Accessibility of the Boundary of the Thurston Set

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 $\lambda \in \mathrm{\partial }\mathcal{M}$ or $\mathrm{\partial }{\mathcal{M}}_0$ to guarantee it is path accessible from the complement $\mathbb{D}\setminus \mathcal{M}$.