Almost Regular Closedness of the Connectedness Locus for Pairs of Affine Maps on $\mathbb{R}^2$

TL;DR

This paper investigates the topological structure of the connectedness locus for pairs of affine maps in the plane, showing it is mostly regular closed except possibly at isolated points, using the method of traps.

Contribution

It proves the connectedness locus is regular closed away from the diagonal, extending understanding of its topological properties and applying the method of traps.

Findings

Connectedness locus is regular closed away from the diagonal.

Potential isolated points in the locus are conjectured not to exist.

The method of traps is central to the proof.

Abstract

We study the connectedness locus $\mathcal{N}$ for the family of iterated function systems of pairs of homogeneous affine-linear maps in the plane. We prove this set is regular closed (i.e., it is the closure of its interior) away from the diagonal, except possibly for isolated points, which we conjecture do not exist. We provide an overview of the "method of traps", introduced by Calegari et al. (2017), which lies at the heart of our proof.