7 citations · 38 across the 19 of their papers we have counts for
21 papers
Endpoints of smooth plane dendroids
David S. Lipham
Let be a smooth dendroid in the plane . We show that each endpoint of is arcwise accessible from , and that the space of endpoints $E(X…
A dichotomy for spaces near dimension zero
David S. Lipham
We prove that the classes of weakly -dimensional and almost -dimensional spaces are disjoint. The result has applications to hereditarily locally connected spaces, $\mathbb R…
Meager composants of tree-like continua
David S. Lipham
A subset of a continuum is called a \textit{meager composant} if is maximal with respect to the property that every two of its points are contained in a nowhere dense s…
Totally disconnected subsets of chainable continua
David S. Lipham
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 answ…
The -product of complete Erdős space
David S. Lipham
We show that the -product of complete Erdős space is homeomorphic to the rational product , answering a ques…
Exponential iteration and Borel sets
David S. Lipham
We determine the exact Borel class of the points whose iterates under tend to infinity. We also prove that the sets of non-escaping Julia points for many of these funct…
A characterization of Erdős space factors
David S. Lipham
We prove that an almost zero-dimensional space is an Erdős space factor if and only if has a Sierpiński stratification of C-sets. We apply this characterization to spaces w…
Distinguishing endpoint sets from Erdős space
David S. Lipham
We prove that the set of all endpoints of the Julia set of which escape to infinity under iteration of is not homeomorphic to the rational Hilbert space $\math…
Erdős space in Julia sets
David S. Lipham
We prove that the rational Hilbert space , known as Erdős space, surfaces in complex dynamics via iteration of .
On cohesive almost zero-dimensional spaces
Jan J. Dijkstra, David S. Lipham
We investigate C-sets in almost zero-dimensional spaces, showing that closed C-sets are C-sets. As corollaries, we prove that every rim--compact almost zero-dimensional space…
The topological dimension of radial Julia sets
David S. Lipham
We prove that the meandering set for is homeomorphic to the space of irrational numbers whenever belongs to the Fatou set of . This extends recent results b…
Approximations by disjoint continua and a positive entropy conjecture
David Sumner Lipham
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 o…
Compactification of cut-point spaces
David S. Lipham
We show that if is a separable locally compact Hausdorff connected space with fewer than non-cut points, then embeds into a dendrite …
A note on the topology of escaping endpoints
David Sumner Lipham
We study topological properties of the escaping endpoints and fast escaping endpoints of the Julia set of complex exponential when . We show neither…
Dispersion points and rational curves
David Sumner Lipham
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 Grispolak…
Singularities of meager composants and filament composants
David Sumner Lipham
Suppose is a continuum, , and is the union of all nowhere dense subcontinua of containing . Suppose further that there exists such that every connec…
Embedding irreducible connected sets
David Sumner Lipham
We show that every connected set which is irreducible between two points and embeds into the Hilbert cube in a way that is irreducible between and …
One-to-one composant mappings of and
David Lipham
Knaster continua and solenoids are well-known examples of indecomposable continua whose composants (maximal arcwise-connected subsets) are one-to-one images of lines. We show that…
On indecomposability of
David Sumner Lipham
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 de…
Large widely-connected spaces
David Lipham
We show that there are widely-connected spaces of arbitrarily large cardinality, answering a question by David Bellamy.
Widely-connected sets in the bucket-handle continuum
David Sumner Lipham
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 construct…