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Iztok Banic

Publications and source records attributed to Iztok Banic.

13 recordsLinked to original sources
arXiv2025

Retract or Not: A Tale of Two Fans

Let $X$ be a Lelek fan or a Cantor fan and let $Y$ be a Lelek fan or a Cantor fan. In this paper, we study embeddings $ f: X \to Y $ that admit retractions from $ Y $ onto $ f(X)$. In 1989, W. J. Charatonik and J. J. Charatonik proved that if $X$ is a Lelek fan and $Y$ is a Cantor fan, then no embedding $f$ of $X$ into $Y$ admits a retraction from $Y$ onto $f(X)$. They also showed that if both $X$ and $Y$ are Cantor fans, then every embedding $f$ of $X$ into $Y$ admits such a retraction. In this paper, we address the two remaining cases. First, we consider the situation where $X$ is a Cantor fan and $Y$ is a Lelek fan. We prove that in this case, every embedding $f$ of $X$ into $Y$ admits a retraction from $Y$ onto $f(X)$. Second, we examine the case where both $X$ and $Y$ are Lelek fans. Here, we show that there exist embeddings $f$ that do admit a retraction from $Y$ onto $f(X)$, as well as embeddings that do not. For this latter case, we also identify additional properties of embeddings that ensure the existence of a retraction from $Y$ onto $f(X)$.

math.GN↗
arXiv2025

Lelek-like Fans: Endpoint-dense Continua Supporting Topologically Mixing Maps

The Lelek fan is the only smooth fan that has a dense set of end-points. In this paper, we study non-smooth fans with this property; i.e., we construct an uncountable family of pairwise non-homeomorphic such fans. Furthermore, we prove that each of them admits a topologically mixing non-invertible mapping as well as a topologically mixing homeomorphism.

math.DS↗
arXiv2023

Quotients of dynamical systems and chaos on the Cantor fan

Let $(X,f)$ be a dynamical system. Using an equivalence relation $\sim$ on $X$, we introduce the quotient $(X/_{\sim},f^{\star})$ of the dynamical system $(X,f)$. In the first part of the paper, we give new results about sensitive dependence on initial conditions of $(X/_{\sim},f^{\star})$, transitivity of $(X/_{\sim},f^{\star})$, and periodic points in $(X/_{\sim},f^{\star})$. In the second part of the paper, we use these results to study chaotic functions on the Cantor fan. Explicitly, we study functions $f$ on the Cantor fan $C$ such that (1) $(C,f)$ is chaotic in the sense of Devaney, (2) $(C,f)$ is chaotic in the sense of Robinson but not in the sense of Devaney, and (3) $(C,f)$ is chaotic in the sense of Knutzen but not in the sense of Devaney. We also study chaos on the Lelek fan.

math.DS↗
arXiv2023

Chaos and mixing homeomorphisms on fans

We construct a mixing homeomorphism on the Lelek fan. We also construct a mixing homeomorphism on the Cantor fan. Then, we construct a family of uncountably many pairwise non-homeomorphic (non-)smooth fans that admit a mixing homeomorphism.

math.DS↗
arXiv2023

An uncountable family of non-smooth fans that admit transitive homeomorphisms

Recently, many examples of smooth fans that admit a transitive homeomorphism have been constructed. For example, a family of uncountably many pairwise non-homeomorphic smooth fans that admit transitive homeomorphisms was constructed. In this paper, we construct a family of uncountably many pairwise non-homeomorphic non-smooth fans that admit transitive homeomorphisms.

math.DS↗
arXiv2022

Sufficient conditions for non-zero entropy and finite relations

We introduce the notions of returns, dispersions and well-aligned sets for closed relations on compact metric spaces and then we use them to obtain non-trivial sufficient conditions for such a relation to have non-zero entropy. In addition, we give a characterization of finite relations with non-zero entropy in terms of Li-Yorke and DC2-chaos.

math.DS↗
arXiv2022

The Lelek fan as the inverse limit of intervals with a single set-valued bonding function whose graph is an arc

We consider a family of inverse limits of inverse sequences of closed unit intervals with a single upper semi-continuous set-valued bonding function whose graph is an arc; it is the union of two line segments in $[0,1]^2$, both of them contain the origin $(0, 0)$, have positive slope, and extend to the opposite boundary of $[0,1]^2$. We show that there is a large subfamily $\mathcal F$ of these bonding functions such that for each $f\in \mathcal F$, the inverse limit of the inverse sequence of closed unit intervals using $f$ as a single bonding function, is homeomorphic to the Lelek fan.

math.GN↗
arXiv2022

Minimal dynamical systems with closed relations

We introduce dynamical systems $(X,G)$ with closed relations $G$ on compact metric spaces $X$ and discuss different types of minimality of such dynamical systems, all of them generalizing minimal dynamical systems $(X,f)$ with continuous function $f$ on a compact metric space $X$.

math.DS↗
arXiv2022

Closed relations with non-zero entropy that generate no periodic points

The paper is motivated by E. Akin's book about dynamical systems and closed relations [A], and by J. Kennedy's and G. Erceg's recent paper about the entropy of closed relations on closed intervals [EK]. In present paper, we introduce the entropy of a closed relation G on any compact metric space X and show its basic properties. We also introduce when such a relation G generates a periodic point or finitely generates a Cantor set. Then we show that periodic points, finitely generated Cantor sets, Mahavier products and the entropy of closed relations are preserved by topological conjugations. Among other things, this generalizes the well-known results about the topological conjugacy of continuous mappings. Finally, we prove a theorem, giving sufficient conditions for a closed relation G on [0,1] to have a non-zero entropy. Then we present various examples of closed relations G on [0,1] such that (1) the entropy of G is non-zero, (2) no periodic point or exactly one periodic point is generated by G, and (3) no Cantor set is finitely generated by G.

math.DS↗
arXiv2021

Characterizations of $\mathcal P$-like continua that do not have the fixed point property

We give two characterizations of $\mathcal P$-like continua $X$ that do not have the fixed point property. Both characterizations are stated in terms of sequences of open covers of $X$ that follow fixed-point-free patterns. We use these to characterize planar tree-like continua that do not have the fixed point property in terms of infinite sequences of tree-chains in the plane that follow fixed-point-free patterns. We also establish a useful relationship between these tree-chains and commutative simplicial diagrams that we use later to construct a finite sequence (of any given length) of tree-chains in the plane that follows a fixed-point-free pattern. An earlier characterization of $\mathcal P$-like continua with the fixed point property was given in 1994 by Feuerbacher based on a 1963 result by Mioduszewski. The Mioduszewski-Feuerbacher characterization is expressed in terms of almost commutative inverse diagrams. In contrast, our approach is more geometric, and it may potentially lead to new methods in the elusive search for a planar tree-like continuum without the fixed-point property.

math.DS↗
arXiv2014

Integrations on rings

In calculus, an indefinite integral of a function $f$ is a differentiable function $F$ whose derivative is equal to $f$. In present paper, we generalize this notion of the indefinite integral from the ring of real functions to any ring. The main goal of the paper is to focus on the properties of such generalized integrals that are inherited from the well-known basic properties of indefinite integrals of real functions.

math.RA↗