Local connectedness of Julia sets for transcendental entire functions

Shunsuke Morosawa

Department of Mathematics and Information Science, Faculty of Science, Kochi University, Kochi, 780-8520, Japan e-mail morosawa@math.kochi-u.ac.jp

1 Introduction

Let f be a transcendental entire function. We denote the n-th iteration of f by f^{n}. The maximal open subset of the complex plain ℂ where {f^{n}} is normal is called the Fatou set of f and is denoted by F (f). The complement of F (f) in ℂ̂ is called the Julia set of f and is denoted by J(f). Note that we regard ∞ is a point in the Julia set. Since J(f) is completely invariant, J(f) \ {∞} is unbounded in ℂ. Kisaka considered a topological property of the boundaries of unbounded Fatou components ([7]). In this paper, we consider bounded Fatou components. Stallard ([11]) showed that if a transcendental entire function has small growth order, then every Fatou component is bounded.

We call ζ ∈ ℂ a singular value if f is not a smooth covering map over any neighborhood of ζ. It plays an important role in studying its dynamics. We denote the set of them by sing(f^{−1}) and set

S = {f | f is transcendental entire and sing(f^{−1}) is a finite set}.

For f ∈ S, every point of sing(f^{−1}) is either an asymptotic value of f or a critical value of f. It is known that if f ∈ S, then F (f) does not contain neither a wandering domain nor a Baker domain ([4]). Hence if f ∈ S, then we have {z | f^{n}(z) → ∞} ⊂ J(f).

In § 2, we give two conditions on singular values to treat Julia sets easier just like those of hyperbolic rational functions. Under those conditions, we show that the boundary of a bounded Fatou component is a Jordan curve and further that if all the cyclic components of the Fatou set are bounded, then its Julia set is locally connected. In § 3, we consider the family {f_{λ}(z) = λze^{z} | λ ∈ ℂ}, which is contained in S. By using curves in Julia sets, we show that some functions in the family have only bounded Fatou components. Curves in Julia sets of exponential maps were first studied by Devaney and Krych ([2]) as Cantor bouquets. Finally, in § 4, we give an example of a transcendental entire function whose Julia set is a Sierpinski carpet.

Some pictures of Fatou set stated in this note can be seen in the author’s home page^{1}.

2 Local connectedness

For a transcendental entire function f, we give two conditions on singular values.

[SV1] If ζ ∈ F (f) ∩ sing(f^{−1}), then it is a critical value and is absorbed by an attracting cycle.

[SV2] If ζ ∈ J(f) ∩ sing(f^{−1}), then

\overline{∪_{n≥0}f^{n}(ζ)} ∩ ∂D = ∅

for any Fatou component D.

Set

SV = {f ∈ S | f satisfies [SV1] and [SV2] }.

Note that, for f ∈ SV, every cyclic component of F (f) is attracting. We also remark that, for f ∈ SV, it is not necessary that its cyclic components are bounded. For example, let f(z) = λ sin z with λ ∈ ℝ, |λ| < 1. Then f ∈ SV and F (f) has an unbounded attracting component.

Theorem 1 Let D be a cyclic Fatou component of f ∈ SV. If D is bounded, then ∂D is a Jordan curve.

Proof. It is enough to show the case that D is a fixed component. By using the argument similar to that used to prove that the boundary of a simply connected attracting component of a hyperbolic rational function is a closed curve ( see e.g. [1]), we prove the theorem.

First, we show

lim_{n→∞} inf_{z∈∂D} |(f^{n})′(z)| = +∞.

Assume that there exist ϵ > 0 and an infinite set N of positive integers such that, for n ∈ N, z_{n} ∈ ∂D and branches g_{n} of (f^{n})^{−1} at z_{n} with g_{n}(z_{n}) ∈ ∂D satisfying |g′_{n}(z_{n})| > ϵ. If necessary, taking subsequence, we may assume z_{n} → ζ ∈ ∂D. Take a neighborhood U of ζ so that

\overline{∪_{n≥0}f^{n}(sing(f^{−1}))} ∩ U = ∅.

