arXiv subjects

David Sumner Lipham

Publications and source records attributed to David Sumner Lipham.

7 recordsLinked to original sources

arXiv2020

Approximations by disjoint continua and a positive entropy conjecture

E.D. Tymchatyn constructed a hereditarily locally connected continuum which can be approximated by a sequence of mutually disjoint arcs. We show the example re-opens a conjecture of G.T. Seidler and H. Kato about continua which admit positive entropy homeomorphisms. We prove that every indecomposable semicontinuum can be approximated by a sequence of disjoint subcontinua, and no composant of an indecomposable continuum can be embedded into a Suslinian continuum. We also prove that if $Y$ is a hereditarily unicoherent Suslinian continuum, then there exists $\varepsilon>0$ such that every two $\varepsilon$-dense subcontinua of $Y$ intersect.

math.GN↗

arXiv2020

A note on the topology of escaping endpoints

We study topological properties of the escaping endpoints and fast escaping endpoints of the Julia set of complex exponential $\exp(z)+a$ when $a\in (-\infty,-1)$. We show neither space is homeomorphic to the whole set of endpoints. This follows from a general result stating that for every transcendental entire function $f$, the escaping Julia set $I(f)\cap J(f)$ is first category.

math.DS↗

arXiv2019

Dispersion points and rational curves

We construct two connected plane sets which can be embedded into rational curves. The first is a biconnected set with a dispersion point. It answers a question of Joachim Grispolakis. The second is indecomposable. Both examples are completely metrizable.

math.GN↗

arXiv2019

Embedding irreducible connected sets

We show that every connected set $X$ which is irreducible between two points $a$ and $b$ embeds into the Hilbert cube in a way that $X\cup \{c\}$ is irreducible between $a$ and $b$ for every point $c$ in the closure of $X$. Also, a connected set $X$ is indecomposable if and only if for every compactum $Y\supseteq X$ and $a\in X$ there are two points $b$ and $c$ in the closure of $X$ such that $X\cup \{b,c\}$ is irreducible between every two points from $\{a,b,c\}$. Following the proofs of these theorems, we illustrate a cube embedding of the main example from "On indecomposability of $βX$". We prove the example embeds into the plane.

math.GN↗

arXiv2018

Singularities of meager composants and filament composants

Suppose $Y$ is a continuum, $x\in Y$, and $X$ is the union of all nowhere dense subcontinua of $Y$ containing $x$. Suppose further that there exists $y\in Y$ such that every connected subset of $X$ limiting to $y$ is dense in $X$. And, suppose $X$ is dense in $Y$. We prove $X$ is homeomorphic to a composant of an indecomposable continuum, even though $Y$ may be decomposable. An example establishing the latter was given by Christopher Mouron and Norberto Ordoñez in 2016. If $Y$ is chainable or, more generally, an inverse limit of identical topological graphs, then we show $Y$ is indecomposable and $X$ is a composant of $Y$. For homogeneous continua we explore similar problems which are related to a 2007 question of Janusz Prajs and Keith Whittington.

math.GN↗

arXiv2018

On indecomposability of $βX$

The following is an open problem in topology: Determine whether the Stone-Čech compactification of a widely-connected space is necessarily an indecomposable continuum. Herein we describe properties of $X$ that are necessary and sufficient in order for $βX$ to be indecomposable. We show that indecomposability and irreducibility are equivalent properties in compactifications of indecomposable separable metric spaces, leading to some equivalent formulations of the open problem. We also construct a widely-connected subset of Euclidean $3$-space which is contained in a composant of each of its compactifications. The example answers a question of Jerzy Mioduszewski.

math.GN↗

arXiv2018

Widely-connected sets in the bucket-handle continuum

A connected topological space is said to be widely-connected if each of its non-degenerate connected subsets is dense in the entire space. The object of this paper is the construction of widely-connected subsets of the plane. We give a completely metrizable example that answers a question of Paul Erdős and Howard Cook, while a similar example answers a question of Jerzy Mioduszewski.

math.GN↗