Let ${\lambda }_{-}$ and ${\lambda }_{+}$ be two bounded measured laminations on the hyperbolic disk ${\mathbb{H}}^2$, which “strongly fill” (definition below). We consider the left earthquakes along ${\lambda }_{-}$ and ${\lambda }_{+}$, considered as maps from the universal Teichmüller space $\mathcal{T}$ to itself, and we prove that the composition of those left earthquakes has a fixed point. The proof uses anti-de Sitter geometry. Given a quasi-symmetric homeomorphism $u:{\mathbb{RP}}^1\to {\mathbb{RP}}^1$, the boundary of the convex hull in $Ad{S}^3$ of its graph in ${\mathbb{RP}}^1\times {\mathbb{RP}}^1\simeq \partial Ad{S}^3$ is the disjoint union of two embedded copies of the hyperbolic plane, pleated along measured geodesic laminations. Our main result is that any pair of bounded measured laminations that “strongly fill” can be obtained in this manner.

中文翻译：

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反德西特拟圆凸包上的弯曲叠片

让 ${\lambda }_{-}$ 和 ${\lambda }_{+}$ 是双曲圆盘上的两个有界测量叠片 ${\mathbb{H}}^2$，“强填充”（定义如下）。我们考虑沿左地震${\lambda }_{-}$ 和 ${\lambda }_{+}$，被视为来自通用 Teichmüller 空间的映射 $\mathcal{吨}$到它自己，我们证明那些左地震的组成有一个固定点。证明使用反德西特几何。给定一个准对称同胚$你：{\mathbb{RP}}^1\to {\mathbb{RP}}^1$，凸包的边界在 $\mathrm{一种}d{秒}^3$ 它的图在 ${\mathbb{RP}}^1\times {\mathbb{RP}}^1\simeq \partial \mathrm{一种}d{秒}^3$是双曲平面的两个嵌入副本的不相交联合，沿着测量的测地线叠层折叠。我们的主要结果是可以通过这种方式获得任何一对“强烈填充”的有界测量叠片。