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Journal ArticleDOI

On the Julia set of analytic self-maps of the punctured plane

01 Jan 1995-Analysis (OLDENBOURG WISSENSCHAFTSVERLAG)-Vol. 15, Iss: 3, pp 251-256

Abstract: Let f be a non-constant and non-linear entire function, g an analytic self-map of C\{0}, and suppose that exp ◦f = g ◦ exp. It is shown that z is in the Julia set of f if and only if e is in the Julia set of g. 1991 Mathematics Subject Classification: 30D05, 58F23
Topics: Filled Julia set (73%), Julia set (68%), External ray (67%), Plane (geometry) (53%), Mathematics Subject Classification (53%)

Summary (1 min read)

1 Introduction and main result

  • The main objects studied in complex dynamics are the Fatou set F (f) which is defined as the set where the family {fn} of iterates of f is normal and the Julia set J(f) := D\F (f).
  • It is less obvious that the authors also have exp−1 J(g) ⊂ J(f)(4) and hence, together with (3), exp−1 J(g) = J(f).
  • This result is stated in [12, Lemma 1.2] and [20, Lemma 2.2], but I have been unable to follow the arguments for (4) given there.
  • The authors theorem may be useful to obtain results for analytic self-maps of C∗ from those for entire functions.
  • The authors also mention that it was used in [4, 7, 18] that (5) holds for certain particular examples of functions f and g satisfying (1).

2 Lemmas

  • The second claim is deduced from the first one by induction.
  • Then all components of F (f) are simply connected.
  • The conclusion now follows from [6, Theorem 10].

3 Points that tend to infinity under iteration

  • Eremenko [10] considered for transcendental entire f the set I(f) = {z : lim n→∞ |fn(z)| =∞} and proved that J(f) = ∂I(f).
  • Eremenko’s proof that I(f) 6= ∅ is based on the theory of Wiman and Valiron on the behavior of entire functions near points of maximum modulus [17, 30].
  • The authors summarize the above discussion as follows.

4 Proof of the theorem

  • The authors have already shown in the introduction (and also after Lemma 1) that (3) holds.
  • The conclusion then follows since J(f) is closed.
  • Combining this with (7) the authors find that |fnk(z2)| = o(Mnk)(8) as n→∞.
  • Hence <fn(z2)→∞ and <fn+1(z2)/<fn(z2)→∞ as n→∞ by (2).
  • The authors deduce that |fn(z2)| ≥ <fn(z2) ≥Mn for all large n, contradicting (8).

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On the Julia set of analytic self-maps of the
punctured plane
Walter Bergweiler
1
Received:
Abstract Let f be a non-constant and non-linear entire function, g an analytic self-map
of C\{0}, and suppose that exp f = g exp. It is shown that z is in the Julia set of f if
and only if e
z
is in the Julia set of g.
1991 Mathematics Subject Classification: 30D05, 58F23
1 Introduction and main result
Let f be an analytic self-map of a domain D
b
C, where
b
C = C {∞} denotes the
Riemann sphere. The main objects studied in complex dynamics are the Fatou set F (f)
which is defined as the set where the family {f
n
} of iterates of f is normal and the Julia
set J(f) := D\F (f). By Montel’s theorem J(f) = if
b
C\D contains more than two points.
Thus it suffices to consider the cases D =
b
C, D = C, and D = C
:= C\{0}. If D =
b
C, then
f is rational. This case was studied in long memoirs by Fatou [15] and Julia [19] between
1918 and 1920 and has been the object of much research in recent years, see the books
[5, 9, 29] for an introduction. The case that D = C so that f is entire was considered first
by Fatou [16] in 1926 and since then by many other authors, see [6] and [11, §4] for surveys.
This paper is concerned with the case D = C
which was studied first by Radstr¨om [27] in
1953 and more recently in [2, 8, 12–14, 20–26].
Given an analytic self-map g of C
there exists an entire function f satisfying
exp f(z) = g(e
z
)(1)
1
Supported by a Heisenberg Fellowship of the Deutsche Forschungsgemeinschaft

