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Continuum Theory
Events

Submissions closed on 2025-02-14 11:59PM [Eastern Time (US & Canada)].

Organizers: Goran Erceg, Rodrigo Hernández Gutiérrez, and Logan Hoehn

Accepted Submissions:

A Dendrite Equivalence Relation on Loop Space — Spencer Arnesen

This talk will discuss how to turn a loop space into a group by factoring through dendrites. Inspired by the fact that group homomorphisms between fundamental groups of one-dimensional spaces induce, up to conjugation, a continuous map, and that path homotopies on one-dimensional spaces factor through a dendrite we show that homotopy through a dendrite is an equivalence relation and induces a group structure on a subset of loops. This group is always locally free.

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A notable contractible dendroid — Alejandro Illanes

A dendroid is an arcwise connected continuum such that the intersection of any two of its subcontinua is connected. In 1985, Tadeusz Mackiowiak constructed a contractible non-selectible dendroid X. Through the years the originality of the structure of this dendroid has been useful to produce several counterexamples. In this talk we will mention some other important properties of X and some of the examples that have constructed using it, including a new one related to the hyperspace of subcontinua with empty interior of a continuum.

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An Uncountable Family of Generalized Inverse Limit Spaces Which are Pointwise Self Homeomorphic (Updated). — Faruq Mena

In this talk, we will discuss how we found uncountable families of generalized inverse sequences on intervals and also on finite trees such that the inverse limit spaces of these sequences are pointwise self-homeomorphic. We give several examples of pointwise self-homeomorphic continua obtained in this manner including the dendrite D3 and a dendrite containing Dω.

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Big and large continua — Teja Kac

We generalize the notion of generalized inverse limits of inverse sequences of closed intervals with upper semicontinuous bonding functions to inverse limits of inverse sequences over directed graphs. We show that under certain conditions such inverse limits contain big/large continua.

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Building R-trees — Curtis Kent

We discuss a natural way to build actions of the fundamental group of one-dimensional spaces (which might not have universal covers) on R-trees. We will then discuss how the tools from the study of one-dimensional spaces can be adapted to more general spaces to build actions of locally free groups on R-trees with prescribed orbit spaces.

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Cantor fences in plane continua — David Lipham

David Bellamy constructed a surprising example of a smooth dendroid in the plane with a connected set of endpoints. In this talk, I will present the new result that any planable smooth dendroid with 1-dimensional endpoint set must contain a Cantor fence (a copy of 2ω×[0,1]) or a Bellamy dendroid (a smooth dendroid whose endpoint set is connected). This is false outside the plane, and it is unknown whether every Bellamy dendroid contains a Cantor fence. More generally, a continuum is said to be non-Suslinian if it contains an uncountable family of pairwise disjoint, non-degenerate subcontinua. I will discuss some open problems about this property in Julia sets and other plane continua with rich dynamical structures. Among these are: If a plane continuum admits a mixing homeomorphism, then is it non-Suslinian? Is the Sierpiński carpet the only locally connected plane continuum that admits a mixing homeomorphism?

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Coselectibility regarding symmetric products — Patricia Pellicer-Covarrubias

In this talk we consider a concept which is the dual to the concept of a selectible space, namely, a Λ-coselection space (Λ may be any given hyperspace of a space X). We consider this concept when Λ is the nth symmetric product Fn(X). We present sufficient conditions for a continuum to be either an F2(X)-coselection space or an F3(X)-coselection space.

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Dynamics on Hereditarily Decomposable Tree-like Continuum — Christopher Mouron

In this talk give an example of a hereditarily decomposable tree-like continuum that admits homeomorphisms that have the following dynamic properties: mixing, the specification property, and continuum-wise turbulence. I will also give results about topological properties (or lack of properties) that prevent hereditarily decomposable tree-like continuum from admitting homeomorphisms with some of the previous properties.

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Fan homogeneity — Rene Gril Rogina

We present recent results regarding different types of homogeneity for fans and discuss ongoing research into the topic. We define a larger class of fans with a specific property and use it to prove our results. This is joint work will Will Brian of UNC Charlotte.

