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Topological Dynamics and Continuum Theory
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Submissions closed on 2025-06-15 11:59PM [Central Time (US & Canada)].

Talks related to topological dynamics, continuum theory, and the interaction between them. Organizers: Lori Alvin, James Kelly.

Accepted Submissions:

k-type chaos of Zd-actions — Anshid Aboobacker Icon: submission_accepted

In this talk, we define and study the notions of k−type proximal pairs, k−type asymptotic pairs and k−type Li Yorke sensitivity for dynamical systems given by Zd actions on compact metric spaces. We prove the Auslander-Yorke dichotomy theorem for k−type notions. The preservation of some of these notions under conjugacy is also studied. We also study relations between these notions and their analogous notions in the usual dynamical systems.

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Building Continua with non-trivial self covers — Mathew Timm Icon: submission_accepted

We will look at several methods for building continua with non-trivial self covers and discuss their relationships with some problems in topology and group theory.

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Chaos on Peano continua — Klára Karasová Icon: submission_accepted

Among all notions of chaos, there are three widely accepted: Devaney chaos, Li-Yorke chaos and (positive) topological entropy. It is known that exact Devaney chaos, i.e. an exact map with dense set of periodic points, satisfies all these three notions. Various results establish the existence of maps with properties related to chaos (e.g., transitivity) for specific spaces such as the interval, the Cantor set or the Lelek fan, as well as for broader classes, including manifolds and dendrites. Furthermore, chaotic behavior often emerges as a generic phenomenon in the sense of Baire category. Together with Benjamin Vejnar, we prove that every Peano continuum (i.e. a locally connected continuum) admits exact Devaney chaos. Additionally, we generalize some prior results by showing that if a Peano continuum X satisfies the condition that selfmaps locally constant on some dense open subset form a dense subset of all selfmaps, then: • exactly Devaney chaotic maps form a dense subset of chain transitive self- maps of X, • mixing is generic among chain transitive self-maps of X, • shadowing is generic among all self-maps of X.

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Elliptic sectors and heteroclinic regions in real time holomorphic flows — Nicolas Kainz Icon: submission_accepted

The geometric description of the phase space of holomorphic dynamical systems with real time is a crucial research field. In this context, the globalization of local structures is of particular interest. For example, the local structure of an equilibrium of order m∈N∖{1} has already been sufficiently investigated and characterized. Locally, there exist 2m−2 elliptic sectors, all consisting of homoclinic trajectories tending to the equilibrium in both time directions. Now the question arises how this local structure can be globalized using analytical tools, as is the case, for example, with the basin of attraction of nodes and foci. I present a method to define a global elliptic sector based on so-called “sector-forming orbits” and show some topological properties for it: The global elliptic sector is open, flow-invariant, path-connected, and simply connected. Moreover, all orbits are nested inside each other, consistent with the intuitive notion of an elliptic sector. Furthermore, for the case m≥3, it coincides with the naively defined global elliptic sector, which merely contains all homoclinic trajectories with adjacent definite directions. This gives us an analytic precise definition of a globalization of a locally defined elliptic sector, together with important and useful topological properties. Moreover, it is possible to investigate the geometrical structure that can occur between two global elliptic sectors with no common boundary near the equilibrium. In this context, the question also arises as to how many so-called “heteroclinic regions” can appear between two such sectors.

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Equivalence of equicontinuity and distality for real non-autonomous systems — Sushmita Yadav Icon: submission_accepted

This talk will focus on the topological dynamics of a non-autonomous dynamical system (X,F), where X is a compact metric space and F={f1,f2,…} is a sequence of continuous surjective functions on X. In particular we will discuss equicontinuity and distality for non-autonomous systems on the interval. We will discuss the distality of the system using the enveloping cover E0(X)={ωk:k∈Z}― (where ωn=fn∘fn−1...∘f1). We use analytical tools to establish the equivalence of distality and equicontinuity for non-autonomous systems on the interval.

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Generalized Proinov-type contractions using simulation functions with applications to fractals — Ramesh Kumar Devaraj Icon: submission_accepted

The intention of this article is to introduce a generalization of Proinov-type contraction via simulation functions. We name this generalized contraction map as Proinov-type Z-contraction. This article establishes the existence and uniqueness of fixed points for these contraction mappings in quasi-metric space and also, include explanatory examples with graphical interpretation. As an application, we generate a new iterated function system (IFS) consisting of Proinov-type Z-contractions in quasi-metric spaces. At the end of the paper, we prove the existence of a unique attractor for the IFS consisting of Proinov-type Z-contractions.

