Filled Julia sets

In the case that the Fatou set of the limit function g_λ has no parabolic cycles and no wandering domains, we are able to prove a slightly stronger result than the Hausdorff convergence of the Julia set. In this section we prove the Hausdorff convergence of the filled Julia set under the assumptions above. Since in this case Siegel discs or other attracting basins are admissible, we use parts of the results stated below to prove Theorem 3.27 on the Hausdorff convergence of Julia sets in the following section.

In Section 3.3 we proved the kernel convergence under the assumption that the Fatou set F(g_λ) consists only of a Baker domain or contains a parabolic basin.

We approximated those components by basins of attraction ofgλ,µ. The cases not considered until now are thatg_λ has an attracting basinA₀containing the unique free critical pointc₀or a Siegel disc, in which case an irrational fixed point belongs to the Fatou set.

Kriete consider in [47] a family of meromorphic functionsfn converging uniformly on C to a meromorphic function f. He proved the Hausdorff convergence of the filled Julia set, whenf fulfils certain conditions, that is, whenf has no wandering domains, Baker domains or rationally indifferent cycles. In our case, the functions g_λ do not fulfil these conditions. Hence we can not directly apply his result but we are able to adapt his definitions and results for our particular case. We begin by defining the filled Julia set in an analogue way to the well-known filled Julia set.

In the case of polynomials, ∞ is always an attracting fixed point. Therefore the filled Julia set is defined as the complement of the basin of attraction of ∞.

For rational maps the point at infinity is a regular point and it can present any behaviour. For transcendental entire functions the point at ∞ is an essential singularity and is always contained in the Julia set. In our case, F(g_λ) has a Baker domain B_λ for every λ where the iterates converge to ∞. Hence it make sense to define thefilled Julia set as the complement of B_λ

K(g_λ) :=C\ B_λ. For the approximating functionsgλ,µ we define

K(g_λ,µ) :=C\ A_µ,

where A_µ is the attracting basin converging to B_λ asµ tends to zero. Note that K(g_λ) andK(g_λ,µ) contain the Julia setsJ(gλ) andJ(gλ,µ) respectively. We state the result for these sets as follows.

3.4. Filled Julia sets 121

Theorem 3.26. Let λ ∈ C^∗ be arbitrary but fixed such that g_λ has no wander-ing domains and no parabolic cycles. Then K(g_λ,µ) converges to K(g_λ) in the Hausdorff metric as µ tends to zero.

Proof. Recall thatK(g_λ) andK(g_λ,µ) are closed and completely invariant underg_λ and g_λ,µ respectively. From the Kernel convergence ofA_µtoB_λ there exists aµ∗

such that if an open setU ⊂ B_λ, thenU is contained inA_µfor everyµ≤µ∗. For someε >0 consider the neighbourhood Uε(K(g_λ)). The latter argument implies that K(g_λ,µ) ⊂ U_ε(K(g_λ)) for every µ ≤ µ∗. Therefore to prove the Hausdorff convergence we only need to show that K(g_λ)⊂Uε(K(g_λ,µ)) for any given ε >0.

Assume that this is false. Then there exists some ε > 0 and µ0 ∈ 0,¹₂ such that K(g_λ) * U_ε(K(g_λ,µ)) for every µ ≤ µ₀. Since K(g_λ) has no isolated points (otherwiseJ(gλ) would have some) there existsz0∈ K(g_λ) withU_ε/2(z0)∩K(g_λ)6=

∅ and U_ε/2(z0)∩ K(g_λ,µ) = ∅ for every µ ≤ µ0. Let {µ_k}_k∈N be an arbitrary sequence of parameters in (0, µ₀] such that µ_k → 0 as k → ∞. For each µ_k we consider points zµk ∈ K(g_λ,µk) and define the set of accumulation points of sequences {z_µk}_k∈Ndenoted by E.

We have thatE ⊂ K(g_λ) butE 6=K(g_λ). Using the lower-semicontinuity described in Section 1.9 we obtain thatJ(gλ)⊂Uε(J(gλ,µ)) for each givenε >0 and every µ≤µ0. Combining the three previous statements yieldsJ(g_λ)⊂ E, recalling that J(g_λ) =∂K(g_λ).

