Continuity of Hausdorff Dimension of Julia-lavaurs Sets as a Function of the Phase

Let f0(z) = z2 + 1/4 and E0 the set of phases σ such that the critical point 0 escapes in one step by the Lavaurs map gσ ; it is a topological strip in the cylinder of phases whose boundary consists of two Jordan curves symmetric wrt R/Z. We prove that if σn ∈ E0 converges to σ ∈ ∂E0 in such a way that gσn (0) converges to gσ(0) along an external ray, then the Hausdorff dimension of the Julia-Lavaurs set J(f0, gσn) converges to the Hausdorff dimension of J(f0, gσ).