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

TL;DR

This paper investigates the connectedness locus for pairs of affine maps in the plane, providing bounds, structural properties, and conjectures about its connectedness and geometric features, with implications for self-affine attractors.

Contribution

It establishes rigorous bounds for the connectedness locus using power series analysis and explores its topological and geometric properties, including connectedness, local connectedness, and cusp corners.

Findings

Large portion of the set N is connected and locally connected

Established bounds for the set N based on power series analysis

Identified zero angle cusp corners at algebraic points

Abstract

We study the connectedness locus 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 N were obtained in joint work with P. Shmerkin (2006). Here we establish rigorous bounds for the set N based on the study of power series of special form. We also derive some bounds for the region of "*-transversality" which have applications to the computation of Hausdorff measure of the self-affine attractor. We prove that a large portion of the set N is connected and locally connected, and conjecture that the entire connectedness locus is connected. We also prove that the set N has many zero angle "cusp corners," at certain points with algebraic coordinates.