Available via license: CC BY 4.0
Content may be subject to copyright.
arXiv:1803.08368v1 [math.DS] 22 Mar 2018
A Short Comparison of Classical Complex
Dynamics and Holomorphic Semigroup Dynamics
Bishnu Hari Subedi and Ajaya Singh
Abstract. This is an expository plus research paper which mainly exposes pre-
liminary connection and contrast between classical complex dynamics and semi-
group dynamics of holomorphic functions. Classically, we expose some existing
results of rational and transcendental dynamics and we see how far these results
generalized to holomorphic semigroup dynamics as well as we also see what new
phenomena occur.
1. Introduction
It is quite natural to extend the Fatou-Julia-Eremenko theory of the iteration
of single holomorphic map in complex plane Cor extended complex plane C∞
to composite of the family of holomorphic maps. So, the purpose of this paper
is to expose the theory of complex dynamics not only for the iteration of single
holomorphic map on Cor C∞but also for the composite of the family Fof such
maps. Let Fbe a space of holomorphic maps on Cor C∞. For any map φ∈F,
Cor C∞is naturally partitioned into two subsets: the set of normality and its
complement. We say that a family Fis normal if each sequence from the family
has a subsequence which either converges uniformly on compact subsets or diverges
uniformly to ∞. The set of normality or Fatou set F(φ) of the map φ∈Fis the
largest open set on which the iterates φn=φ◦φ◦...◦φ(n-fold composition of φ
with itself) is a normal family. The complement J(φ) is the Julia set. A maximally
connected subset of the Fatou set F(φ) is a Fatou component. The main concern
of such an iteration theory is to describe the nature of the components of Fatou set
and the structure and properties of the Julia set.
In our study, classical complex dynamics refers the iteration theory of single
holomorphic map and holomorphic semigroup dynamics refers the dynamical the-
ory generated by various classes of holomorphic maps. In holomorphic semigroup
dynamics, algebraic structure of semigroup naturally attached to the dynamics and
hence the situation is largely complicated. The principal aim of this paper is to
2010 Mathematics Subject Classification. 37F10, 30D05.
Key words and phrases. Holomorphic semigroup, Fatou set, Julia set, escaping set.
This research work of the first author is supported by PhD faculty fellowship from University
Grants Commission, Nepal.
1
2 B. H. SUBEDI AND A. SINGH
see how far classical complex dynamics applies to holomorphic semigroup dynamics
and what new phenomena appear in holomorphic semigroup settings.
2. The notion of holomorphic semigroup
Semigroup Sis a very classical algebraic structure with a binary composition
that satisfies associative law. It naturally arose from the general mapping of a set
into itself. So a set of holomorphic maps on Cor C∞naturally forms a semigroup.
Here, we take a set Aof holomorphic maps and construct a semigroup Sconsists
of all elements that can be expressed as a finite composition of elements in A. We
say such a semigroup Sby holomorphic semigroup generated by set A. For our
simplicity, we denote the class of all rational maps on C∞by Rand class of all
transcendental entire maps on Cby E. Our particular interest is to study of the
dynamics of the families of above two classes of holomorphic maps. For a collection
F={fα}α∈∆of such maps, let
S=hfαi
be a holomorphic semigroup generated by them. Here Fis either a collection Rof
rational maps or a collection Eof transcendental entire maps. The index set ∆ to
which αbelongs is allowed to be infinite in general unless otherwise stated. Here,
each f∈Sis a holomorphic function and Sis closed under functional composition.
Thus, f∈Sis constructed through the composition of finite number of functions
fαk,(k= 1,2,3, . . . , m). That is, f=fα1◦fα2◦fα3◦ · · · ◦ fαm. In particular, if
fα∈R, we say S=hfαiarational semigroup and if fα∈E, we say S=hfαia
transcendental semigroup.
A semigroup generated by finitely many holomorphic functions fi,(i= 1,2,...,
n) is called finitely generated holomorphic semigroup. We write S=hf1, f2,...,fni.
If Sis generated by only one holomorphic function f, then Sis cyclic semigroup.
We write S=hfi. In this case, each g∈Scan be written as g=fn, where fnis
the nth iterates of fwith itself. Note that in our study of semigroup dynamics, we
say S=hfiatrivial semigroup.
The following result will be clear from the definition of holomorphic semigroup.
