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arXiv · 2607.26812

Limit cubic laminations

A. Blokh·L. Oversteegen·V. Timorin
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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 $\mathcal{P}=(\overline{0 \frac13}, \overline{0 \frac23})$ 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.

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A. Blokh, L. Oversteegen, V. Timorin. 2026-07-29. Limit cubic laminations. https://arxiv.org/abs/2607.26812

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