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L. G. Oversteegen

Publications and source records attributed to L. G. Oversteegen.

5 recordsLinked to original sources
arXiv2019

Shortest paths in arbitrary plane domains

Let $Ω$ be a connected open set in the plane and $γ: [0,1] \to \overlineΩ$ a path such that $γ((0,1)) \subset Ω$. We show that the path $γ$ can be ``pulled tight'' to a unique shortest path which is homotopic to $γ$, via a homotopy $h$ with endpoints fixed whose intermediate paths $h_t$, for $t \in [0,1)$, satisfy $h_t((0,1)) \subset Ω$. We prove this result even in the case when there is no path of finite Euclidean length homotopic to $γ$ under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a ``shortest'' path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.

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arXiv2018

Extension of isotopies in the plane

Let $A$ be any plane set. It is known that a holomorphic motion $h: A \times \mathbb{D} \to \mathbb{C}$ always extends to a holomorphic motion of the entire plane. It was recently shown that any isotopy $h: X \times [0,1] \to \mathbb{C}$, starting at the identity, of a plane continuum $X$ also extends to an isotopy of the entire plane. Easy examples show that this result does not generalize to all plane compacta. In this paper we will provide a characterization of isotopies of uniformly perfect plane compacta $X$ which extend to an isotopy of the entire plane. Using this characterization, we prove that such an extension is always possible provided the diameters of all components of $X$ are uniformly bounded away from zero.

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arXiv2016

A canonical parameterization of paths in $\mathbb{R}^n$

For sufficiently tame paths in $\mathbb{R}^n$, Euclidean length provides a canonical parametrization of a path by length. In this paper we provide such a parametrization for all continuous paths. This parametrization is based on an alternative notion of path length, which we call $\mathsf{len}$. Like Euclidean path length, $\mathsf{len}$ is invariant under isometries of $\mathbb{R}^n$, is monotone with respect to sub-paths, and for any two points in $\mathbb{R}^n$ the straight line segment between them has minimal $\mathsf{len}$ length. Unlike Euclidean path length, the $\mathsf{len}$ length of any path is defined (i.e., finite) and $\mathsf{len}$ is continuous relative to the uniform distance between paths. We use this notion to obtain characterizations of those families of paths which can be reparameterized to be equicontinuous or compact. Finally, we use this parametrization to obtain a canonical homeomorphism between certain families of arcs.

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arXiv2016

A complete classification of homogeneous plane continua

We show that every non-degenerate homogeneous plane continuum is homeomorphic to either the unit circle, the pseudo-arc, or the circle of pseudo-arcs. It follows that any planar homogenous compactum has the form $X \times Z$, where $X$ is a either a point or one of these three homogeneous plane continua, and $Z$ is a finite set or the Cantor set. The main technical result in this paper is a new characterization of the pseudo-arc: a non-degenerate continuum is homeomorphic to the pseudo-arc if and only if it is hereditarily indecomposable and has span zero.

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