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.