arXiv subjects

E. D. Tymchatyn

Publications and source records attributed to E. D. Tymchatyn.

10 recordsLinked to original sources

arXiv2019

Inverse systems with simplicial bonding maps and cell structures

For a topologically complete space $X$ and a family of closed covers $\mathcal A$ of $X$ satisfying a "local refinement condition" and a "completeness condition," we give a construction of an inverse system $\mathbf{ N}_{\mathcal A}$ of simplicial complexes and simplicial bonding maps such that the limit space $N_{\infty} = \varprojlim \mathbf{N}_{\mathcal A}$ is homotopy equivalent to $X$. A connection with cell structures [2],[3] is discussed

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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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arXiv2012

Fixed point theorems in plane continua with applications

We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation + 1. A fixed point theorem for positively oriented, perfect maps of the plane is obtained. This generalizes results announced by Bell in 1982. A continuous map of an interval to the real line which sends the endpoints in opposite directions has a fixed point. We generalize this to maps on non-invariant continua in the plane under positively oriented maps of the plane (with appropriate boundary conditions). These methods imply that in some cases non-invariant continua in the plane are degenerate. This has important applications in complex dynamics. E.g., a special case of our results shows that if $X$ is a non-separating invariant subcontinuum of the Julia set of a polynomial $P$ containing no fixed Cremer points and exhibiting no local rotation at all fixed points, then $X$ must be a point. It follows that impressions of some external rays to polynomial Julia sets are degenerate.

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arXiv2008

Extending Isotopies of Planar Continua

In this paper we solve the following problem in the affirmative: Let $Z$ be a continuum in the plane $\complex$ and suppose that $h:Z\times [0,1]\to\complex$ is an isotopy starting at the identity. Can $h$ be extended to an isotopy of the plane? We will provide a new characterization of an accessible point in a planar continuum $Z$ and use it to show that an accessible point is preserved during the isotopy. We show next that the isotopy can be extended over hyperbolic crosscuts. The proof makes use of the notion of a metric external ray, which mimics the notion of a conformal external ray, but is easier to control during an isotopy.

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arXiv2008

The plane fixed point problem

In this paper we present proofs of basic results, including those developed so far by H. Bell, for the plane fixed point problem. Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation + 1. We develop a prime end theory through hyperbolic chords in maximal round balls contained in the complement of a non-separating plane continuum $X$. We define the concept of an {\em outchannel} for a fixed point free map which carries the boundary of $X$ minimally into itself and prove that such a map has a \emph{unique} outchannel, and that outchannel must have variation $=-1$. We also extend Bell's linchpin theorem for a foliation of a simply connected domain, by closed convex subsets, to arbitrary domains in the sphere. We introduce the notion of an oriented map of the plane. We show that the perfect oriented maps of the plane coincide with confluent (that is composition of monotone and open) perfect maps of the plane. We obtain a fixed point theorem for positively oriented, perfect maps of the plane. This generalizes results announced by Bell in 1982 (see also \cite{akis99}). It follows that if $X$ is invariant under an oriented map $f$, then $f$ has a point of period at most two in $X$.

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arXiv2008

Characterizing indecomposable plane continua from their complements

We show that a plane continuum X is indecomposable iff X has a sequence (U_n) of not necessarily distinct complementary domains satisfying what we call the double-pass condition: If one draws an open arc A_n in each U_n whose ends limit into the boundary of U_n, one can choose components of U_n minus A_n whose boundaries intersected with the continuum (which we call shadows) converge to the continuum.

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arXiv2005

On intersection of simply connected sets in the plane

Several authors have recently attempted to show that the intersection of three simply connected subcontinua of the plane is simply connected provided it is non-empty and the intersection of each two of the continua is path connected. In this note we give a very short complete proof of this fact. We also confirm a related conjecture of Karimov and Repovs.

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arXiv2002

On simultaneous linear extensions of partial (pseudo)metrics

We consider the question of simultaneous extension of (pseudo)metrics defined on nonempty closed subsets of a compact metrizable space. The main result is a counterpart of the result due to Künzi and Shapiro for the case of extension operators of partial continuous functions and includes, as a special case, Banakh's theorem on linear regular operators extending (pseudo)metrics.

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