It shows that every element of holomorphic semigroup can be written as finite
composition of the sequence of fα
Proposition 2.1.Let S=hfαibe an arbitrary holomorphic semigroup. Then
for every f∈S,fm(for all m∈N) can be written as fm=fα1◦fα2◦fα3◦ · · · ◦ fαp
for some p∈N.
This proposition 2.1 tells us that it need not to apply same map over and over
again in holomorphic semigroup dynamics. But instead, we may start with family
of maps and we consider dynamics over iteratively defined composition of sequence
of maps.
Next, we define and discuss some special collection and sequences of holomorphic
functions. Note that all notions of convergence that we deal in this paper will be
with respect to the Euclidean metric on the complex plane Cor spherical metric
on the Riemann sphere C∞.
A SHORT COMPARISON OF CLASSICAL COMPLEX DYNAMICS AND HOLOMORPHIC... 3
The family Fof complex analytic maps forms a normal family in a domain
Dif given any composition sequence (fα) generated by the member of F, there
is a subsequence (fαk) which is uniformly convergent or divergent on all compact
subsets of D. If there is a neighborhood Uof the point z∈Csuch that Fis normal
family in U, then we say Fis normal at z. If Fis a family of members from the
semigroup S, then we simply say that Sis normal in the neighborhood of zor Sis
normal at z.
Let fbe a holomorphic map. We say that fiteratively divergent at z∈Cif
fn(z)→αas n→ ∞, where αis an essential singularity of f. A sequence (fk)k∈Nof
holomorphic maps is said to be iteratively divergent at zif fn
k(z)→αkas n→ ∞
for all k∈N, where αkis an essential singularity of fkfor each k. Semigroup S
is iteratively divergent at zif fn(z)→αfas n→ ∞, where αfis an essential
singularity of each f∈S. Otherwise, a function f, sequence (fk)k∈Nand semigroup
Sare said to be iteratively bounded at z.
3. Fatou set, Julia set and Escaping set
In classical complex dynamics, each of Fatou set, Julia set and escaping set
are defined in two different but equivalent ways. In first definition, Fatou set is
defined as the set of normality of the iterates of given function, Julia set is defined
as the complement of the Fatou set and escaping set is defined as the set of points
that goes to essential singularity under the iterates of given function. The second
definition of Fatou set is given as a largest completely invariant open set and Julia
set is given as a smallest completely invariant close set whereas escaping set is a
completely invariant non-empty neither open nor close set in C. Each of these
definitions can be naturally extended to the settings of holomorphic semigroup S
but extension definitions are not equivalent. Based on above first definition (that
is, on the Fatou-Julia-Eremenko theory of a complex analytic function), the Fatou
set, Julia set and escaping set in the settings of holomorphic semigroup are defined
as follows.
Definition 3.1 (Fatou set, Julia set and escaping set).Fatou set of the
holomorphic semigroup Sis defined by
F(S) = {z∈C:Sis normal in a neighborhood of z}
and the Julia set J(S)of Sis the compliment of F(S). If Sis a transcendental
semigroup, the escaping set of Sis defined by
I(S) = {z∈C:Sis iteratively divergent at z}
We call each point of the set I(S)by escaping point.
It is obvious that F(S) is the largest open subset (of Cor C∞) on which the
family Fin S(or semigroup Sitself) is normal. Hence its compliment J(S) is a
smallest closed set for any semigroup S. Whereas the escaping set I(S) is neither
an open nor a closed set (if it is non-empty) for any semigroup S. Any maximally
connected subset Uof the Fatou set F(S) is called a Fatou component.
4 B. H. SUBEDI AND A. SINGH
If S=hfi, then F(S), J(S) and I(S) are respectively the Fatou set, Julia set
and escaping set in classical complex dynamics. In this situation we simply write:
F(f), J(f) and I(f).
The main motivation of this paper comes from seminal work of Hinkkanen and
Martin [5] on the dynamics of rational semigroup and the extension study of K.
K. Poon [8] to the dynamics of transcendental semigroup. Both of them naturally
generalized classical complex dynamics to the dynamics of the sequence of different
functions by means of composition. Another motivation of studying escaping set
of transcendental semigroup comes from the work of Dinesh Kumar and Sanjay
Kumar[6, 7] where they defined escaping set and discussed how far escaping set of
classical transcendental dynamics can be generalized to semigroup dynamics.