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Generalization of the specification property to CR-dynamical systems — Ivan Jelić

We will recall the definition and basic properties of the notion of the specification property in the case of a standard topological dynamical system (X,f). We will then define a CR-dynamical system (X,F) and introduce different generalizations of the specification property for this type of dynamical system. More precisely, we will introduce and investigate the notions of (strong/weak) specification property and compare them together with their "initial" versions.

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Generalizations of the notion of a hereditarily equivalent continuum — Bryant Rosado Silva

We say that a continuum X is a hereditarily equivalent continuum (HEC) if every non-degenerate subcontinuum of it is homeomorphic to X. We can weaken this condition in three different levels: If considered in the hyperspace of continua of X, denoted by Cont(X), being hereditarily equivalent means that Cont(X)∖{{x} | x∈X}={K∈Cont(X) | K≃X}. This is an open and dense set, hence comeager, thus the first way to weaken it is to ask for the set of homeomorphic copies of X to be a comeager subset of Cont(X). A continuum with this property we call a generically hereditarily equivalent continuum (GHEC). However, we can go further and consider the hyperspace of maximal order arcs MOA(X). In the case of an HEC, any maximal order arc is made of an initial unitary set called the root and homeomorphic copies of X, hence we can say that - GCHEC holds for a space X if comeager many elements of MOA(X) have this property of being a chain made of copies of X apart from the root. - GCGHEC holds for X if comeager many elements of MOA(X) contain comeager many copies of X. In this talk, we partially address two natural questions that arise from these definitions: "What kind of spaces satisfy these properties?" and "How are these properties related?"

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Generalized inverse limits with Markov set-valued functions on finite graphs — Hayato Imamura

In this talk, we introduce definitions of Markov set-valued functions on finite graphs and the same pattern between two Markov set-valued functions. These functions are defined using the framework of cell complexes. They allow for infinite Markov partitions and have graphs that may contain 2-cells. We also show that two generalized inverse limits with bonding functions that are Markov set-valued functions following the same pattern are homeomorphic. This is joint work with E. Matsuhashi and Y. Oshima.

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Hereditarily Decomposable Continua have Non-Block Points — Daron Anderson

We expand upon our earlier results, to show that every nondegenerate hereditarily decomposable Hausdorff continuum has two or more non-block points, i.e points whose complements contain a continuum-connected dense subset. The celebrated non-cut point existence theorem states that all nondegenerate Hausdorff continua have two or more non-cut points, and the corresponding result for non-block points is known to hold for metrizable continua. It is also known that there are consistent examples of Hausdorff continua with no non-block points, but that non-block point existence holds for Hausdorff continua that are either aposyndetic, irreducible, or separable.

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Knaster continua in the plane — Ana Anusic

We show that for every Knaster continuum X, and every countable set C of composants of X, there exists a planar embedding of X in which the whole set C is accessible. I will also show that some of these embeddings can be done in dynamically significant way by using a generalization of Barge-Martin construction. This is a joint work with Logan Hoehn.

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More on the hyperspace of non-cut subcontinua of a continuum — Jorge E. Vega

We give conditions under which the Vietoris hyperspace of non-cut subcontinua is the same as the hyperspace of all subcontinua. Also, we give in the class of finite graph conditions under which the hyperspace of non-cut subscontinua is connected. This is joint work with A. Illanes and V. Martínez-de-la-Vega.

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On increasing and persistent Whitney properties — Hugo Villanueva

In 2009, increasing Whitney properties were defined by F. Orozco and give results and examples of topological increasing Whitney properties. In this talk we define Whitney persistent and locally Whitney persistent properties. We present results and examples of continua and topological properties to establish relations between these concepts and those of Whitney and increasing Whitney properties. This is a joint work with José Gerardo Ahuatzi-Reyes and Norberto Ordoñez-Ramírez.

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Planarity of compactifications of R with arc-like remainder — Andrea Ammerlaan

In 1972, Nadler and Quinn asked if for any arc-like continuum X, and point x∈X, there exists a plane embedding of X in which x is accessible. A continuum X is arc-like if it can be expressed as an inverse limit on arcs and, if X is in the plane R2, a point x∈X is called accessible if there exists an arc A⊂R2 such that A∩X= {x}. The question was recently answered in the positive (AA, Anušić, Hoehn 2024). This talk will discuss some consequences of the result: if X is an arc-like continuum, then any continuum which is the disjoint union of X and a ray R, with cl(R)∖R⊆X, is embeddedable in the plane, as is any compactification of a line having remainder X. Joint work with Logan Hoehn.

