In the hyperbolic case, the Julia-Lavaurs sets can be defined as possible limits of the Julia sets of the polynomials $z^2+c$ in the Hausdorff metric, when the parameter $c$ tends to $1/4$. We will study continuity of the Hausdorff dimension of the Julia-Lavaurs sets on the boundary of the hyperbolic component containing the real line.
The talk will follow the speaker's paper in Fund. Math. 214 (2011), no. 2, 119-133.