+k+M−1)+∑i=m+k+M∞a¯n+iqi.To Sm, it has the distance(8.16)∥f¯(n)−Sm∥=∥∑i=m+k+M∞a¯n+iqi−γRm+k+M−1∥=∥∑i=m+k+M∞(a¯n+i−γ)qi∥≤∑i=m+k+M∞|a¯n+i−γ|∥qi∥=∑i=m+k+M∞|a¯n+i−γ|1i!.The sequence (|a¯i−γ|)i∈ℕ is bounded by c∈ℝ+,
thus the latter term (8.17)≤c∑i=m+k+M∞1i!.Because of (8.14), the sum
converges to 0 for n→∞.
Thus, (8.18)∥f¯(n)−Sm∥→0forn→∞,n∈I,Sm∈ω(f¯)=ω(f).Statement (C)
If there is only one lacuna cluster in the cluster
sequence, then statement (B) implies withβ=γ that the attractor is countably infinite.
If there are infinitely many lacuna clustersα1,α2,α3,… in the cluster sequence, then statement (A)
implies for each couple αj,αj+1 countably infinitely many elements of the
attractor. Using countable× countable = countable, we conclude (C).


### 8.5. Proof of Theorem7.1

From (7.4), we find for k=1,2,…,n(8.19)csn=1n+1,csn+k=k+1n+1,csn−k=n+1−kn.Statement (A)
Using (8.19) and 0<ci≤1,
we have for sufficiently large n,(8.20)∥C(sn)∥=∥∑i=0ni+1n+1xii!+∑i=n+1∞csn+ixii!∥≤1n+1∥∑i=0n(i+1)xii!∥+∥∑i=n+1∞csn+ixii!∥<1n+1∑i=0∞(i+1)1i!+∥∑i=n+1∞xii!∥=2e1n+1+∥Rn∥→0forn→∞.Statement (B)
Letn,m∈ℕ0,m<n.
For Tm,
see (6.1) and (7.5), using (2.2) and (8.19), we have(8.21)C(sn−m−1)(x)=∑i=0∞csn−m−1+ixii!=∑i=0mn−m+inxii!+∑i=m+1n+m+1i−mn+1xii!+∑i=n+m+2∞csn−m−1+ixii!=Tm(x)+1n∑i=0m(i−m)xii!+1n+1∑i=m+1n+m+1(i−m)xii!+∑i=n+m+2∞csn−m−1+ixii!.This leads to  (8.22)∥C(sn−m−1)−Tm∥≤1n∥∑i=0m(i−m)xii!∥+1n+1∥∑i=m+1n+m+1(i−m)xii!∥+∥∑i=n+m+2∞csn−m−1+ixii!∥≤1n∑i=0m|i−m|i!+1n+1∑i=m+1n+m+1|i−m|i!+∥Rn+m+1∥→0forn→∞.Statement (C)
Statement (A) implies the statement is valid forα=0.
From statement (B), we conclude
Tm∈ω(C) for each m∈ℕ0.
Because of limm→∞Tm=e1 and Theorem 5.1, we find
e1∈ω(C).
Thus, the statement is true for α=1.
Let0<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.
Forε>0, we choose k∈ℕ such that(8.23)ek<ε3,∥Rk∥<ε3.We define m:=kp−1 and n:=kq−1.
This leads ton−m=k(q−p)≥k and(8.24)∥Rn−m∥≤∥Rk∥<ε3.Furthermore, we have(8.25)|m+1n+1−α|=|pq−α|<ε3e.Using (2.2), (8.19), (8.23), (8.24),
(8.25), we deduce(8.26)∥C(sn+m)−eα∥=∥∑i=0∞csn+m+ixii!−α∑i=0∞xii!∥=∥∑i=0n−mm+i+1n+1xii!−α∑i=0n−mxii!+∑i=n−m+1∞csn+m+ixii!−α∑i=n−m+1∞xii!∥=∥∑i=0n−m(m+1n+1−α)xii!+∑i=0n−min+1xii!+∑i=n−m+1∞(csn+m+i−α)xii!∥≤∑i=0n−m|m+1n+1−α|1i!+∑i=1n−m1n+11(i−1)!+∑i=n−m+1∞|csn+m+i−α|1i!<ε3e∑i=0n−m1i!+1kq∑i=1n−m1(i−1)!+∑i=n−m+1∞1i!<ε3ee1+1kqe1+∥Rn−m∥<ε3+ε3+ε3=ε.Thus, we have proved statement
(C). Statements  (D) and (E) follow by using (A), (B), and
(C).


## 8.1. Proof of Theorem5.1

Property (A)
Using (2.2),(8.1)∥f(n)∥=maxx∈E|∑i=0∞an+ixii!|=maxx∈E|an+∑i=1∞an+ixii!|≥|an|because 0∈E.
Thus, (|an|)n∈ℕ0 is an unbounded minorizing sequence.Property (B)
It follows from
(C) withn=1 and β0=α.Property (C)
By assumption
the cluster sequence(a¯i)i∈ℕ0 of the Taylor sequence becomes a¯i=βimodn. Consider p∈𝒜 defined by(8.2)p(x):=∑i=0∞a¯ixii!=∑i=0∞βimodnxii!=∑k=0n−1βk∑i=0∞xin+k(in+k)!=∑k=0n−1βkHn(n−k).Proposition 3.1 implies that the
orbit (p(n))n∈ℕ0 is a periodic orbit. Using (4.6), the orbit (f(n))n∈ℕ0 is asymptotically periodic to
its attractor ω(f)={p,p(1),p(2),…,p(n−1)}.


