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Relation between Fatou, Julia and Escaping Sets of
a Holomorphic Semigroup and its Proper Subsemi-
groups
Bishnu Hari Subedi
Tribhuvan University, Central Department of Mathematics, Kathmandu, Nepal
Email: subedi.abs@gmail.com
Abstract
In this poster presentation, we show under what conditions the Fatou, Julia and escaping sets of a holomorphic semigroup
are respectively equal to the Fatou, Julia and escaping sets of its proper subsemigroups.
Key words: Fatou set, Julia set, escaping set, holomorphic semigroup, finite index, cofinite index, Rees index.
1 Introduction
•We confine our study to the Fatou, Julia and escaping sets of a holomorphic semigroup and its subsemi-
groups.
•A semigroup Sis a very classical algebraic structure with a binary composition that satisfies the associative
law. Semigroups arose naturally from the general mappings of a set into itself.
•Let F={fα}α∈∆.The holomorphic semigroup generated by Fis denoted by S=hfαi.The index
set ∆is allowed to be infinite in general unless stated otherwise.
•If ∆is a finite set, then we say Sis finitely generated. If F={f}, then Sis called a cyclic semigroup,
and we write S=hfi. In this case, each g∈Scan be written as g=fn, where fnis the nth iterate of
fwith itself.
•Sis called a rational semigroup or a transcendental semigroup depending on whether Fis a collection of
rational functions or transcendental entire functions.
•The family Fof holomorphic functions forms a normal family in a domain Dif every sequence (fα)⊆F
has a subsequence (fαk)which is uniformly convergent or divergent on all compact subsets of D.
•We say that semigroup Sis iteratively divergent at z∈Cif fn(z)→ ∞ as n→ ∞ for all f∈S.
•Like in classical holomorphic dynamics, the Fatou, Julia and escaping sets in the settings of a holomorphic
semigroup are defined as follows:
Definition 1.1 (Fatou, Julia and escaping sets).The 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 complement 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)an escaping point.
•Any maximally connected subset Uof the Fatou set F(S)is called a Fatou component.
•If S=hfi, then F(S), J(S)and I(S)are respectively denoted by F(f), J(f)and I(f). Let f∈S.
Then T=hfiis a cyclic subsemigroup of Sgenerated by f.
•The general relation between Fatou, Julia and escaping sets of a semigroup Sand its cyclic subsemigroup
Tis the following assertion.
Theorem 1.1. Let Sbe a holomorphic 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).
2 The relation between Fatou, Julia and escaping sets of a holomorphic
semigroup and its cyclic subsemigroups
•It is possible that the Fatou, Julia or escaping sets of a holomorphic semigroup may be equal, respectively,
to the Fatou, Julia or escaping sets of its cyclic subsemigroup. We investigate such type of holomorphic
semigroups on the basis of the following concepts.
•Let fbe a holomorphic function. Then
CV (f) = {w∈C:w=f(z)for some zsuch that f0(z) = 0}
is the set of critical values.
•The set AV (f)consisting of all w∈Csuch that there exists a curve Γ : [0,∞)→Cso that Γ(t)→ ∞
and f(Γ(t)) →was t→ ∞ is the set of asymptotic values of f.
•SV (f) = (CV (f)∪AV (f)) is the set of singular values of f.
•If SV (f)finite, then fis said to be of finite type. If SV (f)bounded, then fis said to be of bounded
type. The sets B={f:fis said to be bounded type }and S={f:fis said to be finite type }are
respectively known as Eremenko-Lyubich class and Speicer class.
•A holomorphic semigroup Sis said to be bounded type (or finite type) if each of its generators is of bounded
type (or finite type).
•Post-singular set of a holomorphic function fis defined by P(f) = Sn≥0fn(SV (f)).fis said to be
post-singularly bounded(or finite) if P(f)is bounded (or finite) set.
•A transcendental semigroup Sis said to be post-singularly bounded (or post-singularly finite) if each g∈S
is post-singularly bounded (or post-singularly finite).
•Holomorphic function fis said to be hyperbolic if P(f)is compact subset of F(f). A holomorphic
semigroup Sis said to be hyperbolic if each g∈Sis hyperbolic.
•The proposed relation holds for such a transcendental semigroup which is post singularly bounded or hy-
perbolic as shown in the following results.
