Accepted (17):
1/2-Indecomposability and positive entropy for inverse limits of Markov set-valued functions — James Kelly <james.kelly@cnu.edu>
We discuss a class of upper semi-continuous set-valued functions called Markov set-valued functions. Such set-valued functions have a corresponding symbolic dynamical system. Through examples, partial results, and open questions we explore the relationship between the topology of the set-valued function's inverse limit and the topological entropy of the symbolic system. In particular, we establish some conditions where positive topological entropy is equivalent to the inverse limit containing a 1/2-indecomposable subcontinuum.
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Building Spaces with Non-trivial Self Covers — Mathew Timm <mtimm@bradley.edu>
We consider a dynamical systems approach for building spaces which have non-trivial self covers and a connection to self similar groups.
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Continua that admit an inscribed polygon: the Euclidean and hyperbolic settings — Ulises Morales-Fuentes <ulises.morales@uaem.mx>
A plane continuum $X$ is said to admit an inscribed polygon, $P$, if every embedding of $X$ into $\mathbb{R}^2$ (the Euclidean plane) contains the vertices of a polygon similar to $P$. In this talk we adapt this definition to the hyperbolic geometry setting: A plane continuum quasi-inscribes a polygon $Q$ in the hyperbolic plane $\mathbb{H}$, if given any embedding $\gamma:X\hookrightarrow\mathbb{H}$, we have that for all $\varepsilon>0$, $\gamma(X)$ admits a polygon whose inner angular sum is $\varepsilon$-close to the sum in $Q$; and both polygons share geometric structure. In particular, we show that there is a wide class of continua that quasi-inscribe rectangles. We will also include some results, obtained in the Euclidean setting, regarding the inscription of squares and rectangles in continua, in this case we will focus on continua that are ray compactifications.
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Decomposability of inverse limits of positive entropy systems on the Gehman dendrite — Jakub Tomaszewski <tomaszew@agh.edu.pl>
The problem of the decomposability of inverse limits of dynamical systems on continua has been of long-standing interest to many researchers in topological dynamics. A cornerstone result by Barge and Martin ([1](https://doi.org/10.1090/S0002-9947-1985-0779069-7)) shows that if a topological dynamical system on a unit interval has positive entropy, then the inverse limit of this system must contain an indecomposable continuum. Since then, Ingram ([2](https://doi.org/10.1090/S0002-9939-1989-0984796-1)), Ye ([3](https://doi.org/10.1016/0166-8641(94)00035-0)), Mouron ([4](https://doi.org/10.1090/S0002-9939-2010-10783-9)), and others have carried out a number of studies, e.g., investigating maps exhibiting a local periodic behavior of a special kind on arc-like continua, and especially homeomorphisms of such spaces with positive entropy. A result by Darji and Kato ([5](https://doi.org/10.1016/j.aim.2016.09.012)) states that the inverse limit of a topological dynamical system on a graph-like continuum with positive entropy must contain an indecomposable continuum. In this talk, we will show that if we consider the Gehman dendrite, then it is possible to construct a system with arbitrarily large entropy whose inverse limit is hereditarily decomposable.
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- Plenary
Embeddings of tree-like continua in the plane — Logan Hoehn <loganh@nipissingu.ca>
There are a number of interesting open problems in continuum theory that hinge on determining which tree-like continua can be embedded in the plane. Up to now, there are very few techniques available to show that a given tree-like continuum cannot be embedded in the plane. However, for a special class of tree-like continua, those which are inverse limits of simplicial inverse systems of trees, there is some hope that an algorithm may exist for checking planarity. I will describe this state of affairs, and pose some questions and computational challenges. At the same time, recent results are revealing that more tree-like continua can be embedded in the plane than perhaps were expected. I will discuss two such results: 1) Suppose $Y$ is any continuum of the form $Y = X \cup R$, where $X$ is an arc-like continuum, $R$ is a ray, $X \cap R = \emptyset$, and $\overline{R} \setminus R \subseteq X$. Then $Y$ can be embedded in the plane. 2) Suppose $Y$ is any continuum of the form $Y = K \cup \bigcup_{n=1}^\infty A_n$, where $K$ is a Knaster continuum and $\{A_n: n = 1,2,\ldots\}$ is a family of pairwise disjoint arcs, each intersecting $K$ in a single point, with $\mathrm{diam} A_n \to 0$. Then $Y$ can be embedded in the plane. This is joint work with Andrea Ammerlaan and Ana Anušić.