The fundamental contrast between classical complex dynamics and semigroup
dynamics appears by different algebraic structure of corresponding semigroups. In
fact, non-trivial semigroup (rational or transcendental) need not be, and most often
will not be abelian. However, trivial semigroup is cyclic and therefore abelian. As
we discussed before, classical complex dynamics is a dynamical study of trivial
(cyclic) semigroup whereas semigroup dynamics is a dynamical study of non-trivial
holomorphic semigroup.
The following immediate result holds good from definition 3.1 of escaping set.
Theorem 3.1.Let Sbe a transcendental semigroup and let z∈Cis an escaping
point under S. Then every sequence (gk)k∈Nin Sis iteratively divergent at zand
every subsequence of (gk)k∈Nis also iteratively divergent at z.
The following characterization of escaping set will be clear from the definition
3.1 of escaping set and proposition 2.1, which can be an alternative definition.
Theorem 3.2.If a complex number z∈Cis escaping point of any transcen-
dental semigroup S, then every sequence in Shas a subsequence which diverges to
∞at z.
On the basis of the theorem 3.2, we can say that our definition 3.1 of escaping
set is more general than that of the definition of Dinesh Kumar and Sanjay Kumar
[6, Definition 2.1]. That is, our definition of escaping set implies the definition of
Dinesh Kumar and Sanjay Kumar.
4. Basic Comparison of classical and holomorphic semigroup dynamics
The following immediate relations hold for any f∈Sfrom the definition 3.1.
Indeed, it shows certain connection between classical complex dynamics and semi-
group dynamics.
Theorem 4.1.Let Sbe a semigroup. Then
(1) F(S)⊂F(f)for all f∈Sand hence F(S)⊂Tf∈SF(f).
(2) J(f)⊂J(S)for all f∈S.
(3) I(S)⊂I(f)for all f∈Sand hence I(S)⊂Tf∈SI(f)in the case of
transcendental semigroup S.
A SHORT COMPARISON OF CLASSICAL COMPLEX DYNAMICS AND HOLOMORPHIC... 5
Hinkkanen and Martin proved the following results ([5, Lemma 3.1 and Corollary
3.1]).
Theorem 4.2.Let Sbe a rational semigroup. Then Julia set J(S)is perfect
and J(S) = Sf∈SJ(f)
K. K. Poon proved the following results ([8, Theorems 4.1 and 4.2]).
Theorem 4.3.Let Sbe a transcendental semigroup. Then Julia set J(S)is
perfect and J(S) = Sf∈SJ(f)
From the theorem 4.1 ((1) and (3)), we can say that the Fatou set and the
escaping set may be empty. For example, the escaping set of semigroup S=hf, gi
generated by functions f(z) = ezand g(z) = e−zis empty (the particular function
h=g◦fk∈S(say) is iteratively bounded at any z∈I(f)). We know that Fatou
set may be empty but escaping set is non-empty in classical complex dynamics.
This is a contrast feature of escaping set in classical complex dynamics and semi-
group dynamics. From the same theorem part (2), we can say that in classical and
semigroup dynamics, Julia set is non-empty.
Dinesh Kumar and Sanjay Kumar [6, Theorem 2.5] have identified the following
transcendental semigroup S, where I(S) is an empty set.
Theorem 4.4.The transcendental semigroup S=hf1, f2igenerated by two
functions f1and f2from respectively two parameter family {e−z+γ+cwhere γ, c ∈
Cand Re(γ)<0, Re(c)>1}and {ez+µ+d, where µ, d ∈Cand Re(µ)<
0, Re(d)6−1}of functions has empty escaping set I(S)
Proof. Under the conditions stated in the theorem, for any function ffrom the
first family we have I(f)⊂ {z∈C:Rez < 0,(4k−3) π
2< Imz < (4k−1)π
2, k ∈Z}
(see for instance [6, Lemma 2.2]) and for any function gfrom the second family
we have I(g)⊂ {z∈C:Rez > 0,(4k−1) π
2< Imz < (4k+ 1)π
2, k ∈Z}(see for
instance [6, Lemma 2.3]). From theorem 4.1, for any f, g ∈S=hf1, f2i, we have
I(S)⊂I(f)∩I(g) = ∅.
There are several transcendental semigroups where escaping set I(S)6=∅. The
following examples of Dinesh Kumar and Sanjay Kumar [6, Examples 2.6 and 2.7]
are better to mention here.
Example 4.1.Let S=heλz, esλz + 2πi/λifor all λ∈C− {0}and s∈N. Then
I(S) = I(f)6=∅.
