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 $\lambda \in \mathbb{D}\setminus \{0\}$ 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 $\mathbb{D}\setminus \mathcal{M}$.

arXiv comment22 pages, 6 figures. Added an Appendix and updated references. It will appear on Experimental Mathematics