Fix θ0 = [a1, a2, ...] (continued fraction) of bounded type with all coefficients ai greater

than the Shishikura-Inou constant S. Choosing n and N great enough (in this order),

take θ = [a1, ..., an, N, S, S, S, ...]. Set fθ(z) = e

2iπθ

z + z

2

, denote by Kθ and ∆θ its filled

Julia set and its Siegel disc; define in the same way f0, K0 and ∆0.

Choose δ very small and define the annulus W = {z ∈ C|2δ < d(z, ∆0) < 10δ} , so

that a point in ∆0 cannot escape to infinity under fθ without stepping in W . By JSPM

II, we can suppose that (∀z ∈ ∆θ)d(z, ∆0) <

1

2

δ. Denote by Xp the set of points in ∆0

which bounce at least p times between ∆0 and W under fθ, and set X∗

p = Xp − Xp+1.

Using Mc Mullen’s density theorem, Koebe distortion estimates and JSPM III, we get

estimates on the measure of Xp and X∗

p

, and finally show that m(∆0 \Kθ) =

Pm(X∗

2p+1)

can be made arbitrarily small. From this the construction of a τ such that Kτ has empty

interior but positive measure follows easily.