Example 4.2.Let S=hf, gi, where f(z) = λsin z(λ∈C− {0}) and g(z) =
fn+ 2πfor all n∈N. Then I(S) = I(f)6=∅.
If escaping set I(S)6=∅, then Eremenko’s result ∂I(f) = J(f) [4] of classical
transcendental dynamics can be generalized to semigroup settings. The following
results is due to Dinesh Kumar and Sanjay Kumar [6, Lemma 4.2 and Theorem
4.3] which yield the generalized answer in semigroup settings.
Theorem 4.5.Let Sbe a transcendental semigroup such that I(S)6=∅. Then
6 B. H. SUBEDI AND A. SINGH
(1) int(I(S)) ⊂F(S)and ext(I(S)) ⊂F(S), where int and ext respectively
denote the interior and exterior of I(S).
(2) ∂I (S) = J(S), where ∂I(S)denotes the boundary of I(S).
Proof. (1) We refer for instance lemma 4.2 of [6].
(2) The facts int(I(S)) ⊂F(S) and ext(I(S)) ⊂F(S) yield J(S)⊂∂I(S).
The fact ∂I (S)⊂J(S) is obvious.
From this theorem 4.5, the fact J(S)⊂I(S) follows trivially. If I(S)6=∅,
then we prove the following result which is a generalization of Eremenko’s result
I(f)∩J(f)6=∅[4, Theorem 2] of classical transcendental dynamics to holomorphic
semigroup dynamics.
Theorem 4.6.Let Sbe a transcendental semigroup such that F(S)has a mul-
tiply connected component. Then I(S)∩J(S)6=∅
Following result of Baker [1, Theorem 3.1] is better to worth mention.
Lemma 4.1.Let fbe a transcendental entire function and Ube a multiply
connected component of F(f). Then fn(z)→ ∞ locally uniformly on U.
Proof of the Theorem 4.6. Suppose F(S) has a multiply connected com-
ponent U. Then by theorem 4.1 (1), Uis also multiply connected component of
F(f) for each f∈Sand by lemma 4.1, for each f∈S,fn(z)→ ∞ locally uni-
formly on Uand also that fn(z)→ ∞ on ∂U. It follows by normality (that is,
by theorem 3.2) that every sequence in Shas a subsequence which diverges to ∞
locally uniformly on Uand ∂U. This proves that fn(z)→ ∞ for all z∈Uand
z∈∂U for each f∈Sand hence by theorem 4.1 (3) , U⊂I(S). Since ∂U ⊂J(f)
for all f∈S. By theorem 4.1 (2), ∂U ⊂J(S). This proves that I(S)∩J(S)6=∅.
The one of the most important result of classical complex dynamics is either
J(f) = Cor C∞or J(f) has empty interior for any holomorphic map fon Cor C∞
(see [3, Lemma 3]). There are lot of examples of transcendental entire functions
and rational functions whose Julia set is entire complex plane or extended complex
plane. Note that while J(f) = Cor C∞is possible for some holomorphic map f,
we always have F(f)6=Cor C∞. On the other hand, the analogous result is not
hold in semigroup dynamics. Hinkkanen and Martin [5, Example-1] provided the
following example that shows that Julia set of a rational semigroup Smay have
non-empty interior even if J(S)6=C∞.
Example 4.3.Rational semigroup S=hz2, z2/ai, where a∈C,|a|>1has
Fatou set F(S) = {z:|z|<1or |z|>|a|} and Julia set J(S) = {z: 1 6|z|6|a|}.
Let Ube a component of Fatou set F(f). Then f(U) is contained in some
component Vof F(f). Note that if fis rational map then V=f(V). If fis
transcendental, then it is possible that V6=f(U). Let us recall the following result
of Bergweiler and Rohde [3] of classical complex dynamics.
A SHORT COMPARISON OF CLASSICAL COMPLEX DYNAMICS AND HOLOMORPHIC... 7
Theorem 4.7.If fis entire, then V−f(U)contains at most one point which
an asymptotic value of f.
The following example of Huang Zhigang [10, Example 2] shows that above
result (theorem 4.7) can not be preserved for general semigroup dynamics. This is
a contrast between classical complex dynamics and semigroup dynamics.
Example 4.4.Let S=hzn, azni, where n > 2and |a|>1. The Fatou set F(S)
contains following components
U=(n
s1
|a|<|z|<n
v
u
u
tn−1
s1
|a|)and V={|z|>1}.