Theorem 2.1 ([4, Theorems 3.8 and 3.14]).Let f∈B(or f∈S) periodic with period p and post
singularly bounded (or finite). Let g=fn+p, n ∈N. Then Then S=hf, giis post singularly
bounded (or finite) and all components of I(S)are unbounded.
Theorem 2.2 ([4, Theorem 3.16]).Let f∈Bperiodic with period p and hyperbolic. Let g=
fn+p, n ∈N. Then S=hf, giis hyperbolic and all components of I(S)are unbounded.
We generalize these results to the following assertion
Theorem 2.3 ([7, Theorem 3.1]).Let S=hf1, f2,...,fniis an abelian bounded (or finite) type tran-
scendental semigroup in which each fiis hyperbolic for i= 1,2,...,n. Then semigroup Sis hyperbolic,
and F(S) = F(f), J(S) = J(f)&I(S) = I(f)for all f∈S.
3 The relation between Fatou, Julia and escaping sets of a holomor-
phic semigroup and its general subsemigroups
–Theorem 1.1 can be extended for any general subsemigroup of a holomorphic semigroup as shown in the
following results.
Theorem 3.1. For any subsemigroup Tof a holomorphic semigroup S, we have F(S)⊂
F(T), J(S)⊃J(T)and I(S)⊂I(T).
–Fatou, Julia and escaping sets of a holomorphic semigroup may coincide with Fatou, Julia and escaping
sets of its subsemigroup on the basis of the following definition.
Definition 3.1 (Finite index and cofinite index).A subsemigroup Tof a holomorphic semigroup S
is said to be of finite index if there exists finite collection {f1, f2,...,fn}of elements of S1, where
S1=S∪ {Identity}, such that
S= (f1◦T)∪(f2◦T)∪. . . ∪(fn◦T)(3.1)
The smallest nthat satisfies 3.1 is called the index of Tin S. Similarly a subsemigroup Tof a holo-
morphic semigroup Sis said to be of cofinite index if there exists finite collection {f1, f2,...,fn}of
elements of S1such that for any f∈S, there is i∈ {1,2,...,n}such that
fi◦f∈T(3.2)
The smallest nthat satisfies 3.2 is called the cofinite index of Tin S.
–Note that the size of a subsemigroup Tof a semigroup Sis measured in terms of index. It is used to
measure the difference between a semigroup and a subsemigroup.
–If the subsemigroup Tis big enough in semigroup S, then Sand Tshare many properties as shown in
the following result:
Theorem 3.2 ([7, Theorem 1.1]).If a subsemigroup Thas finite index or cofinite index in an abelian
transcendental semigroup S, then I(S) = I(T), J(S) = J(T)and F(S) = F(T).
–There is another notion of index which is defined as follows.
Definition 3.2 (Rees index).Let Sbe a semigroup and Tbe a subsemigroup. The Rees index of Tin S
is defined as |S−T|+ 1. In this case, we say Tis large subsemigroup of Sand Sis a small extension
of T.
–This definition was first introduced by A. Jura [3] in the case when Tis an ideal of the semigroup S. In
such a case, the Rees index of Tin Sis the cardinality of factor semigroup S/T .
–From Definition 3.2, it is clear that the Rees index of Tin Sis the size of the complement S−T.
–For a subsemigroup to have finite Rees index in its parent semigroup is a fairly restrictive property, and it
occurs naturally in semigroups. (For instance, all ideals in the additive semigroup of positive integers are
of finite Rees index.)
–Note that Rees index does not generalize group index, and even the notion of finite Rees index does not
generalize finite group index. That is, if Gis an infinite group and Hproper subgroup, the group index
of Hin Gmay be finite even though the Rees index is infinite. In fact, let Gbe an infinite group and H
a subgroup of G; then Hhas finite Rees index in Gif and only if H=G.
–We prove the following dynamical similarity of a holomorphic semigroup and its subsemigroup.
Theorem 3.3 ([7, Theorem 3.1]).Let Tbe a large subsemigroup of finitely generated holomorphic semi-
group S. Then I(S) = I(T),F(S) = F(T)and J(S) = J(T).
4 Acknowledgments
I would like to thank CIMPA-CRM for a fellowship to take part in the intensive research program in low
dimensional dynamical systems and applications.
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