Choose a branch h_{n} of (f^{n})^{−1} in U so that h_{n}(z_{n}) = g_{n}(z_{n}) for z_{n} ∈ U. Since U intersects the Julia set, an arbitrary locally uniform limit of a subsequence of {h_{n}} is constant.

1http://www.math.kochi-u.ac.jp/morosawa/indexe.html

Further, it is not infinity, for D is bounded. Hence we have h′_{n}(z) → 0 locally uniformly in U, which implies g′_{n}(z_{n}) → 0. This contradicts our assumption.

Take w ∈ ∂D and set z_{n} = f^{n}(w). Choose a branch g_{n} of (f^{n})^{−1} with g_{n}(z_{n}) = w. We have

f^{n}(g_{n}(z)) = z and |(f^{n})′(g_{n}(z))||g′_{n}(z)| = 1

for all z in some neighborhood of z_{n}. From the above, for all ϵ > 0, there exists n_{0} such that

1 = |(f^{n})′(w)||g′_{n}(z_{n})| < ϵ|(f^{n})′(w)|

for all n ≥ n_{0}. Hence we have |(f^{n})′(w)| > 1/ϵ.

Since D is an attracting component of a transcendental entire function, it is connected. The assumption that f ∈ SV implies that f is a finite branched covering of D. We denote the local degree of f in D by p. Hence f|_{D} is conformally conjugate to a finite Blaschke product of degree p in the unit disk. Thus there exist a compact set E of D, A = {z | r < |z| < 1} and a homeomorphism φ from A onto D \ E which satisfies

f ∘ φ(z) = φ(z^{p})

for z ∈ {z | r^{1/p} < |z| < 1}. Since f is expanding on some neighborhood of ∂D as we show above, φ can be extended continuously to a map from A′ = \{z | r < |z| ≤ 1\} onto \overline{D} \ E. We again denote it by φ. Hence ∂D = φ(S^{1}) is a closed curve, where S^{1} = {z | |z| = 1}.

Assume that there exist distinct points z_{1} and z_{2} in S^{1} such that φ(z_{1}) = φ(z_{2}). Take a simple curve γ in A′ whose end points are z_{1} and z_{2}. Then Γ = φ(γ) is a simple closed curve in \overline{D}. Since \{f^{n}\} is bounded on Γ, from the maximum modulus theorem, it is also bounded on the bounded complementary component G of Γ. This contradicts that G intersects the Julia set. Hence ∂D is a Jordan curve.

Theorem 2 Let f be in SV. If all the cyclic components of F (f) are bounded, then J(f) is locally connected.

Proof. A theorem due to Whyburn ([14]) shows that a closed set in ℂ̂ is locally connected if and only if it satisfies following two conditions.

(1) The boundary of each its complementary component is locally connected.

(2) For an arbitrary ϵ > 0, the number of the complementary components whose diameters with respect to the spherical distance exceed ϵ is finite.

From Theorem 1, the boundary of each cyclic component is locally connected. Since f is in SV, there exist no wandering components. Hence, an arbitrary component of F (f) is mapped to some cyclic component by finitely many iteration of f. Thus its boundary is locally connected because of the condition [SV2].

Next, we show J(f) satisfies the condition (2). Take a cyclic component D. Since \overline{D} does not contain any asymptotic value, an arbitrary component of f^{−n}(D) is bounded for all n.

We show that, for an arbitrary ϵ > 0, the number of the components of the backward orbit of D whose diameters exceed ϵ is finite, because the number of the cyclic component of F (f) is finite for f ∈ S. Assume that there exist a component D_{0} of F (f) and components D_{n} (n ≥ 1) of F (f) each of whose diameter exceeds ϵ such that

f^{j(n)}(D_{n}) = D_{0} and (∪_{n≥1}D_{n}) ∩ (∪f^{n}(sing(f^{−1})) = ∅.

From the fact that ∂D_{0} is a Jordan curve and the assumption that f satisfies [SV2], we can take a neighborhood U of \overline{D_{0}} such that a branch g_{n} on U of f^{−j(n)} satisfying g_{n}(D_{0}) = D_{n}. Then {g_{n}} is a normal family on U. Since U intersects the Julia set, every its limit function is constant. This is a contradiction.