Bergweiler
for all z C. This function f is unique up to an additive constant which is a multiple of
2πi. It follows from (1) that
exp f
n
(z) = g
n
(e
z
)(2)
for n N. From (2) we can easily deduce that if exp z
0
F (g), then z
0
F (f ); that is,
exp
1
F (g) F (f).(3)
In fact, let U be a neighborhood of z
0
and V = exp U. If g
n
k
0 or g
n
k
in V as
k , then <f
n
k
−∞ or <f
n
k
and hence |f
n
k
| in U as k . If
g
n
k
ϕ 6≡ 0, in V , then |f
n
k
(z) ψ(e
z
)| = 2πim
k
+ o(1) for z U as k , where
m
k
Z and exp ψ = ϕ. Again we find that {f
n
k
} has a convergent subsequence. We thus
conclude that {f
n
} is normal in U if {g
n
} is normal in V . Hence (3) holds.
It is less obvious that we also have
exp
1
J(g) J(f)(4)
and hence, together with (3),
exp
1
J(g) = J(f).(5)
Consider for example f(z) = 2z and g(z) = z
2
. Then (1) holds and {f
n
} is normal in C
so
that J(f) = {0}, but J(g) = {z : |z| = 1}. Note, however, that in this example f is linear,
a case which is usually excluded in complex dynamics.
Theorem Let f be entire, g an analytic self-map of C
, and suppose that (1) holds. If f
is not linear or constant, then (5) holds.
This result is stated in [12, Lemma 1.2] and [20, Lemma 2.2], but I have been unable to
follow the arguments for (4) given there. The proofs of (3) in [12] and [20] are different
from the one given above. On the other hand, the question whether (5) holds was raised in
[24] and certain partial results were obtained. In particular, the above theorem answers the
question asked in [24] whether J(g) = C
implies that J(f) = C.
Our theorem may be useful to obtain results for analytic self-maps of C
from those
for entire functions. For example, it was proved in [2, 24] that if g is an analytic self-map
of C
, then the components of F (g) are simply or doubly connected. This result follows
immediately from our theorem and Lemma 3 below. As another example we mention the
results of Baker and Weinreich [4] on the boundary of unbounded invariant components of
the Fatou set of transcendental entire functions. Our theorem immediately yields analogous
results for analytic self-maps of C
. A further application concerns the Lebesgue measure
and the Hausdorff dimension of Julia sets of entire functions and analytic self-maps of C
,
see [13]. We also mention that it was used in [4, 7, 18] that (5) holds for certain particular
examples of functions f and g satisfying (1).
Acknowledgment I would like to thank Norbert Terglane and Steffen Rohde for useful
discussions and Yubao Guo for translating [12].
2 Lemmas
Lemma 1 Let f be a (non-constant and non-linear) rational function, entire function, or
analytic self-map of C
. Then J(f) is the closure of the set of repelling periodic points of f.

On the Julia set of analytic self-maps of the punctured plane
Recall that z
0
is called a repelling periodic point of f if f
n
(z
0
) = z
0
and |(f
n
)
0
(z
0
)| > 1 for
some n N, with a slight modification if f is rational and z
0
= . Lemma 1 is due to
Fatou [15, §30, p. 69] and Julia [19, p. 99, p. 118] for rational functions, Baker [1] for entire
functions, and Bhattacharyya [8, Theorem 5.2] for analytic self-maps of C
. A different
proof (that applies to all three cases) has recently been given by Schwick [28].
We note that Lemma 1 also gives a short proof of (3) because if z
0
is a repelling periodic
point of f, then exp z
0
is a repelling periodic point of g. This is the proof of (3) given in [12].
Lemma 2 Let f and g be as in the theorem. Then there exists ` Z such that f(z + 2πi) =
f(z) + `2πi. For n N and m Z we have f
n
(z + m2πi) = f
n
(z) + `
n
m2πi.
The first claim follows easily from (1). The second claim is deduced from the first one by
induction.
Lemma 3 Let f and g be as in the theorem. Then all components of F (f) are simply
connected.
To prove Lemma 3 we note that Lemma 2 implies that |f(it)| = O(t) as t , t > 0. The
conclusion now follows from [6, Theorem 10].
3 Points that tend to infinity under iteration
Eremenko [10] considered for transcendental entire f the set
I(f) = {z : lim
n→∞
|f
n
(z)| = ∞}
and proved that
J(f) = I(f ).(6)
The main difficulty is to prove that I(f) 6= . Once this is kown, (6) is not difficult to
deduce. Eremenko’s proof that I(f) 6= is based on the theory of Wiman and Valiron
on the behavior of entire functions near points of maximum modulus [17, 30]. His proof
shows that there exists z I(f) such that |f
n+1
(z)| M(|f
n
(z)|, f) as n , where
M(r, f) = max
|ζ|=r
|f(ζ)|. Since log M(r, f)/ log r as r for transcendental entire
f it follows that log |f
n+1
(z)|/ log |f
n
(z)| as n . We define
I
0
(f) =
(
z : lim
n→∞
log |f
n+1
(z)|
log |f
n
(z)|
=
)
and deduce that I
0
(f) 6= . Again it follows that J(f) = I
0
(f).
The theory of Wiman-Valiron and the arguments of Eremenko based on it do not require
that f is transcendental entire but only that f is analytic in a neighborhood of and that
is an essential singularity of f . In particular, the above arguments remain valid for analytic
self-maps of C
with an essential singularity at . We summarize the above discussion as
follows.
Lemma 4 Let f be a transcendental entire functions or an analytic self-map of C
with an
essential singularity at . Then I
0
(f) 6= and J(f) = I
0
(f).

