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

Omer Rosler

TL;DR

This work studies the connectedness locus ℵ for pairs of homogeneous affine maps on ℝ^2 and proves that ℵ is almost regular closed away from the diagonal by extending the method of traps. It introduces the subset ℵ′_U of parameter pairs whose associated power-series zeros are sign changes with a unique-perturbation property and shows traps are dense in this subset, yielding interior points. Since the surjective perturbation mechanism fails in full generality for 𝒩, the authors obtain a partial regular-closedness result for ℵ′_U and then deepen the analysis to show that any potential outliers are algebraic and isolated, with a conjecture that no outliers exist. The work combines a robust combinatorial description of cylinder-sets, geometry of the attractor via its convex hull, and analytic control of zeros to advance understanding of the topology of connectedness loci for plane affine IFS, with implications for the structure of cusp corners and higher-dimensional analogs.

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.

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

TL;DR

This work studies the connectedness locus ℵ for pairs of homogeneous affine maps on ℝ^2 and proves that ℵ is almost regular closed away from the diagonal by extending the method of traps. It introduces the subset ℵ′_U of parameter pairs whose associated power-series zeros are sign changes with a unique-perturbation property and shows traps are dense in this subset, yielding interior points. Since the surjective perturbation mechanism fails in full generality for 𝒩, the authors obtain a partial regular-closedness result for ℵ′_U and then deepen the analysis to show that any potential outliers are algebraic and isolated, with a conjecture that no outliers exist. The work combines a robust combinatorial description of cylinder-sets, geometry of the attractor via its convex hull, and analytic control of zeros to advance understanding of the topology of connectedness loci for plane affine IFS, with implications for the structure of cusp corners and higher-dimensional analogs.

Abstract

We study the connectedness locus 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.

Paper Structure

This paper contains 20 sections, 32 theorems, 50 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Summary of the Results
  3. Notation and Preliminaries
  4. General IFS
  5. Choice of Normalization for $\mathcal{M}$ and $\mathcal{N}$
  6. Overview of the method of traps
  7. Short-hop Paths
  8. The Homogeneous Self-Affine Case
  9. Applying the Method of Traps for $\mathcal{N}$
  10. Convex Hull of the Attractor for $\mathcal{N}$
  11. $\mathbb{R}^2$ Version of the Surjective Perturbation Lemma
  12. Auxiliary results required
  13. Proof for subset of $\mathcal{N}$
  14. Regular Closedness of a subset
  15. Extending Regular Closedness using Perturbations
  16. ...and 5 more sections

Key Result

Lemma 1.1

If $b$ is cyclic then the attractor of $\{Tx,Tx+b\}$ is given by power series in the following sense:

Figures (6)

  • Figure 1: $\mathcal{M}\cap[0,1]^2$ in black and gray. The gray part indicates the trivial part.
  • Figure 2: The set $\mathcal{N}$ in black and gray. The gray part indicates the trivial part.
  • Figure 3: Zoomed in parts of $\mathcal{N}$
  • Figure 4: The attractor for the linear family in $\mathcal{N}$. The extremal points are $\pm(\frac{1}{1-\gamma},\frac{1}{1-\lambda})$
  • Figure 5: An example of a trap
  • ...and 1 more figures

Theorems & Definitions (71)

  • Definition 1.1
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3: Folklore
  • proof
  • Corollary 1.3.1
  • Remark
  • Theorem 2.0: Traps are dense in $\N'_U$
  • ...and 61 more