We study a conformal map ƒ of the unit disk D onto a hyperbolically convex set in D, in particular the behaviour of ƒ on the preimage T ƒ = {z ∊ ∂D:|ƒ(z)| = 1} of ∂D. The main problem is how much the Hausdorff dimension can increase for sets on T ƒ. The case that ƒ(D) is bounded by full circles is treated in more detail. In this case ƒ can be written as a composition sequence of mappings onto halfplanes.