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
n−1
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
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