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The weak Extension Principle — Alessandro Vignati

We study the weak Extension Principle wEP allowing us to completely understand maps between \v{C}ech-Stone remainders of locally compact noncompact second countable spaces, generalising work of Farah in the 2000s. In short, the wEP asserts that all maps between such remainders come from maps between the underlying spaces. We show that once assuming fairly mild axioms (namely the Open Colouring Axiom and Martin's Axiom) the wEP holds, while this is not the case if the Continuum Hypothesis holds. This is joint work with D. Yilmaz.

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Unary Topological Algebras on Continua, and Some Associated Function Algebras — Matt Insall

Let D2 denote the unit disk in the plane, and let C denote the set of (continuous) self-maps of D2. Using ∘, as is common, to denote the binary operation on C that takes a pair of continuous functions to another continuous function, we study some properties of the following algebras and their subalgebras: A=⟨D2;C⟩, and F=⟨C;∘⟩. The algebra A can be naturally endowed with a topology, and we will suppress any notation the choice of topology on it, because we are interested in only the usual topology, so we think of A as a topological algebra; it is a {\bf multi-unary topological algebra} on the continuum D2. As is well-known, there are various reasonable topologies that can be given to the algebra F, but we will treat it only as an algebra for now. Note that the algebra F is a semigroup, and it is a subalgebra of a function algebra (an algebra of functions over a set that is closed under composition and contains the projection functions) over D2. Recall that in a semigroup, a left translation, λa is a self-map of the semigroup defined using a parameter a, an element of the given semigroup, using the formula λa(f)=af. In our case, the parameters are continuous functions on D2 and the semigroup operation is composition, so instead of juxtaposition of symbols, we will write λa(f)=a∘f. Similarly, a right translation is defined by the other order of the ``multiplication'': ρa(f)=f∘a. Given an element a∈C, we call the set Λa={(f,g)∈C2|λa(f)=λa(g)}={(f,g)∈C2|a∘f=a∘g} the {\bf kernel} of λa. kernels of right translations are defined similarly: Pa={(f,g)∈C2|ρa(f)=ρa(g)}={(f,g)∈C2|f∘a=g∘a}. These are {\bf congruences} of the algebra (semigroup) F; i.e. they are equivalence relations θ on the set C that are compatible with the semigroup operation (composition). The compatibility property can be described via the containment {(b∘f,c∘g)|(b,c),(f,g)∈θ}⊆θ. On a set X, two special equivalence relations are congruences for any structure on X, namely the {\bf identity relation}, ΔX={(p,p)|p∈X}, and the {\bf all relation}, ∇X={(p,q)|p,q∈X}. It is clear that all kernels of left translations on a semigroup are congruences on that semigroup, and similarly, kernels of rigjt translations on a semigroup are congruences on that semigroup. We will sketch a proof of the following: Theorem. The algebra F has only three kinds of congruences, namely the identity relation, the all relation, and kernels of left translations by members of C. Our proof of the above result will employ nonstandard methods and results from the theory of function algebras on finite sets, and interestingly, the above immediately entails the below consequence Corollary. In the semigroup F, every right translation equalizer is a left translation equalizer, and vice versa. This is joint work with Malgorzata Marciniak (mmarciniak@lagcc.cuny.edu)

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Uncountable family of Lelek-like fans — Judy Kennedy

Defining an appropriate equivalence relation on a Lelek fan L we construct an uncountable family of pairwise non-homeomorphic Lelek-like fans. In this talk plan is to explain the construction of that family. This is joint work with Iztok Banič, Goran Erceg, and Ivan Jelić.

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Uniqueness of cones for some not locally connected continua — Daria Michalik

A continuum X has unique cone provided that the following property holds: if Y is a continuum and Cone(X) is homeomorphic to Cone(Y), then X is homeomorphic to Y. In this talk we consider the problem of the uniqueness of cones for some not locally connected continua, e.g. the indecomposable continua and the compactifications of the ray.

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