Bergweiler
4 Proof of the theorem
If g is rational, then g(z) = cz
k
where c C
and k Z. It follows that f is linear or
constant, contradicting the hypothesis. Thus g is transcendental and there is no loss of
generality in assuming that is an essential singularity of g.
We have already shown in the introduction (and also after Lemma 1) that (3) holds.
It remains to prove (4). We thus assume that w
0
= exp z
0
J(g) and have to show that
z
0
J(f). Let U be a neighborhood of z
0
. We shall show that U J(f ) 6= . The conclusion
then follows since J(f) is closed.
By Lemma 1 and Lemma 4 there exist z
1
, z
2
U such that w
1
= exp z
1
is a repelling
periodic point of g, say g
k
(w
1
) = w
1
, and w
2
= exp z
2
I
0
(g). If z
1
J(f) or z
2
J(f),
then we are done. If z
1
and z
2
lie in different components of F (f), then we connect them by
a path in U. This path meets J(f) and again we have U J(f) 6= . Thus we may assume
that z
1
and z
2
lie in the same component of F(f). Since w
2
I
0
(g) I(g) we deduce from
(2) that z
2
I(f) and hence z
1
I(f).
By a result of Baker ([3, Lemma 1], see also [6, Lemma 7]) and Lemma 3 there exists a
constant C such that
|f
n
(z
2
)| C|f
n
(z
1
)|(7)
for all large n.
Since g
k
(w
1
) = w
1
we have exp f
k
(z
1
) = exp z
1
and hence f
k
(z
1
) = z
1
+ m2πi for some
m Z. By Lemma 4 we have
f
2k
(z
1
) = f
k
(z
1
+ m2πi) = f
k
(z
1
) + `
k
m2πi = z
1
+ m(1 + `
k
)2πi
and induction shows that
f
nk
(z
1
) = z
1
+ m
n1
X
j=0
`
jk
2πi.
We deduce that if M > max{1, |`|}, then
|f
nk
(z
1
)| = o(M
nk
)
as n . Combining this with (7) we find that
|f
nk
(z
2
)| = o(M
nk
)(8)
as n .
On the other hand, |g
n
(w
2
)| and log |g
n+1
(w
2
)|/ log |g
n
(w
2
)| as n by
the choice of w
2
. Hence <f
n
(z
2
) and <f
n+1
(z
2
)/<f
n
(z
2
) as n by (2). We
deduce that
|f
n
(z
2
)| <f
n
(z
2
) M
n
for all large n, contradicting (8). Thus z
1
and z
2
cannot lie in the same component of F (f)
and the proof of the theorem is complete.

On the Julia set of analytic self-maps of the punctured plane
References
[1] I. N. Baker, Repulsive fixpoints of entire functions, Math. Z. 104 (1968), 252-256.
[2] I. N. Baker, Wandering domains for maps of the punctured plane, Ann. Acad. Sci.
Fenn. Ser. A I Math. 12 (1987), 191-198.
[3] I. N. Baker, Infinite limits in the iteration of entire functions, Ergodic Theory Dynam-
ical Systems 8 (1988) 503-507.
[4] I. N. Baker and J. Weinreich, Boundaries which arise in the dynamics of entire func-
tions, Rev. Roumaine Math. Pures Appl. 36 (1991), 413-420.
[5] A. F. Beardon, Iteration of rational functions, Springer, New York, Berlin, Heidelberg
1991.
[6] W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N. S.)
29 (1993), 151-188.
[7] W. Bergweiler, Invariant domains and singularities, Math. Proc. Cambridge Philos.
Soc., to appear.
[8] P. Bhattacharyya, Iteration of analytic functions, PhD Thesis, University of London,
1969.
[9] L. Carleson and T. W. Gamelin, Complex dynamics, Springer, New York, Berlin,
Heidelberg 1993.
[10] A. E. Eremenko, On the iteration of entire functions, in Dynamical systems and ergodic
theory, Banach Center Publications 23, Polish Scientific Publishers, Warsaw 1989, 339-
345.
[11] A. E. Eremenko and M. Yu. Lyubich, The dynamics of analytic transforms, Leningrad
Math. J. 1 (1990), 563-634; translation from Algebra i Analiz 1 (1989).
[12] L. Fang, Complex dynamical systems on C
(Chinese), Acta Math. Sinica 34 (1991),
611-621.
[13] L. Fang, Area of Julia sets of holomorphic self-maps of C
, Acta Math. Sinica (N. S.)
9 (1993), 160-165.
[14] L. Fang, On the iteration of holomorphic self-maps of C
, preprint.
[15] P. Fatou, Sur les ´equations fonctionelles, Bull. Soc. Math. France 47 (1919), 161-271;
48 (1920), 33-94, 208-314.
[16] P. Fatou, Sur l’it´eration des fonctions transcendantes enti`eres, Acta Math. 47 (1926),
337-360.
[17] W. K. Hayman, The local growth of power series: a survey of the Wiman-Valiron
method, Canad. Math. Bull. (3) 17 (1974), 317-358.

Citations
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01 May 1999
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01 Jan 1998
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