$R^1$, $R^2$ and $R^3$-continua were defined by S. T. Czuba in 1980, in particular he showed that the existence of any one of these sets in a continuum $X$ implies the noncontractibility of $X$. Also, $R^i$-continua have proved to be useful when studying noncontractibility of hyperspaces. In this talk we recall these concepts and we present some relations between them in continua and hyperspaces.
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We show how the Wa\.zewski universal dendrite of order $m$, for any positive integer $m$ greater than 2, can be obtain as the generalized inverse limit of a single set-valued upper semi-continuous bonding function on $[0,1]$ whose graph consists of exactly $m$ line segments. $D_m$ has been obtained previously as a generalized inverse limit of a single bonding function but in that case the bonding function was extremely complicated consisting of infinitely many line segments. This is joint work with Faruq Mena.
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Suppose $X$ is a compact metric space and $F$ is a closed relation on $X$. For a classical dynamical property $\mathcal{P}$, we introduce a natural trichotomy of CR-dynamical properties associated with a closed relation $F$: $CR-\mathcal{P}, CR-post\mathcal{P}, CR-pre\mathcal{P}.$ These three notions are designed so that $(X,F)$ satisfies $CR-\mathcal{P}$ exactly when the shift system $(X_F^+,\sigma_F^+)$ satisfies $\mathcal{P}$; whenever the shift system has property $\mathcal{P}$, the relation $F$ has $CR-post\mathcal{P}$; and whenever $F$ has $CR-pre\mathcal{P}$. the shift system has $\mathcal{P}$. We apply this to minimality, dense-orbit transitivity, and transitivity, establishing precise equivalences in each case. Our examples show that, in general, the three CR-versions of a property form a strict hierarchy, with none of the implications reversible without additional assumptions. This is joint work with Iztok Banic, Matevz Crepnjak, Goran Erceg, Ivan Jelic.
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In this talk we show that for a continuum X, the following conditions are equivalent: (i) the continuum X is locally connected, (ii) each non-cut set of X has arbitrarily small open neighborhoods whose complements are connected, (iii) each non-cut set of X has continuum-wise connected complement, (iv) the continuum X is aposyndetic with respect to each of its non-cut sets, and (v) the continuum X is aposyndetic with respect to each of its nonempty closed sets. Co-authors: Raúl Escobedo and Eduardo García-Muñoz.
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Given metric continuum X we consider the n-th symmetric product, Fn(X) defined as the hyperspace of nonempty subsets with at most n elements. The continuum X is an Fn-coselection space (n≥2) if for each ε > 0, there exists a mapping gε : X →Fn(X) \F1(X) such that x ∈ gε(x) and diameter(gε(x)) <ε for each x∈X. Answering two questions by Patricia Pellicer-Covarrubias, in this talk we present two significant examples: (a) we prove that a Cook continuum is not an Fn-coselection space for any n ≥2, and (b) there exist two no homeomorphic compactifications of the ray [0,∞) with remainder a simple closed curve which are F2-coselection spaces.
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Given a positive integer n, a non-empty topological space is said to be 1/n-homogeneous provided there are exactly n orbits for the action of the group of homeomorphisms of the space onto itself, in which case we say that the degree of homogeneity of X, is n. In this talk, I will present our recent contributions to this lines of research.
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This is a continuation of Van Nall's talk {\it Specification on the Lelek Fan}. I will be discussing examples and non-examples of homeomorphisms of the Lelek and Cantor fans with the following properties: transitivity, mixing, shadowing, the specification property and maybe a few more.
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We call a fence any compact metric space whose connected components are either points or arcs. In this talk I will present a very general method for raising maps of the Cantor space to various fences with dense set of endpoints, such as the Lelek Fence (in Complex Dynamics known also as the Hairy Cantor set) and Fraïssé Fence, while preserving the dynamics of the base homeomorphism of the Cantor space. As simple corollaries we obtain that Lelek Fan admits homeomorphisms with various dynamical properties.
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Using as inspiration known and new results about the Fraïssé fence and the flow of its homeomorphism group on the space of chains of compacta, we suggest a new space by using pseudo-arcs instead of arcs and briefly discuss some of our questions and motivations. This is an ongoing project with Benjamin Vejnar (Charles University).
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We show that the family of functions in the paper by Sherzad and Mena that give $G_3$ as the inverse limit space, can be expanded to include upper semi-continuous functions whose graphs have a finite number (or even one) of ``short'' line segments of the form $[x_1,\alpha]\times \{a_i\}$ and $[\alpha,x_2] \times \{b_i\}$ where $0 < x_1 < \alpha < x_2 <1$. This is joint work with Sarezh R. Rasul.
