Continuum Theory

A class of 1-dimensional continua: Shape Theory meets Dynamics — Krystyna Kuperberg

The notion of movability was introduced by K. Borsuk in 1967 as one of the basic notions in shape theory. A compactum X embedded in an absolute neighborhood retract (ANR), such as the Hilbert cube or the Euclidean space, is movable if for any neighborhood U of X there is a smaller neighborhood V of X such that V can be moved by a homotopy within U into any neighborhood W of X. Movability does not depend on the choice of the ANR or the embedding. A continuum that is locally homeomorphic to the Cartesian product of the Cantor set and an open interval is called a lamination. In this talk we consider movable and non-movable laminations appearing as invariant sets in aperiodic continuous dynamical systems, as well as the flow around them and the larger 1-dimensional invariant compacta containing the laminations. A non-movable invariant lamination in a 3-dimensional Euclidean space is often contained in an invariant compactum that is movable.

View Submission

Accessible points of arc-like continua — Andrea Ammerlaan

This talk will discuss the Nadler-Quinn problem. Posed in 1972, the problem asks if, given any arc-like continuum $X$ and any point $x \in X$ , we can embed $X$ in the plane with $x$ accessible. In 2001, Minc constructed a particularly simple example of an arc-like continuum $X$ and point $p \in X$ for which it was not known whether $p$ could be made accessible in a plane embedding of $X$ . In 2020, Anusic proved that $X$ can, in fact, be embedded with $p$ accessible. I will give an overview of this proof and briefly introduce a more recent approach to the problem.

View Submission

Buried points of plane continua — David Lipham

Sets on the boundary of a complementary component of a continuum in the plane have been of interest since the early 1920’s. Curry and Mayer defined the buried points of a plane continuum to be the points in the continuum which were not on the boundary of any complementary component. Motivated by their investigations of Julia sets, they asked what happens if the set of buried points is totally disconnected and non-empty. Curry, Mayer and Tymchatyn showed that in that case the continuum is Suslinian, i.e. it does not contain an uncountable collection of non-degenerate pairwise disjoint subcontinua. In an answer to a question of Curry et al, van Mill and Tuncali constructed a plane continuum whose buried point set was totally disconnected, non-empty and one-dimensional at each point of a countably infinite set. In this talk I will present proof that the van Mill-Tuncali example was the best possible in the sense that whenever the buried set is totally disconnected, it is one-dimensional at each of at most countably many points. I will also discuss a few related problems about plane continua and endpoints of dendroids. This talk is based on joint work with Jan van Mill, Murat Tuncali, Ed Tymchatyn, and Kirsten Valkenburg.

View Submission

Cantor Sets and Topological Entropy for Set-valued Functions on Countable Domains — James Kelly

We characterize when an inverse limit of a set-valued function is a Cantor set. Given a set-valued function $F : X \to 2^{X}$ , we define the set $D (F) = ⋂_{n = 1}^{\infty} F^{n} (X)$ . It is known that $\lim_{\leftarrow} F = \lim_{\leftarrow} F |_{D (F)}$ , so we only need to consider the $F |_{D (F)}$ . When $D (F)$ is finite, $\lim_{\leftarrow} F$ is a shift of finite type, so we focus on the case where $D (F)$ is infinite, and we give a characterization for $\lim_{\leftarrow} F$ to be a Cantor set for this context. We go on to examine the entropy of a set-valued function on a countable domain and how that relates to the inverse limit being a Cantor set. This includes joint work with L. Alvin and S. Greenwood.

View Submission

Dendrites and path-homotopies — Curtis Kent

In a one-dimensional space, any nullhomotopic loop factors through a dendrite. Analogously, we can say that two paths $f$ and $g$ , with the same endpoints, are equivalent if $f * \overset{―}{g}$ factors through a loop in a dendrite, where $\overset{―}{g}$ is the path $g$ traversed backwards. We will show that the equivalence relation generated by this relation is the same as the path-homotopy and discuss its consequences. This is joint work with Greg Conner, Jeremy Brazas, and Paul Fabel.

View Submission

End-point-generated smooth fans — Rene Gril Rogina

We define end-point-generated smooth fans and give known examples. We also define combs and use them to answer previously open problems about specific Mahavier products and endpoint-generated smooth fans as well as construct an uncountable family of such fans. This is joint work with Will Brian of UNC Charlotte.

View Submission

Gehman Dendrite G_4 as Generalize Inverse Limit Spaces of Upper Semi Continuous Bonding Functions on [0,1] — Faruq Mena

In this talk we prove that the Gehman dendrite G_4 can be obtained as a generalized inverse limit space with a single upper semi-continuous bonding function on [0,1]. This answers a question of Farhan and Mena. Moreover, we find an uncountable family of inverse sequences on [0,1] whose inverse limit spaces are homeomorphic to the Gehman dendrite G_4.