3 Julia sets of f_{λ}(z) = λze^{z}

We set

D = {f_{λ}(z) = λze^{z} | λ ∈ ℂ}.

Iteration of elements of D are studied by some mathematicians (e.g. [5], [6], [8] and [12]). In the case that λ = 0, we regard that f_{0} has an attracting fixed point 0 for convenience. For f_{λ}(z) ∈ D, we see

sing(f_{λ}^{−1}) = \{f_{λ}(−1) = \frac{λ}{e}, 0\}

and the set of the fixed points of f_{λ} is

{0} ∪ {−Log λ + 2nπi}_{n∈ℤ}.

We define

B = {λ ∈ ℂ | f_{λ}^{n}(−1) ↛ ∞}

and further

B_{−1} = {λ ∈ ℂ | |λ| < 1} and B_{1} = {λ ∈ ℂ | |1 − Log λ| < 1}.

Proposition 3 If λ ∉ B, then J(f_{λ}) = ℂ̂. The function f_{λ}(z) has an attracting fixed point if and only if λ ∈ B_{−1} ∪ B_{1}. Further, if λ ∈ B_{−1}, then 0 is the attracting fixed point and f_{λ}^{n}(−1) tends to 0 as n tends to infinity and if λ ∈ B_{1}, then −Log λ is the attracting fixed point.

Proof. Since fixed points of f_{λ} are 0 and −Log λ+2nπi, it is clear that f_{λ} has an attracting fixed point if and only if λ ∈ B_{−1} ∪ B_{1} and if λ ∈ B_{−1}, then 0 is the attracting fixed point and if λ ∈ B_{1}, then −Log λ is the attracting fixed point.

We denote a neighborhood of the origin by U_{ϵ} = {z | |z| < ϵ}. In the case that λ ∈ B_{−1}, we can choose ϵ so small that f^{−1}(U_{ϵ}) consists of two components and one of which contains U_{ϵ} and the other does not intersect U_{ϵ}, because the origin is an asymptotic value and has only one inverse image itself. This means the attracting component containing the origin is completely invariant, since f_{λ} is not univalent on it. Hence it contains the critical point of f, which is −1.

If λ ∉ \overline{B_{−1}}, then the asymptotic value 0 is a repelling fixed point. Since f_{λ} has only two singular values f_{λ}(−1) and 0, we have J(f_{λ}) = ℂ̂ for λ ∉ B by the relations between cyclic Fatou components and singular values.

Let

Λ = {λ ∈ ℂ | |Im λ| ≥ eArg λ}.

In [5], Fagella showed the following.

Theorem 4 If λ ∈ Λ, then there exists a curve L(t) (t ∈ [0, ∞)) satisfying the following conditions.

(1) f_{λ}(L(t)) = L(t).

(2) For sufficiently large h

L(t) ∩ H_{h} ⊂ {z | −π − θ ≤ Im z ≤ π − θ},

where H_{h} = {z | Re z ≥ h} and θ = Arg λ.

(3) L(0) = 0.

(4) For t ≠ 0

lim_{n→∞} f_{λ}^{n}(L(t)) = ∞.

Theorem 5 If λ ∈ Λ and f_{λ}(z) = λze^{z} has an attracting cycle whose period is greater than one, then each component of F (f_{λ}) is bounded and J(f_{λ}) is locally connected.

Proof. Assume that the period of the attracting cycle is p(> 1). Let D_{0} be the attracting component of this cycle containing −1. If ∂D_{0} contains the fixed point 0, then so does each ∂f_{λ}^{k}(D_{0}) for k = 1, 2, ⋯, p − 1. Hence each f_{λ}^{k}(D_{0}) contains a curve terminating at 0 satisfying that the curves are permuted by f_{λ}. From Theorem 4, there exists an invariant curve whose end point is 0, which we denote by L. This is a contradiction. Hence there exists a neighborhood U_{ϵ} = {z | |z| < ϵ} of 0 such that U_{ϵ} does not intersect each of the attracting cyclic components. If necessary, we take ϵ sufficiently small so that f_{λ}^{−1}(U_{ϵ}) consists of two simply connected components. We denote the unbounded one by H, which is in the left half plane. Each component of the inverse image of L tends to infinity in both direction and intersects with H in one direction. Further, for all sufficiently large h and n ∈ ℤ \ {0}