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Hedgehogs or how to make a continuum rigid — Teja Kac <teja.kac1@um.si>
For any Peano continuum $X$, we construct uncountable families of rigid, $\frac{1}{n}$-rigid, and $0$-rigid continua, of which all spaces contain a homeomorphic copy of $X$. We also show that for any continuum from the mentioned uncountable families, it holds that for every sequence of continuous surjective functions from the continuum into itself, the inverse limits of such a continuum with the described sequence are homeomorphic to the continuum itself.
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Inscription problems in plane continua — Cristina Villanueva-Segovia <cristina@im.unam.mx>
Given a plane continuum $X$ and an annulus $A\subseteq\mathbb{R}^2$, we say that $X$ $A$-inscribes a polygon $P$ if every essential embedding of $X$ into the annulus $A$ contains a similar copy of (the vertices of) $P$. In this talk we will present conditions on $X$ that guarantee that $X$ $A$-inscribes squares for some fixed annulus $A$. Moreover, we will analyze how ubiquitous this property is among continua that separate the plane.
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Interruptions and Chaos on Non-smooth fans — Jimmy Zakeršnik <jimmy.zakersnik1@um.si>
In this talk, we present the construction and dynamical properties of a family of arcwise connected continua known as fans. Firstly, we present a construction that, for any smooth fan $X$ containing a top and at least one accumulating leg, produces an uncountable family of pairwise non-homeomorphic non-smooth fans, such that the set of endpoints of each of them is homeomorphic to the set of endpoints of $X$. Secondly, we show that this construction preserves certain dynamical properties such as topological mixing, and some types of chaos. Finally, we apply the results of the paper on a few well-known examples.
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- Plenary
Non-existence of common models for certain classes of continua — Eiichi Matsuhashi <matsuhashi@riko.shimane-u.ac.jp>
A continuum $X$ is called a \emph{common model} for a class $\mathcal{C}$ of continua if every member of $\mathcal{C}$ is a continuous image of $X$. One of the natural questions in continuum theory is whether a given class of continua admits a common model, and if not, how the non-existence of common models can be established. In this talk, we will discuss several recent results concerning the non-existence of common models for classes of continua arising in hyperspace theory and the theory of indecomposable continua. The main tool is a recent theorem on meandering continua, which provides a general method for establishing non-existence results. As applications, we will present new classes of continua associated with Whitney properties and Whitney reversible properties, together with several classes related to indecomposable continua, and show that these classes do not admit common models.
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On chains of compacta — Bryant Rosado Silva <bryantrs99@hotmail.com>
In this talk, we are going to discuss what the typical maximal chain of compacta in the Cantor space looks like and use it as inspiration to discuss chains on the pseudoarc. This is a work in progress with Benjamin Vejnar.
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On the hyperspace of completely regular curves — Paweł Krupski <pawel.krupski@pwr.edu.pl>
A nondegenerate continuum $X$ is _completely regular_ if each nondegenerate subcontinuum of $X$ has nonempty interior. The class of completely regular continua contains all nontrivial connected finite topological graphs and is contained in the class of all regular curves. Let $CR(I^n)$ denote the hyperspace of completely regular subcontinua of the cube $I^n$, $2\le n\le\infty$, considered as a subspace of the Vietoris hyperspace $C(I^n)$ of all subcontinua of $I^n$. We will discuss the descriptive complexity of $CR(I^n)$: the hyperspace is a Borel subset of $C(I^n)$ which is not $F_{\sigma\delta\sigma}$. In fact, in the spirit of the theory of absorbing sets, one can show that $CR(I^n)$ is an absolute retract which is strongly $G_{\delta\sigma\delta}$-universal in the topological Hilbert cube $C(I^n)$.
