Accessibility of the Boundary of the Thurston Set

TL;DR

This paper investigates the boundary accessibility of the Thurston set related to a specific iterated function system, providing conditions under which boundary points are reachable from outside the set.

Contribution

It introduces sufficient conditions for boundary points of the Thurston set to be accessible from the exterior, based on the asymptotic similarity of related attractors.

Findings

Boundary points are path accessible under certain conditions.

Asymptotic similarity aids in analyzing boundary accessibility.

Provides a framework for understanding the boundary structure of the Thurston set.

Abstract

Consider two objects associated to the Iterated Function System (IFS) $\{1+\lambda z,-1+\lambda z\}$: the locus $\mathcal{M}$ of parameters $\lambda\in\mathbb{D}\setminus\{0\}$ for which the corresponding attractor is connected; and the locus $\mathcal{M}_0$ of parameters for which the related attractor contains $0$. The set $\mathcal{M}$ can also be characterized as the locus of parameters for which the attractor of the IFS $\{1+\lambda z, \lambda z, -1+\lambda z\}$ contains $\lambda^{-1}$. Exploiting the asymptotic similarity of $\mathcal{M}$ and $\mathcal{M}_0$ with the respective associated attractors, we give sufficient conditions on $\lambda\in\partial\mathcal{M}$ or $\partial\mathcal{M}_0$ to guarantee it is path accessible from the complement $\mathbb{D}\setminus\mathcal{M}$.