{z | Re z > h and 2nπ − θ < Im z < 2(n + 1)π − θ}

intersects a component of the inverse image of L, where recall θ = Arg λ. Choose l_{1} and l_{2} from the components of the inverse image of L such that the domain bounded by them contains −1 and does not contain any its component. Since l_{1} and l_{2} is in J(f_{λ}), D_{0} is contained in the domain bounded by l_{1}, l_{2} and ∂H. Assuming that D_{0} is unbounded and hence so is D_{k} = f_{λ}^{k}(D_{0}) for k = 1, 2, ⋯, p − 1, we have

(∪_{k=0}^{p−1}D_{k}) ∩ {z | Re z > h} ⊂ {z | Re z > h and |Im z| < M}

for some M. Choose h_{1} and set

R(h_{1}) = {z | Re z > h_{1} and |Im z| < M}

so that

|f_{λ}(z)| = |λ||z|e^{Re z} > Re z + 2M

for every z ∈ R(h_{1}). Hence we have Re f_{λ}^{p}(z) > Re z for every z ∈ D_{0}∩R(h_{1}). This implies lim_{n→∞} f^{np}(z) = ∞ for z ∈ D_{0} ∩ R(h_{1}). On the other hand, for z ∈ D_{0}, f^{np}(z) tends to the attracting fixed point of f^{p} in D_{0}. This is a contradiction. Thus D_{0} is bounded and so is D_{k}. Since sing(f_{λ}^{−1}) = {0, f_{λ}(−1)} and the asymptotic value is a repelling fixed point, {D_{k}}_{k=0}^{p−1} is the only one set of cyclic components F (f_{λ}) contains. By Theorem 2, we see that J(f_{λ}) is locally connected.

We call the subset of the Julia set whose points do not lie on the boundary of any component of the Fatou set the residual Julia set. Residual Julia sets for rational functions are considered in [10] and those for entire functions are considered in [3].

Theorem 6 If λ ∈ Λ and f_{λ}(z) = λze^{z} has an attracting cycle whose period is greater than one, then L(t) defined in Theorem 4 is contained in the residual Julia set of J(f_{λ}).

Proof. As was shown in the proof of Theorem 5, a fixed point L(0) = 0 is not on any cyclic component of F (f_{λ}). Since any component of F (f_{λ}) is eventually mapped into the immediate basin containing −1, 0 is not on the boundary of any Fatou component.

Theorem 5 shows that every component of F (f_{λ}) is bounded. Further it is not wandering. On the other hand, for t ≠ 0, f_{λ}^{n}(L(t)) tends to infinity. Hence L(t) does not intersect the boundary of any component of F (f_{λ}).

4 Another example

We say that a closed subset in ℂ̂ is a Sierpinski carpet if it is the complement of a countable dense family of open topological discs whose diameters tend to zero and whose closure are pairwise disjoint closed topological discs. Note that any two Sierpinski carpets are homeomorphic. Some rational functions whose Julia sets are Sierpinski carpets are known (see [9] [13]). We give an example of a transcendental entire function whose Julia set is a Sierpinski carpet by the argument similar to that in the previous section.

Theorem 7 Let

g_{a}(z) = ae^{a}{z − (1 − a)}e^{z}

for a > 1. Then J(g_{a}) is locally connected. Moreover, it is a Sierpinski carpet.

Proof. It is clear that sing(g_{a}^{−1}) = {−a, 0}, where −a is a critical value and 0 is an asymptotic value. Hence g_{a} ∈ S. Since g_{a}(−a) = −a, −a is a super-attracting fixed point. We denote the attracting component containing −a by D. By using the graphical analysis, it is easy to see that lim_{n→∞} g_{a}^{n}(x) = ∞ for all x > b, where b is the fixed point satisfying −a < b < 0. Hence the interval [b, ∞) in ℝ is contained in J(g_{a}) and especially the asymptotic value 0 is contained in J(g_{a}).

Set

l^{+} = \{z = x + iy | x = 1 − a − y \frac{cos y}{sin y} , π < y < 2π\} and
l^{−} = \{z = x + iy | x = 1 − a − y \frac{cos y}{sin y} , −2π < y < −π\} .

