Order trees and laminations of the plane
The authors define the concept of a lamination \(\lambda\) of \(\mathbb R^2\). The leaves of such a \(\lambda\) are smoothly embedded 1-manifolds which are never isolated. For an essential lamination, no leaves are circles and there are local charts whose intersection with every leaf is connected. They define also order trees and \(\mathbb R\)-order trees. One type of order tree is an \(\mathbb R\)-tree [\textit{J. W. Morgan} and \textit{P. B. Shalen}, Ann. Math., II. Ser. 120, 401-476 (1984; Zbl 0583.57005)]. For every lamination \(\lambda\) of \(\mathbb R^2\) there is an associated order tree \(T(\lambda)\) whose vertices are the closures of the complementary domains of \(\lambda\) along with the nonboundary leaves of \(\lambda\). For essential laminations, the natural map \(\nu:\mathbb R^2\rightarrow T(\lambda)\) is a quotient map and \(T(\lambda)\) is an \(\mathbb R\)-order tree. It is proved (Theorem 4.2) that every cyclically ordered \(\mathbb R\)-order tree \(T\) is isomorphic to the associated (order) tree of some essential lamination of \(\mathbb R^2\). There is a construction called ``splitting along leaves'' (section 5) which is used to replace non-boundary leaves by closed complementary regions. Whenever \(\lambda'\) is obtained from \(\lambda\) by such a process, then \(T(\lambda)\cong T(\lambda')\). In section 6, it is shown that each isomorphism of cyclically ordered \(\mathbb R\)-order trees is induced by a homeomorphism. Two later sections deal with the relation of essential laminations of \(\mathbb R^2\) to foliations of \(\mathbb R^2\) and essential laminations of \(\mathbb R^3\).
scientific article; zbMATH DE number 1082985