On the closure of a plane ray that limits onto itself
We show that the closure of any self-entwined ray in the plane must contain a Cantor set of mutually disjoint continua. This is false in dimension three.
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arXiv subjects
David S. Lipham
Publications and source records attributed to David S. Lipham.
On a local property of fences and fans
Two closely related classes of topological spaces are fences and fans. A fence is a compact metric space whose components are either arcs or singletons. A fan is a continuum formed by joining arcs at a common vertex, in such a way that intersections of subcontinua are always connected. We prove that every fence can be embedded in the plane and that both fences and fans admit a basis consisting of pierced open sets. This resolves a question by Iztok Banič, Goran Erceg, Ivan Jelić, Judy Kennedy, and Van Nall.
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Endpoints of smooth plane dendroids
Let $X$ be a smooth dendroid in the plane $\mathbb R^2$. We show that each endpoint of $X$ is arcwise accessible from $\mathbb R^2\setminus X$, and that the space of endpoints $E(X)$ has the property of a circle. In the event that $E(X)$ is connected, we call $X$ a *Bellamy dendroid*. We prove that if $E(X)$ is 1-dimensional, then $X$ contains a Bellamy dendroid or a Cantor set of arcs. In particular, if $E(X)$ totally disconnected and 1-dimensional, then $X$ is non-Suslinian. An example is constructed to show that this is false outside the plane.
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The $σ$-product of complete Erdős space
We show that the $σ$-product of complete Erdős space $\mathfrak E_{\mathrm{c}}$ is homeomorphic to the rational product $\mathbb Q\times \mathfrak E_{\mathrm{c}}$, answering a question by Rodrigo Hernández-Gutiérrez and Alfredo Zaragoza.
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Exponential iteration and Borel sets
We determine the exact Borel class of the points whose iterates under $\exp(z)+a$ tend to infinity. We also prove that the sets of non-escaping Julia points for many of these functions are topologically equivalent.
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A dichotomy for spaces near dimension zero
We prove that the classes of weakly $1$-dimensional and almost $0$-dimensional spaces are disjoint. The result has applications to hereditarily locally connected spaces, $\mathbb R$-trees, and endpoints of smooth fans.
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Erdős space in Julia sets
We prove that the rational Hilbert space $\mathfrak E$, known as Erdős space, surfaces in complex dynamics via iteration of $e^z-1$.
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Meager composants of tree-like continua
A subset $M$ of a continuum $X$ is called a \textit{meager composant} if $M$ is maximal with respect to the property that every two of its points are contained in a nowhere dense subcontinuum of $X$. Motivated by questions of Bellamy, Mouron and Ordoñez, we show that no tree-like continuum has a proper open meager composant, and that every tree-like continuum has either $1$ or $2^{\aleph_0}$ meager composants. We also prove a decomposition theorem: If $X$ is tree-like and every indecomposable subcontinuum of $X$ is nowhere dense, then the partition of $X$ into meager composants is upper semi-continuous and the space of meager composants is a dendrite.
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Totally disconnected subsets of chainable continua
We show that the endpoint set of a Suslinian chainable continuum must be zero-dimensional at some point. In particular, it cannot be homeomorphic to complete Erdős space. This answers a question of Jerzy Krzempek.
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Distinguishing endpoint sets from Erdős space
We prove that the set of all endpoints of the Julia set of $f(z)=\exp(z)-1$ which escape to infinity under iteration of $f$ is not homeomorphic to the rational Hilbert space $\mathfrak E$. As a corollary, we show that the set of all points $z\in \mathbb C$ whose orbits either escape to $\infty$ or attract to $0$ is path-connected. We extend these results to many other functions in the exponential family.
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The topological dimension of radial Julia sets
We prove that the meandering set for $f_a(z)=e^z+a$ is homeomorphic to the space of irrational numbers whenever $a$ belongs to the Fatou set of $f_a$. This extends recent results by Vasiliki Evdoridou and Lasse Rempe. It implies that the radial Julia set of $f_a$ has topological dimension zero for all attracting and parabolic parameters, including all $a\in (-\infty,-1]$. Similar results are obtained for Fatou's function $f(z)=z+1+e^{-z}$.
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A characterization of Erdős space factors
We prove that an almost zero-dimensional space $X$ is an Erdős space factor if and only if $X$ has a Sierpiński stratification of C-sets. We apply this characterization to spaces which are countable unions of C-set Erdős space factors. We show that the Erdős space $\mathfrak E$ is unstable by giving strongly $σ$-complete and nowhere $σ$-complete examples of almost zero-dimensional $F_{σδ}$-spaces which are not Erdős space factors. This answers a question by Dijkstra and van Mill.
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On cohesive almost zero-dimensional spaces
We investigate C-sets in almost zero-dimensional spaces, showing that closed $σ$C-sets are C-sets. As corollaries, we prove that every rim-$σ$-compact almost zero-dimensional space is zero-dimensional and that each cohesive almost zero-dimensional space is nowhere rational. To show these results are sharp, we construct a rim-discrete connected set with an explosion point. We also show every cohesive almost zero-dimensional subspace of $($Cantor set$)$$\times\mathbb R$ is nowhere dense.
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Compactification of cut-point spaces
We show that if $X$ is a separable locally compact Hausdorff connected space with fewer than $\mathfrak c$ non-cut points, then $X$ embeds into a dendrite $D\subseteq \mathbb R ^2$, and the set of non-cut points of $X$ is a nowhere dense $G_δ$-set. We then prove a Tychonoff cut-point space $X$ is weakly orderable if and only if $βX$ is an irreducible continuum. Finally, we show every separable metrizable cut-point space densely embeds into a reducible continuum with no cut points. By contrast, there is a Tychonoff cut-point space each of whose compactifications has the same cut point. The example raises some questions about persistent cut points in Tychonoff spaces.
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