## 8.2. Proof of Theorem6.1

Using the Li/Yorke-property (4.1).(A)⇒ (B) Let (bi)i∈ℕ0⊂ℂ be a periodic sequence. Then Theorem 5.1(C)
implies that the sequence (p(n))n∈ℕ,
defined by p(x)=∑i=0∞bi(xi/i!), is a periodic orbit of system (2.1).Letδ:=lim supi→∞|ai−bi|, assuming δ>0. Using (2.2),(8.3)∥f(n)−p(n)∥=maxx∈E|∑i=0∞an+ixii!−∑i=0∞bn+ixii!|=maxx∈E|∑i=0∞(an+i−bn+i)xii!|≥|an−bn|because 0∈E.
For infinitely many n, we have |an−bn|>δ/2.
Thus, (8.4)lim supn→∞∥f(n)−p(n)∥≥δ2>0, where (f(n))n∈ℕ and (p(n))n∈ℕ are bounded, and hence (∥f(n)−p(n)∥)n∈ℕ0 is bounded too.(B)⇒ (A)Let (p(n))n∈ℕ0 be a periodic orbit of system (2.1). Because (∥f(n)∥)n∈ℕ0 and (∥p(n)∥)n∈ℕ0 are bounded, (∥f(n)−p(n)∥)n∈ℕ0 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 Theorem6.2

Statement (A)
It can be
proved directly from elementary combinatorics.Statement (B)
(1) Forα=0,  let I⊂ℕ0 be the set of indices, where a block of zeros
in the cluster sequence (a¯i)i∈ℕ0 starts. Then for n∈I,(8.5)a¯i=0forn≤i<n+kn,kn∈ℕ,kn→∞forn→∞.Using (2.2) and (4.6), we have for n∈I,(8.6)∥f¯(n)∥=maxx∈E|∑i=0∞a¯n+ixii!|=maxx∈E|∑i=kn∞a¯n+ixii!|≤∑i=kn∞|a¯n+i|1i!.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(8.7)≤a∑i=kn∞1i!→0forn→∞.Hence, f¯(n)→e0 for n→∞ and e0∈ω(f¯)=ω(f).
(2) Forα≠0, the function g:=f−eα is lacunary function. Thus, its Taylor
sequence has the  lacuna cluster 0.
Using (4.3) and the case α=0 above imply(8.8)e0∈ω(g)=ω(f−eα)=−eα+ω(f)⟺eα+e0=eα∈ω(f).


## 8.4. Proof of Theorem6.3

Statement (A)
Letm∈ℕ.
For Um=βTm+γRm, we construct a subsequence
(f¯n)n∈I,I⊂ℕ0,
of the orbit (f¯n)n∈ℕ0, converging to Um.
As assumed, the cluster sequence(a¯i)i∈ℕ0 of the Taylor sequence is of
type(8.9)(a¯i)i∈ℕ0=(…,β,γ,…,β,β,γ,γ,…,β,β,β,γ,γ,γ,…).The last β in a row of β's defines an index n∈I,
where(8.10)I:={n∈ℕ:a¯i=βforn−m≤i≤nanda¯i=γforn<i≤n+kwithk≥m}.Note that I is an infinite set. Using (2.2), we find for the
derivative f¯(n−m),(8.11)∥f¯(n−m)−Um∥=maxx∈E|∑i=0∞a¯n−m+ixii!−β∑i=0mxii!−γ∑i=m+1∞xii!|=maxx∈E|∑i=n+k+1∞(a¯n−m+i−γ)xii!|≤∑i=n+k+1∞|a¯n−m+i−γ|1i!. As assumed, the sequence (ai)i∈ℕ0 is bounded, thus (|a¯i−γ|)i∈ℕ0 is bounded by a real number c∈ℝ+,
thus,(8.12)∥f¯(n−m)−Um∥≤c∑i=n+k+1∞1i!→0forn→∞,n∈I.Hence, f¯(n−m)→Um for n→∞,n∈I and Um∈ω(f¯)=ω(f).Statement (B)
Letm∈ℕ.
For Sm, we construct a subsequence (f¯(n))n∈I,I⊂ℕ0,
of the orbit (f¯(n))n∈ℕ0, converging to Sm.
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+Mn−1. We define the set I by(8.13)I:={n∈ℕ0:a¯n+i=βfor0≤i<m,a¯n+i=bi−mform≤i<m+k,a¯n+i=γform+k≤i<m+k+Mn,a¯n+i≠γfori=m+k+Mn},where I contains infinitely many elements. Because γ is a lacuna cluster, we have(8.14)Mn→∞forn→∞.For n∈I,
we consider the nth derivative f¯(n),
using (2.2) and  the abbreviation M:=Mn,(8.15)f¯(n)=∑i=0∞a¯n+iqi=β∑i=0m−1qi+∑i=mm+k−1bi−mqi+γ∑i=m+km+k+M−1qi+∑i=m+k+M∞a¯n+iqi=βTm−1+∑j=0k−1bjqm+j+γ(Rm+k−1−Rm+k+M−1)+∑i=m+k+M∞a¯n+iqi.To Sm, it has the distance(8.16)∥f¯(n)−Sm∥=∥∑i=m+k+M∞a¯n+iqi−γRm+k+M−1∥=∥∑i=m+k+M∞(a¯n+i−γ)qi∥≤∑i=m+k+M∞|a¯n+i−γ|∥qi∥=∑i=m+k+M∞|a¯n+i−γ|1i!.The sequence (|a¯i−γ|)i∈ℕ is bounded by c∈ℝ+,
thus the latter term (8.17)≤c∑i=m+k+M∞1i!.Because of (8.14), the sum
converges to 0 for n→∞.
Thus, (8.18)∥f¯(n)−Sm∥→0forn→∞,n∈I,Sm∈ω(f¯)=ω(f).Statement (C)
If there is only one lacuna cluster in the cluster
sequence, then statement (B) implies withβ=γ that the attractor is countably infinite.
If there are infinitely many lacuna clustersα1,α2,α3,… in the cluster sequence, then statement (A)
implies for each couple αj,αj+1 countably infinitely many elements of the
attractor. Using countable× countable = countable, we conclude (C).