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Research in general topology is important for the study of dynamical systems. The complexity of dynamical systems suggests the existence of complex topological structures in their base spaces. This lecture will discuss the following two topics: (Part 1) Extended Takens-type reconstruction theorems for one-sided dynamical systems, and (Part 2) The existence of indecomposable continua in chaotic dynamical systems.
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Ingram, \cite[Problem 6.56, p.81]{ingram2012introduction}, asked what can be said about the Property of Kelley in the generalized inverse limit space $\varprojlim\{X_i,f_i\}$ where $\{f_i\}$ is a sequence of upper semi-continuous bonding functions. In this work, we give conditions on the projection maps from the graph of the functions $f_i$ to the domain and co-domain such that if the first factor space, in the case of Theorem 2.2, or all factor spaces, in the case of Theorem 2.4, have the Property of Kelley then the generalized inverse limit space $\varprojlim\{X_i,f_i\}$ has the Property of Kelley. Furthermore, we present examples demonstrating that if any condition is dropped then the inverse limit space may not have the Property of Kelley. These results also answers several questions by Charatonik, Mena and Roe \cite{Charatonik2020}. This is joint work with Faruq Mena and Robert Roe.
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It has now been twenty years since the publication of W. T. Ingram and W. S. Mahavier’s landmark paper, “Inverse limits of upper semi-continuous set valued functions” (Houston Journal of Mathematics, 2006, vol. 32, no. 1, p. 119-130). For all these years, generalized inverse limits whose factor spaces are arcs have been studied intensively by researchers around the world. However, generalized inverse limits whose factor spaces are circles have been far less thoroughly studied, and could offer researchers a whole new frontier to explore. We state some questions about these spaces and provide various examples, including an example of a generalized inverse limit on circles (with a single, continuum-valued bonding function) that gives rise to an indecomposable continuum.
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A dynamical system is said to have the shadowing property provided that approximate orbits are well-approximated by true orbits. It has previously been established that for a continuum belonging to certain classes of continua, shadowing is a common, i.e. generic, property in its space of continuous self-maps. In particular, this is known for manifolds and for locally connected one-dimensional continua. We demonstrate that shadowing is a generic property in the space of continuous self-maps for any continuum which admits retractions onto graphs.
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For a continuum, we consider the equivalence relation in which two points are equivalent if they can be joined by an arc. This equivalence relation is analytic in general (i.e. a continuous image of a Polish space). Recently, Debs and Saint Raymond proved that, for planar continua, this equivalence relation is always Borel measurable. We show that for every planar continuum, the arc-connection relation is in fact Borel reducible to the Vitali equivalence relation, where two real numbers are equivalent if their difference is rational. Moreover, the Knaster continuum is an example where this complexity is attained. This is joint work with Michal Hevessy and Yusuf Uyar. We also investigate several related questions concerning continuum-wise connectivity.
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In this talk, we study the dynamical behavior of hyperspace maps induced by continuous functions on dendrites. Our main goal is to show that if $X$ is a dendrite and $f : X \to X$ is a continuous map for which every point of $X$ is periodic, then the induced map \[ 2^f : 2^X \to 2^X \] does not admit Li--Yorke pairs. To establish this result, we analyze two fundamental cases that capture the combinatorial structure of dendrites: closed intervals and trees.
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It is known that under the assumption of chain transitivity, shadowing is equivalent to other, weaker variations of shadowing. For example, a sequence of points in a continuum may act as a pseudo-orbit only on a thick set. We know that such a sequence can be shadowed on a different thick set under the assumption of chain transitivity and shadowing, but we lose information about where the pseudo-orbit begins. To address this, we study a form of shadowing in which the pseudo-orbit is shadowed on a thick set $T \subseteq \mathbb N$ such that $1 \in T$. We discuss the relationship of this form of shadowing with the standard shadowing property in the context of dynamical systems on continua.
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In 1971, Bellamy proved that every continuum can be embedded in an indecomposable continuum as a retract. In 2017, together with Fukaishi, we showed that any continuum 𝑍 can be embedded as an open retract with Cantor set fibers in an indecomposable continuum, which is obtained as the closure of a countable union of topological copies of 𝑍. Building on this construction, together with Ortega, we investigate, for a fixed continuum 𝑍, the class of all indecomposable continua arising in this manner. We present the result that this class admits no common model, in the sense that there exists no single continuum admitting continuous surjections onto all members of the class. If time permits, we will also discuss other classes of continua that admit no common models.