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Persistent Recurrence and Inverse Limits of Unimodal Maps — Lori Alvin <lori.alvin@furman.edu>
Given a unimodal map, the recurrent critical point $c$ is said to be _reluctantly recurrent_ if there exists a $\delta > 0$ such that for every $\ell\in \mathbb{N}$ there is a backward orbit $\mathbf{x} = (x_{-\ell},\cdots x_{-2},x_{-1},x_0)$ in $\omega(c)$ such that $B(x_0,\delta)$ has a monotonic pull-back along $\mathbf{x}$; otherwise we say $c$ is _persistently recurrent_. Given a unimodal map $f$ with an infinite kneading sequence, it is known that the collection of endpoints for the inverse limit space $\varprojlim${$[c_2,c_1],f$} is precisely the collection of folding points if and only if $c$ is persistently recurrent. We revisit this known result and also show that when $c$ is infinitely recurrent and longbranched, it is not possible for $c$ to be persistently recurrent. This is joint work with Jernej Činč.
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Plane continua, canals and dead ends — Rene Gril Rogina <rene.gril@um.si>
Given a continuum X in the Euclidean plane, a canal of X is a way of “approaching” the continuum from outside of X or the bounded components of its complement. Often we search for simple dense canals, which are rays with X as their remainder. While some things are known about planar continua with embeddings that admit such canals, there are still open questions on this topic. In this talk, we first define canals and then “dead ends”, which are used in a construction to obtain new planar continua and new embeddings of these continua, all of which have canals with the desired properties. This is joint work with my PhD advisor Jernej Činč. This work was co-financed by the Slovenian Research and Innovation Agency (ARIS) under Contract No. SN-ZRD/22-27/0552.
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Reciprocating Domains in Classes of Continua — Iztok Banic <iztok.banic@um.si>
In topology, universal objects often serve as models for an entire class of spaces. A space $X$ is called a universal domain in a class $\mathcal C$ if every member of $\mathcal C$ is a continuous image of $X$. Classical examples include the Cantor set among compact metrizable spaces and the arc among Peano continua. In this talk, we give a new notion, called a reciprocating domain. A space $X$ in a class $\mathcal C$ is a reciprocating domain if whenever another space $Y\in\mathcal C$ admits a continuous surjection onto $X$, then $X$ also admits a continuous surjection onto $Y$. Intuitively, such a space cannot be reached from a ``larger'' space in the surjective order without also being able to map back onto that space. We discuss general properties of reciprocating domains and explain their relationship with universal domains. The main part of the talk will focus on examples arising in continuum theory, including chainable continua, tree-like continua, solenoids, circle-like continua, fans, and several related classes. Along the way, we will see that some familiar universal continua are also reciprocating, while in other natural classes reciprocating domains do not exist at all.
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The Shift Map on Mahavier Products — Van Nall <vnall@richmond.edu>
Some interesting continua can be represented in several different ways as a shift invariant subset of the Hilbert cube. In fact, several different representations as Mahavier products can be found for some continua each displaying different dynamical properties. We will review recent results concerning dynamical properties such as transitivity, mixing, shadowing, and specification that are exhibited by the shift map on Mahavier product embeddings of the Cantor fan and the Lelek fan into the Hilbert cube.
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The Specification property on cones and suspensions of the Cantor set. — Christopher Mouron <mouronc@rhodes.edu>
In this talk I will show that if a homeomorphism of the cone over the Cantor set (i.e. the Cantor fan) with the specification property exists, it would be complicated to describe. However, an example of a homeomorphism of the suspension over the Cantor set (i.e. the Cantor fan) with the specification property is given and is easy to describe.
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Ultrafilter orders on chainable continua — Julia Ścisłowska <j.scislowska@uw.edu.pl>
My talk will be devoted to discuss families of ultrafilter orders on a given chainable continuum X (such as e.g. arc, the Warsaw sine curve, the Knaster continuum etc.). These orders depend on a fixed sequence of chains, covering X (obtained from chainability of X), and on a fixed nonprincipal ultrafilter on N. Alternatively ultrafilter orders may be defined using representation of X as an inverse limit of a sequence of arcs and a fixed nonprincipal ultrafilter on N. During the talk I will present some known results in this topic. In particular, I will mention some ideas how we can express the “level of complexity” of a given chainable continuum in the language of ultrafilter orders. This is a joint work with Witold Marciszewski and Benjamin Vejnar, preprint is available at: https://arxiv.org/abs/2510.14577.
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