For a function f(z) = aznin semigroup S,f(U)⊂Vand V−f(U)is an unbounded
domain.
Definition 4.1.Let Sbe a holomorphic semigroup. We define the backward
orbit of any z∈C(or C∞) by
O−(z) = {w∈C∞:there exists f∈Ssuch that f(w) = z}
and the exceptional set of Sis defined by
E(S) = {z∈C∞:O−(z)is finite}
Note that if Sfinitely generated rational semigroup, then E(S)⊂F(S), oth-
erwise we can not assert it. For example ([10, example 1]), semigroup S=hfmi,
where fm(z) = amzn, m ∈N, n >2 and |a|>1, is an infinitely generated polyno-
mial semigroup. Then, E(S) = {0,∞}. It is easy to see that 0 is a limit point of
J(fm) = {|z|=|a|−m
n−1}, and hence 0 ∈J(S). So, in the case of finitely generated
rational semigroup S, we always have E(S)⊂F(S)⊂F(f) for any f∈S. Hence
E(S) contains at most two points. However, if Sfinitely generated transcendental
semigroup, then we can not assert E(S)⊂F(S) in general because for a transcen-
dental function, it is difficult to determine whether Fatou exceptional value belongs
Fatou set or Julia set. For example, 0 is the Fatou exceptional value of f(z) = eλz.
It is known in classical complex dynamics that 0 ∈J(f) if λ > 1/e and 0 ∈F(f)
if λ < 1/e. Poon and Yang [9] gave the following characterization whether a Fatou
exceptional value belongs to Fatou set or Julia set.
Theorem 4.8.Let fis transcendental entire function. If F(f)has no un-
bounded component, then Fatou exceptional value always belongs to Julia set.
So, in the case of finitely generated transcendental semigroup S, if E(f)⊂F(f)
for all f∈S, then we can say E(S)⊂F(S)⊂F(f) for any f∈S. Hence
E(S) contains at most one point. This fact is a generalization of classical complex
dynamics to semigroup dynamics and so it is a nice connection between these two
types of dynamics. Huang Zhigang [10, Proposition 1] proved the following result
which also shows a connection between classical complex dynamics and semigroup
dynamics.
Theorem 4.9.If z /∈E(S), then J(S)⊆O−(z).
8 B. H. SUBEDI AND A. SINGH
References
[1] Baker, I. N.: Wandering domains in the iteration of entire functions, Proc. London Math.
Soc., 49 (1984), 563-576.
[2] Bergweiler, W.: Iteration of meromorphic functions, Bull. (New Series) Amer. Math. Soc.
Vol. 29, Number 2, (1993), 131-188.
[3] Bergweiler, W. and Rohde, S.: Omitted values in the domains of normality, Proc. Amer.
Math. Soc. 123(1995), 1857-1858.
[4] Eremenko, A.: On the iterations of entire functions, Dynamical System and Ergodic Theory,
Banach Center Publication Volume 23, Warsaw, Poland, (1989).
[5] Hinkkanen, A. and Martin, G.J.: The dynamics of semigroups of rational functions- I, Proc.
London Math. Soc. (3) 73, 358-384, (1996).
[6] Kumar, D. and Kumar, S.: The dynamics of semigroups of transcendental entire functions-II,
arXiv: 1401.0425 v3 (math.DS), May 22, 2014.
[7] Kumar, D. and Kumar, S.: Escaping set and Julia set of transcendental semigroups,
arXiv:141.2747 v3 (math. DS) October 10, 2014.
[8] Poon, K.K.: Fatou-Julia theory on transcendental semigroups, Bull. Austral. Math. Soc. Vol-
58(1998) PP 403-410.
[9] Poon, K.K. and Yang, C. C.: Relations between Fatou exceptional value and component of
the Fatou Set, Preprint.
[10] Zhigang, H.: The dynamics of semigroup of transcendental meromorphic functions, Tsinghua
Science and Technology, ISSN 1007-0214 18/21, Vol 9, Number 4 (2004), 472-474.
Central Department of Mathematics, Institute of Science and Technology,
Tribhuvan University, Kirtipur, Kathmandu, Nepal
E-mail address:subedi.abs@gmail.com / subedi bh@cdmathtu.edu.np
Central Department of Mathematics, Institute of Science and Technology,
Tribhuvan University, Kirtipur, Kathmandu, Nepal
E-mail address:singh.ajaya1@gmail.com / singh a@cdmathtu.edu.np
