AI 中文总结
针对单位圆三倍映射的σ₃-不变树状层状结构，在给定极限临界肖像的前提下，研究其极限临界肖像，得到端点非周期时的唯一有限临界集层状结构及极端情形的可数闭临界肖像族，结果可用于构造三次连通轨迹模型。
AI 中文摘要
设σ₃:𝕊→𝕊为单位圆的三倍映射。对于σ₃-不变树状层状结构的序列{ℒᵢ}，在给定其中一个极限临界肖像𝒫=(c̄₀, d̄₀)的前提下，研究其极限临界肖像(c̄, d̄)。若c̄₀和d̄₀的端点是非周期的，则存在唯一的具有有限临界集的层状结构ℒ，使得c̄和d̄可以是与ℒ相容的任意一对临界弦。作为完全相反的极端情形，我们考虑𝒫=(0̄1/3, 0̄2/3)，并描述对应的可数闭的可能临界肖像族(c̄, d̄)以及与它们对应的不同层状结构。这些结果可用于构造三次连通轨迹的模型。
英文摘要
Let $σ_3:\mathbb{S}\to \mathbb{S}$ be the tripling map of the unit circle. For sequences $\{\mathcal{L}_i\}$ of $σ_3$-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.