## 8.5. Proof of Theorem7.1

From (7.4), we find for k=1,2,…,n(8.19)csn=1n+1,csn+k=k+1n+1,csn−k=n+1−kn.Statement (A)
Using (8.19) and 0<ci≤1,
we have for sufficiently large n,(8.20)∥C(sn)∥=∥∑i=0ni+1n+1xii!+∑i=n+1∞csn+ixii!∥≤1n+1∥∑i=0n(i+1)xii!∥+∥∑i=n+1∞csn+ixii!∥<1n+1∑i=0∞(i+1)1i!+∥∑i=n+1∞xii!∥=2e1n+1+∥Rn∥→0forn→∞.Statement (B)
Letn,m∈ℕ0,m<n.
For Tm,
see (6.1) and (7.5), using (2.2) and (8.19), we have(8.21)C(sn−m−1)(x)=∑i=0∞csn−m−1+ixii!=∑i=0mn−m+inxii!+∑i=m+1n+m+1i−mn+1xii!+∑i=n+m+2∞csn−m−1+ixii!=Tm(x)+1n∑i=0m(i−m)xii!+1n+1∑i=m+1n+m+1(i−m)xii!+∑i=n+m+2∞csn−m−1+ixii!.This leads to  (8.22)∥C(sn−m−1)−Tm∥≤1n∥∑i=0m(i−m)xii!∥+1n+1∥∑i=m+1n+m+1(i−m)xii!∥+∥∑i=n+m+2∞csn−m−1+ixii!∥≤1n∑i=0m|i−m|i!+1n+1∑i=m+1n+m+1|i−m|i!+∥Rn+m+1∥→0forn→∞.Statement (C)
Statement (A) implies the statement is valid forα=0.
From statement (B), we conclude
Tm∈ω(C) for each m∈ℕ0.
Because of limm→∞Tm=e1 and Theorem 5.1, we find
e1∈ω(C).
Thus, the statement is true for α=1.
Let0<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.
Forε>0, we choose k∈ℕ such that(8.23)ek<ε3,∥Rk∥<ε3.We define m:=kp−1 and n:=kq−1.
This leads ton−m=k(q−p)≥k and(8.24)∥Rn−m∥≤∥Rk∥<ε3.Furthermore, we have(8.25)|m+1n+1−α|=|pq−α|<ε3e.Using (2.2), (8.19), (8.23), (8.24),
(8.25), we deduce(8.26)∥C(sn+m)−eα∥=∥∑i=0∞csn+m+ixii!−α∑i=0∞xii!∥=∥∑i=0n−mm+i+1n+1xii!−α∑i=0n−mxii!+∑i=n−m+1∞csn+m+ixii!−α∑i=n−m+1∞xii!∥=∥∑i=0n−m(m+1n+1−α)xii!+∑i=0n−min+1xii!+∑i=n−m+1∞(csn+m+i−α)xii!∥≤∑i=0n−m|m+1n+1−α|1i!+∑i=1n−m1n+11(i−1)!+∑i=n−m+1∞|csn+m+i−α|1i!<ε3e∑i=0n−m1i!+1kq∑i=1n−m1(i−1)!+∑i=n−m+1∞1i!<ε3ee1+1kqe1+∥Rn−m∥<ε3+ε3+ε3=ε.Thus, we have proved statement
(C). Statements  (D) and (E) follow by using (A), (B), and
(C).



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*Source: 251298-2009-04-13.xml*