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Inverse Limits with Smith Functions and Indecomposability — Scott Varagona Icon: submission_accepted

We say a set-valued u.s.c. function f from [0,1] to [0,1] is a Smith function if f is surjective, the graph of f is connected, and the graph of f is the union of finitely many horizontal and vertical line segments. The author introduced inverse limits with Smith functions in a presentation at the 2021 Spring Topology and Dynamical Systems Conference. Later, in a 2023 paper, the author answered some questions posed by audience members at that 2021 talk, and he raised some new questions as well. This presentation at the 2025 Summer Topology and Its Applications Conference will discuss our further progress on the study of inverse limits with Smith functions, including some new results and conjectures. Our focus will be the case where the inverse limit is a continuum, in which case we wish to determine when such an inverse limit could be indecomposable.

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Lelek-like fans — Ivan Jelić Icon: submission_accepted

The Lelek fan is the only smooth fan that has a dense set of end-points. In this talk, we study non-smooth fans with this property and construct an uncountable family of pairwise non-homeomorphic such fans.

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Neighborhood N-Shadowing — Elyssa Stephens Icon: submission_accepted

We discuss a variation of the shadowing property, called neighborhood N-shadowing, and various dynamical systems with this property. Specifically, we consider neighborhood 2-shadowing with a focus on shift spaces. We discuss progress on characterizing neighborhood 2-shadowing in shift spaces in terms of the language of the shifts, drawing parallels to the known result that shifts of finite type are exactly those shift spaces with the shadowing property.

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On the set function ℘ — Sergio Macias Icon: submission_accepted

Inspired on the work that Professor Janusz R. Prajs did on homogeneous metric continua in his paper *Mutually Aposyndetic Decomposition of Homogeneous Continua*, [Canad. J. Math., 62 (2010), 182-201] and the version of his work for Hausdorff continua with the uniform property of Effros done by this author, we introduce a new set function, ℘, and present properties of it.

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On various forms of independence and minimality for general triangular systems — Deepanshu Dhawan Icon: submission_accepted

In this talk, we will discuss various notions of independence for general non-autonomous systems. Further, we use the notions to investigate dynamics of a general triangular system. In particular, we investigate the dynamics of a minimal triangular system and relate it to the dynamics of its component systems.

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Plane embeddings of continua, and accessible points — Logan Hoehn Icon: submission_accepted

A point p in a plane continuum X⊂R2 is accessible if there exists an arc A⊂R2 such that A∩X=p. I will describe our recent results about plane embeddings of continua and their accessible points. Specifically, I will discuss arc-like continua (the Nadler-Quinn problem), Knaster continua, and Ingram's atriodic triod-like continuum. This is joint work with Andrea Ammerlaan and Ana Anušić.

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Retract or Not: A Tale of Two Fans — Iztok Banic Icon: submission_accepted

In this talk, we present structural and dynamical aspects of certain arcwise connected continua known as fans. First, we present conditions under which embeddings of the Lelek fan admit retractions, focusing on how features such as wedges and cuts influence retraction properties. Second, we address a classical open question about characterizing fans as unions of arcs intersecting in a single point. This is joint work with Goran Erceg, Sina Greenwood, Ivan Jelic, Judy Kennedy, and Van Nall.

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Specification in Mahavier Systems via Closed Relations — Goran Erceg Icon: submission_accepted

We study two types of specification properties - standard and initial - and extend them to CR-dynamical systems, where the dynamics are given by closed relations instead of continuous functions. Although these properties are often equivalent in classical settings, we show they can behave differently in this broader context. We define new specification-type properties for Mahavier dynamical systems and present several examples that highlight their differences. Each new property matches the classical specification property when applied to continuous functions. This is joint work with Iztok Banič, Ivan Jelić and Judy Kennedy

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Speedups of Toeplitz Flows — Lori Alvin Icon: submission_accepted

Given a minimal Cantor system (X,T), a topological speedup of (X,T) is a dynamical system (X,S) where S is a homeomorphism such that S(x)=Tp(x)(x) for some function p:X→N. We assume the function p is continuous (and thus bounded) and the resulting system (X,S) is minimal. One can ask what properties of the underlying initial system (X,T) are preserved under minimal bounded speedups. We investigate the class of Toeplitz flows, which are minimal symbolic almost one-to-one extensions of odometers. Although the minimal bounded speedup of an odometer is always a conjugate odometer, we demonstrate that the minimal bounded speedup of a Toeplitz flow need not be Toeplitz. We then provide sufficient conditions to guarantee that the minimal bounded speedup will be a Toeplitz flow; in this case, it is never conjugate to the original Toeplitz flow but has the same underlying odometer.

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Turbulent closed relations — Judy Kennedy Icon: submission_accepted

In classical dynamical systems, turbulence has played a pivotal role in understanding chaotic behavior, particularly for interval maps. This talk extends the notion of turbulence from continuous functions to closed relations on compact metric spaces, utilizing Mahavier products and associated shift maps. We define and explore CR-turbulence (Closed Relation Turbulence) and its variants, establishing connections between turbulence and topological entropy in the setting of closed relations. This is joint work with Chris Mouron and Van Nall.

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