View Submission

Hyperspaces of exactly n points — Alejandro Illanes

Given a topological space and a positive integer n, we consider the hyperspace [X]n of subsets of X with exactly n points. In this talk we discuss results we have obtained about the topological properties of [X]n, such as: connectedness, arcwise connectedness, contractibility, existence of selections, spaces [[0,1]]n, etc.

View Submission

Inverse Limits of Finite Path Graphs — Haley Pavlis

The author defines the graph topology for finite graphs. We discuss the properties of a continuous map between graphs and properties of a traditional inverse limit of graphs. Most importantly, that a traditional inverse limit of finite path graphs is non-Hausdorff. We introduce a generalized inverse limit, where the first space is a metric arc and all other spaces are finite path graphs. Using the Bucket Handle continuum as an example, a technique is shown for constructing a generalized inverse limit, where the first space is a metric arc and the others are finite path graphs, that is homeomorphic to a traditional inverse limit of Hausdorff arcs. Using crooked chains, we construct and analyze a non-Hausdorff hereditarily indecomposable continuum. This continuum has some interesting properties, which will be discussed. Ongoing research is discussed and open problems stated.

View Submission

Inverse limits with Markov set-valued functions — Teja Kac

We introduce a new concept of Markov-type set-valued functions on trees allowing the graphs to be $2$ -dimensional. Additionally, we present Markov set-valued functions on compact metric spaces. We establish the conditions under which two inverse limits of inverse sequences of trees or compact metric spaces are homeomorphic. This is joint work with Iztok Banič and Matevž Črepnjak, both University of Maribor.

View Submission

New embeddings of Knaster continuum in the plane — Ana Anusic

Given $n \in N$ , we show that there exists a planar embedding of Knaster continuum with $n$ (fully) accessible composants. This answers a question of Debski and Tymchatin from 1993. This is a joint work with Logan Hoehn.

View Submission

Non-metric Hereditarily Indecomposable Continua — Michel Smith

The author has shown techniques for producing non-metric hereditarily indecomposable continua. Examples are presented. However, attempts to generalize metric construction techniques yield situations in which hereditary indecomposability implies metrizability. We review the author's recent results regarding such situations. Open problems in the area are stated.

View Submission

On open and monotone mappings — Lex Oversteegen

In 1972 A.R. Stralka asked if every open and monotone retraction from a dendroid to an arc is the identity map. In this talk we will review some old results, including a solution to this problem, and connect these results to more recent developments.

View Submission

Projective Fraïssé limits of trees with confluent epimorphisms — Robert Roe

An earlier version of this article had a error in the proof that monotone epimorphisms of finite trees amalgamated. In this talk we will show an example of finite trees that do not amalgamate with monotone epimorphisms. Further, we show how we can use a subfamilies of the family of monotone epimorphisms, that we call simple-monotone and simple*-monotone, to obtain results similar to those in the original paper. We also show the new result that the topological realization of the projective Fraïssé limit of the family of finite trees with simple*-confluent epimorphisms is the Mohler-Nikiel dendroid. This is joint work with W.J. Charatonik, A. Kwiatkowska, and S. Yang.

View Submission

The hyperspace of non-cut subcontinua of graphs and dendrites — Jorge Vega

We give conditions under which the Vietoris hyperspace of non-cut subscontinua is compact, connected, locally connected or totally disconnected for graphs and dendrites. Also, we show that for a dendrite whose set of endpoints is dense this hyperspace is homeomorphic to de Baire space of irrational numbers.

View Submission

Continuum Theory Add Favorite

Accepted Submissions:

A class of 1-dimensional continua: Shape Theory meets Dynamics — Krystyna Kuperberg

Accessible points of arc-like continua — Andrea Ammerlaan

Buried points of plane continua — David Lipham

Cantor Sets and Topological Entropy for Set-valued Functions on Countable Domains — James Kelly

Dendrites and path-homotopies — Curtis Kent

End-point-generated smooth fans — Rene Gril Rogina

Gehman Dendrite G_4 as Generalize Inverse Limit Spaces of Upper Semi Continuous Bonding Functions on [0,1] — Faruq Mena

Hyperspaces of exactly n points — Alejandro Illanes

Inverse Limits of Finite Path Graphs — Haley Pavlis

Inverse limits with Markov set-valued functions — Teja Kac

New embeddings of Knaster continuum in the plane — Ana Anusic

Non-metric Hereditarily Indecomposable Continua — Michel Smith

On open and monotone mappings — Lex Oversteegen

Projective Fraïssé limits of trees with confluent epimorphisms — Robert Roe

The hyperspace of non-cut subcontinua of graphs and dendrites — Jorge Vega