Then each of g_{a}(l^{+}) and g_{a}(l^{−}) is a positive real axis and hence is contained in J(g_{a}). It is easy to see

lim_{y→π+0} 1 − a − y \frac{cos y}{sin y} = −∞  lim_{y→2π−0} 1 − a − y \frac{cos y}{sin y} = +∞
lim_{y→−π−0} 1 − a − y \frac{cos y}{sin y} = −∞  lim_{y→−2π+0} 1 − a − y \frac{cos y}{sin y} = +∞.

Denoting the domain bounded by l^{+} and l^{−} by E, we have E ⊃ D. Choose h so that |g_{a}(z)| ≥ |z| + 4π for all z satisfying Re z > h. Because g_{a}^{n}(z) tend to −a for every z ∈ D, we have Re z < h for z ∈ D. Let N be the smallest number such that g_{a}^{N}(0) > h. Choose ϵ > 0 so small that g_{a}^{N}(U_{ϵ}) ⊂ H_{h}, where U_{ϵ} = {z | |z| < ϵ} and H_{h} = {z | Re z > h}. This implies D ∩ U_{ϵ} = ∅. Let K be an unbounded component of g_{a}^{−1}(U_{ϵ}). Then ∂K intersects with l^{+} and l^{−}. Hence we see that D is bounded.

Since D is the only one cyclic component of F (g_{a}) and F (g_{a}) has no wandering components, every components of F (g_{a}) is eventually mapped to D. From Theorem 2, we see that J(g_{a}) is locally connected, for D is an attracting component.

Assume that there exist two components D_{1} and D_{2} of F (g_{a}) such that ∂D_{1} ∩ ∂D_{2} ≠ ∅ and take ζ ∈ ∂D_{1} ∩ ∂D_{2}. Since they are eventually mapped to D, we can take M such that

g_{a}^{M}(D_{1}) ∩ g_{a}^{M}(D_{2}) = ∅ and g_{a}^{M+1}(D_{1}) ∩ g_{a}^{M+1}(D_{2}) ≠ ∅.

This implies g_{a}^{M}(ζ) is a critical point of g_{a}. This is a contradiction. Therefore J(g_{a}) is a Sierpinski carpet.

References

[1] A. F. Beardon, Iteration of Rational Functions, GTM 132, Springer, 1991.

[2] R. L. Devaney & M. Krych, Dynamics of exp(z), Ergod. Th. & Dynam. Sys., 4(1984), 35-52.

[3] P. Domínguez, Connectedness properties of Julia sets of transcendental entire functions, Complex Variables Th. Appl., 32(1997), 199-215.

[4] A. E. Eremenko & M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier Grenoble, 42(1992), 989-1020.

[5] N. Fagella, Limiting dynamics for the complex standard family, Int. J. Bifurcation and Chaos, 5(1995), 673-699.

[6] C. M. Jang, Julia set of the function z exp(z +μ), Tôhoku Math. J., 44(1992), 271-277.

[7] M. Kisaka, On the connectivity of Julia sets of transcendental entire functions, Ergod. Th. & Dynam. Sys., 18(1998), 189-205.

[8] T. Kuroda & C. M. Jang, Julia set of the function z exp(z + μ) II, Tôhoku Math. J., 49(1997), 577-584.

[9] J. Milnor, Geometry and dynamics of quadratic rational maps, Experimental Math., 2(1993), 37-83.

[10] S. Morosawa, On the residual Julia sets of rational functions, Ergod. Th. & Dynam. Sys., 17(1997), 205-210.

[11] G. M. Stallard, The iteration of entire functions of small growth, Math. Proc. Camb. Phil. Soc., 114(1993), 273-298.

[12] M. Taniguchi, On topological completeness of decorated exponential families, Sci. Bull. Josai Univ., Special Issue, 4(1998), 1-10.

[13] K. M. Pilgrim, Cylinders for iterated rational maps, Thesis, University of California, Berkeley, 1994.

[14] G. T. Whyburn, Analytic Topology, Amer. Math. Soc. Coll. Publ., 28(1942)