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Given a self-map $f$ of a compact metric space $X$, one can associate to it the induced mappings $\overline{f}$ and $\tilde{f}$ on the hyperspace $2^X$ of compact subsets of $X$ and on the hyperspace $C(X)$ of continua in $X$, respectively, both defined in a natural way. Within this framework, it is natural to investigate the relationship between individual and collective dynamics. In this talk, we address the following question. Let $f$ be a self-map of a topological tree $T$, and let $x$ be a periodic point of $f$ with period $p$. What are the possible periods of periodic points of $\left(C(T), \tilde{f}\right)$, that is, of periodic subtrees containing $x$? We then discuss the significance of this result for the study of further properties of the system $\left(C(T), \tilde{f}\right)$. In particular, using this result, we show that the induced system is always almost equicontinuous and we characterize its Birkhoff center. \emph{The talk is based on a joint work with Piotr Oprocha.}
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Given a unimodal map, the recurrent critical point $c$ is reluctantly recurrent if there exists a $\delta > 0$ such that for every $\ell\in \mathbb{N}$ there is a backward orbit $\overline{x} = (x_{-\ell},\ldots, x_{-2},x_{-1},x)$ in $\omega(c)$ such that $B(x,\delta)$ has a monotonic pull-back along $\overline{x}$; otherwise we say that $c$ 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, then it is not possible for $c$ to be persistently recurrent.
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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 canals and also 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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Recent work of Piotr Oprocha and his collaborators has provided a number of delicate examples of dynamical systems separating specification, shadowing, and periodic-point density, primarily in symbolic or totally disconnected spaces. The goal of the present paper is to demonstrated that similar - and in some cases sharper - separations occur on the Lelek fan, a continuum that can be embedded in a Cantor fan. Our constructions rely on Mahavier products of closed relations. By carefully choosing relations on the unit interval, we obtain Mahavier products that are homeomorphic to the Lelek fan whose associated shift maps display diverse dynamical behavior. This approach yields a unified framework for producing and analyzing examples on a familiar continuum.
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A spiral is a compactification of the ray [0,1) with remainder a simple closed curve. In this talk we will discuss how spirals have appeared in important results of Continuum Theory, including some new results.
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A fan X is said to be Lelek-like if it has a dense set of endpoints. We show that there are uncountably many pairwise non-homeomorphic Lelek-like fans, each of which admits a topologically mixing non-invertible map as well as a topologically mixing homeomorphism.
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In 1980, David Bellamy constructed the first example of a tree-like continuum which does not have the fixed-point property. Several others have been constructed since, most recently in 2018 by Rodrigo Hern\'{a}ndez-Guti\'{e}rrez and Logan Hoehn. Their example is expressed as an inverse limit on trees, each of which is an arc with simple triods attached at select points. In this talk, I discuss the construction from Hern\'{a}ndez-Guti\'{e}rrez and Hoehn and give an overview of work towards a similar example where each factor space has branch points of lower degree. Joint work with Logan Hoehn.
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A fan is an arcwise-connected continuum that is hereditarily unicoherent and has exactly one ramification point. Many known examples of fans have been constructed as one-dimensional continua that are unions of arcs intersecting in exactly one point. In 1954, Borsuk proved that every fan is a one-dimensional continuum that can be expressed as a union of arcs intersecting in exactly one point. However, it is still unknown whether this property characterizes fans. In this talk, I will show under which additional assumptions every such union of arcs is indeed a fan. This is joint work with Iztok Banič, Alejandro Illanes, Ivan Jelić, Judy Kennedy, and Van Nall.
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We discuss and illustrate a key ingredient to a recent positive solution \[Brazas, Conner, F, Kent\] to the following problem posed by Jerzy Dydak in 2011. If the continuous surjection $\Pi :X\rightarrow D^{2}$ has unique based path lifting, must $\Pi$ be a homeomorphism, provided $X$ is a connected, locally path connected metric space and $D^{2}$ is the unit disk? The answer is "yes", but ruling out the possibility of a counterexample is nontrivial, and ultimately reduces to the question of whether $X$ could be a certain topological R-tree comprised of all $p$ based irreducible paths in $% D^{2}$. The irreducible paths $\alpha$ in $D^{2}$ are those such that every nonconstant subloop of $\alpha$ fails to lift to some loop in some dendrite. For example piecewise linear, and more geneally, piecewise irreducible paths in $D^{2}$ lift uniquely to $X,$ up to basepoint. The challenge is to exhibit a path in $D^{2}$ which does not lift uniquely to $X.$ Illustrating a method to do this is the main goal of the talk, and the tactic is as follows. Every dendrite is a quotient of a topological disk so that each point preimage intersects the boundary of the disk. However, mating two respective dendrite partitions of two unit half disks, reveals that the join of the quotients need not be a dendrite.
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