Accessibility of the Boundary 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 $\lambda \in \mathbb{D}\setminus \left\{0\right\}$ 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 $\mathbb{D}\setminus \mathcal{M}$.

Keywords:

• Iterated function system
• Fractal geometry
• Self-affine fractals
• Zero-sets of power series

AMS 2010 Subject Classifications:

• 30C15
• 28A80

Conflict of interest

No potential conflict of interest was reported by the author(s).

Notes

1 The word “instar” is used in biology to describe the developmental stage of insects, between each molt until sexual maturity. We chose it because the limit set is obtained by going through (infinitely many) developmental stages, some with a definite larva-like appearance.

2 Solomyak worked with the word $+{(-+++--)}^{\infty }$ which gives the power series $\frac{1-z+z^2+2z^3}{1+z^3}$; in other words, he looked at $f(-z)$. The symmetry of $\mathcal{M}$ allows us to consider f(z) instead.