Connectedness locus for pairs of affine maps and zeros of power series

Abstract

We study the connectedness locus $\mathcal{N}$ for the family of iterated function systems of pairs of affine-linear maps in the plane (the non-self-similar case). First results on the set $\mathcal{N}$ were obtained in joint work with P. Shmerkin [11]. Here we establish rigorous bounds for the set $\mathcal{N}$ based on the study of power series of special form. We also derive some bounds for the region of “$\ast$-transversality” which have applications to the computation of Hausdorff measure of the self-affine attractor. We prove that a large portion of the set $\mathcal{N}$ is connected and locally connected, and conjecture that the entire connectedness locus is connected. We also prove that the set $\mathcal{N}$ has many zero angle “cusp corners,” at certain points with algebraic coordinates.