Limit cubic laminations

Abstract

Let ${\sigma }_3:\mathbb{S}\to \mathbb{S}$ be the tripling map of the unit circle. For sequences $\{{\mathcal{L}}_i\}$ of ${\sigma }_3$-invariant dendritic laminations we study limits $(\overline{c},\overline{d})$ of their critical portraits assuming that one such limit $\mathcal{P}=({\overline{c}}_{\circ },{\overline{d}}_{\circ })$ is given. If the endpoints of ${\overline{c}}_{\circ }$ and ${\overline{d}}_{\circ }$ are non-periodic, then there is a unique lamination $\mathcal{L}$ with finite critical sets such that $\overline{c}$ and $\overline{d}$ can be any couple of critical chords compatible with $\mathcal{L}$. As the extreme opposite case we consider $\mathcal{P}=(\overline{0\frac{1}{3}},\overline{0\frac{2}{3}})$ and describe the corresponding countable closed family of possible critical portraits $(\overline{c},\overline{d})$ and the distinct laminations corresponding to them. These results can be useful for the construction of a model for the cubic connectedness locus.