On the ‘Mandelbrot set’ for pairs of linear maps: asymptotic self-similarity

Abstract

We continue the investigation of iterated function systems (IFS) {λz, λz + 1} in the complex plane, depending on a parameter λ in the open unit disc. Let Aλ be the attractor, and let denote the connectedness locus, that is, the set of λ for which Aλ is connected. We show that the set is locally asymptotically self-similar and asymptotically similar to the attractor of the IFS {λz − 1, λz, λz + 1}, for certain ‘landmark’ points on the boundary of . We also study the parameters for which the attractor Aλ is tree-like, and prove that there are uncountable many such λ, answering a question of Bandt.