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 $\mathbb{ D}\setminus\mathcal{ M}$.
Categories: math.DS