arXiv subjects

Iztok Banič

Publications and source records attributed to Iztok Banič.

8 recordsLinked to original sources

arXiv2026

Towards the complete classification of fans

A fan is an arcwise-connected continuum, which is hereditarily unicoherent and has exactly one ramification point. Many of the known examples of fans were constructed as 1-dimensional continua that are unions of arcs which intersect in exactly one point. Borsuk proved in 1954 that each fan is a 1-dimensional continuum which is the union of arcs intersecting in exactly one point. But it is not yet known if this property is equivalent to being a fan. In this paper, we show that under two additional assumptions, every such union of arcs is a fan.

math.DS↗

arXiv2025

Specification in Mahavier Systems via Closed Relations

We study two fundamental properties of topological dynamical systems, the specification property and the initial specification property, and explore their generalizations to the broader setting of CR-dynamical systems, where the dynamics are governed by closed relations rather than continuous functions. While these two properties are equivalent for many classical systems, we demonstrate that their generalizations to CR-dynamical systems often lead to distinct behaviors. Applying them to Mahavier dynamical systems, we introduce new specification-type properties. These generalized notions extend the classical theory and reveal rich structural differences in dynamical behavior. Moreover, each of the new properties reduces to the standard specification property when restricted to continuous functions.

math.DS↗

arXiv2023

An embedding of the Cantor fan into the Lelek fan

The Lelek fan $L$ is usually constructed as a subcontinuum of the Cantor fan in such a way that the set of the end-points of $L$ is dense in $L$. It easily follows that the Lelek fan is embeddable into the Cantor fan. {It is also a well-known fact that the Cantor fan is embeddable into the Lelek fan, but this is less obvious. When proving this, one usually uses the well-known result by Dijkstra and van Mill that the Cantor set is embeddable into the complete Erdös space, and the well-known fact by Kawamura, Oversteegen, and Tymchatyn that the set of end-points of the Lelek fan is homeomorphic to the complete Erdös space. Then, the subcontinuum of the Lelek fan that is induced by the embedded Cantor set into the set of end-points of the Lelek fan, is a Cantor fan. In our paper, we give an alternative straightforward construction of a Cantor fan into the Lelek fan. We do not use the fact that the Cantor set is embeddable into the complete Erdös space and that it is homeomorphic to the set of end-points of the Lelek fan. Instead, we use our recent techniques of Mahavier products of closed relations to produce an embedding of the Cantor fan into the Lelek fan. Since the Cantor fan is universal for the family of all smooth fans, it follows that also the Lelek fan is universal for smooth fans.

math.GN↗

arXiv2023

Transitive mappings on the Cantor fan

Many continua that admit a transitive homeomorphism may be found in the literature. The circle is probably the simplest non-degenerate continuum that admits such a homeomorphism. On the other hand, most of the known examples of such continua have a complicated topological structure. For example, they are {indecomposable} (such as the pseudo-arc or the Knaster bucket-handle continuum), or they are {not indecomposable} but have some other complicated topological structure, such as a dense set of ramification points (such as the Sierpi\' nski carpet) or a dense set of end-points (such as the Lelek fan). In this paper, we continue our mission of finding continua with simpler topological structures that admit a transitive homeomorphism.} We construct a transitive homeomorphism on the Cantor fan. {In our approach, we use four different techniques, each of them giving a unique construction of a transitive homeomorphism on the Cantor fan:} two techniques using quotient spaces of products of compact metric spaces and Cantor sets, and two using Mahavier products of closed relations on compact metric spaces. {We also demonstrate how our technique using Mahavier products of closed relations may be used to } construct a transitive function $f$ on a Cantor fan $X$ such that $\varprojlim(X,f)$ is a Lelek fan.

math.DS↗

arXiv2023

Span of a Graph: Keeping the Safety Distance

Inspired by Lelek's idea from [Disjoint mappings and the span of spaces, Fund. Math. 55 (1964), 199 -- 214], we introduce the novel notion of the span of graphs. Using this, we solve the problem of determining the \emph{maximal safety distance} two players can keep at all times while traversing a graph. Moreover, their moves must be made with respect to certain move rules. For this purpose, we introduce different variants of a span of a given connected graph. All the variants model the maximum safety distance kept by two players in a graph traversal, where the players may only move with accordance to a specific set of rules, and their goal: visit either all vertices, or all edges. For each variant, we show that the solution can be obtained by considering only connected subgraphs of a graph product and the projections to the factors. We characterise graphs in which it is impossible to keep a positive safety distance at all moments in time. Finally, we present a polynomial time algorithm that determines the chosen span variant of a given graph.

math.CO↗

arXiv2022

A transitive homeomorphism on the Lelek fan

Let $X$ be a continuum and let $φ:X\rightarrow X$ be a homeomorphism. To construct a dynamical system $(X,φ)$ with interesting dynamical properties, the continuum $X$ often needs to have some complicated topological structure. In this paper, we are interested in one such dynamical property: transitivity. By now, various examples of continua $X$ have been constructed in such a way that the dynamical system $(X,φ)$ is transitive. Mostly, they are examples of continua that are not path-connected, such as the pseudo-arc or the pseudo-circle, or they are examples of locally connected continua (and every locally connected continuum is path-connected), Wazewski's universal dendrite and the Sierpinski carpet are such examples. In this paper, we present an example of a dynamical system $(X,φ)$, where $φ$ is a homeomorphism on the continuum $X$ and $X$ is a path-connected but not locally connected continuum. We construct a transitive homeomorphism on the Lelek fan. As a by-product, a non-invertible transitive map on the Lelek fan is also constructed.

math.DS↗

arXiv2022

Transitive points in CR-dynamical systems

We study different types of transitive points in CR-dynamical systems (X,G) with closed relations G on compact metric spaces X. We also introduce transitive and dense orbit transitive CR-dynamical systems and discuss their properties and the relations between them. This generalizes the notion of transitive topological dynamical systems (X, f ).

math.DS↗

arXiv2015

Markov pairs, quasi Markov functions and inverse limits

We show that two inverse limits of inverse sequences of closed intervals and quasi Markov bonding functions are homeomorphic, if the inverse sequences follow the same pattern. This significantly improves Holte's result about when two inverse limits of inverse sequences with Markov interval maps as bonding functions are homeomorphic. Our result improves Holte's result in several directions: (1) it generalizes finite Markov partitions to Markov pairs of sets that may even be uncountable, (2) it generalizes Markov interval maps $I\rightarrow I$ to quasi Markov functions $I\rightarrow J$ (so the domains and the codomains of the bonding functions are not necessarily the same interval), (3) we no longer require the bonding functions to be surjective, and (4) we no longer require the bonding functions to be continuous.

math.DS↗