Thus, there exists some domain G ⊂⊂ K(g_λ)\ E such that G∩ K(g_λ,µ) = ∅ for all µ ≤ µ0. By assumption g_λ has no wandering domains, parabolic cycles and any further Baker domain different from B_λ. SinceG∩ J(g_λ) =∅ andK(g_λ) and E are invariant under gλ we may assume that G lies in a periodic component of F(g_λ). After switching to an appropriate iterate ofg_λ we may assume thatG lies in an invariant component of F(g_λ). The following two cases may occur.

i) G⊂⊂ Afor some component ofF(gλ) containing an attracting periodic point z0, or

ii) Glies in a Siegel disc of F(g_λ).

Now we consider both cases separately.

i) We have thatG⊂⊂ A ⊂int(K(g_λ)) andG⊂⊂Cb\ J(g_λ)⊂C. SinceGis not contained inB_λ, then G⊂ A₀, whereA₀ is the basin of attraction containing the unique free critical point c0. Using the persistence of attracting cycles, we conclude that Glies in the basin of some attracting periodic pointz_0,µ of

122 3. Convergence of the familiesg_λ,µ to g_λ

g_λ,µ different from A_µ. In other words, G⊂ A_0,µ, the basin of attraction of z0,µ, and G⊂Cb\ A_µ=K(g_λ,µ) for every µ≤µ0, which is a contradiction.

ii) IfG lies in a Siegel disc S, then there exists some m ∈N such that afterm iterates Sm

k=0g_λ^m(G) ⊂⊂ K(g_λ)\ K(g_λ,µ) contains a curve γ invariant with respect to g_λ. On the one hand, this implies thatγ andg_λ(γ) are homotopic in C\ K(g_λ,µ) for every µ≤µ₀. This in turn implies that γ and g_λ,µ(γ) are also homotopic in C\ K(g_λ,µ) for every µ ≤ µ₀. Then g_λ,µ(γ) and g_λ,µ² (γ) are homotopic inC\g_λ,µ(K(g_λ,µ)) =C\ K(g_λ,µ). Continuing inductively, we obtain that γ is homotopic tog_λ,µ^k (γ) in C\ K(g_λ,µ) for every k∈N. On the other hand, γ ⊂ A_µ=C\ K(g_λ,µ). Theng_λ,µ^k (γ) is homotopic to a point for sufficiently largek. Finally γ is homotopic to a point in C\ K(g_λ,µ).

Denote byG1the component ofCb\γ containing∂Sand byG2the other com-ponent. In particular, G₂ contains the centre ξ₂ of the Siegel disc S, which is an irrational fixed point of gλ. Recall that∂S ⊂ J(gλ). The density of re-pelling periodic points in the Julia set implies the existence of a rere-pelling point ζ₁ ∈ G₁ ∩ J(g_λ). Now, Rouch´e’s Theorem yields the existence of repelling periodic pointsζµ of gλ,µin G1 for everyµ≤µ0 such that limµ→0ζµ=ζ1. We have thatγ ⊂C\K(g_λ,µ) =A_µwhich impliesG2⊂ A_µ. The homotopy of γ to a point inC\ K(g_λ,µ) together again with Rouch´e’s Theorem implies the existence of fixed points ξµ of gλ,µ which converge toξ2. The latter implies ξµ∈G2 for all µ≤µ0. Henceξµ∈ A_µ and each fixed pointξµ must coincide with the attracting fixed pointz_µ. We obtain limµ→0z_µ= limµ→0ξ_µ=ξ₂, a contradiction.

The theorem above shows that the biggest difficulty to prove convergence in a dynamical sense comes from the presence of wandering domains or parabolic cy-cles in the Fatou set of the limit function. In the present case the convergence of the filled Julia set K(g_λ,µ) toK(g_λ) implies automatically the convergence of the (usual) Julia sets under the assumption that gλ has no parabolic cycles and the Siegel disc is of period one.

Remark 3.2. Concerning the case thatgλ has a Siegel disc S, note that the proof above does not specify through which kind of Fatou component it is approximated.

If the centre of S is an indifferent fixed point, we can approximate it through basins of attracting fixed points. Nevertheless, if the centre is an indifferent period point, we cannot guarantee the existence of a sequence of attracting periodic points