arXiv subjects

Benjamin Vejnar

Publications and source records attributed to Benjamin Vejnar.

At least 19 recordsLinked to original sources

arXiv2026

Classification complexity of homeomorphism group actions

In this paper, we study how the classification complexity of natural orbit equivalence relations changes when the full homeomorphism group of a compact metrizable space is replaced by a dense non-closed subgroup. For a compact space $X$ and a subgroup $G \leq \mathcal{H}(X)$, we consider three canonical actions: the left shift action on $\mathcal{H}(X)$, the induced hyperspace action on $\mathcal{F}(X)$, and the conjugation action on $G$ We first analyze subgroups of the group $\mathcal{H}^+([0,1])$ of increasing interval homeomorphisms, focusing on bi-Lipschitz homeomorphisms, diffeomorphisms, and bi-absolutely continuous homeomorphisms. We show that, in contrast to the behavior of closed subgroups, passing to these subgroups strictly increases the complexities of the associated classification problems or makes them incomparable with the corresponding full-group relations. In the second part, we investigate hyperspace actions of bi-absolutely continuous homeomorphisms on the Cantor space and the Hilbert cube with respect to some Borel probability measure and show that a similar behavior occurs on these spaces as well.

math.LO↗

arXiv2026

The converse to Borsuk's result on fans fails

A fan is an arc-wise connected hereditarily unicoherent continuum with exactly one branching point. By a result of Borsuk, every fan is a 1-dimensional continuum that can be expressed as the union of a family of arcs, each pair of which intersects in the branching point. In this paper, we prove that the converse does not hold by providing a more general result.

math.GN↗

arXiv2026

Dynamics on fences

Homeomorphisms of the Cantor set play a central role in topology, dynamical systems and descriptive set theory. In parallel, several classes of fence-like spaces - such as the hairy Cantor set, hairy arcs, Cantor bouquets in complex dynamics, the Lelek fan in topology and Fra\"iss\'e fence in descriptive set theory - have recently been studied for their rich structural and dynamical properties. In this paper, we introduce a general construction that associates to each homeomorphism of the Cantor set a canonically defined homeomorphism of a corresponding fence space. This construction lifts dynamical properties from the Cantor set to these fence-like spaces, allowing one to systematically transfer features such as minimality, recurrence, and orbit structure. As a consequence, we obtain a unified framework for studying dynamics on a broad class of fence-like spaces and establish new connections between their topological structure and induced dynamical behavior.

math.DS↗

arXiv2026

The Complexity of Connectedness Relations on Polish Spaces

We systematically investigate three different equivalence relations of connectedness: being connected by arcs, being connected by continua and being connected by chains of continua of decreasing diameter. The investigation is conducted from the point of view of Borel reductions, mainly on Polish spaces. All of the studied equivalence relations turn out to be tied together and intimately related to the arc-connection relation. Among other results, it is shown that the arc-connection relation in the plane is Borel reducible to the Vitali equivalence relation and thus of a very low complexity. The same is proven for the chain continuum-connection relation on locally compact subsets of the plane, on which the continuum-connection relation is shown to have higher complexity.

math.GN↗

arXiv2025

Linear orders on chainable continua

We define and study certain linear orders on chainable continua. Those orders depend on a sequence of chains obtained from definition of chainability and on a fixed non-principal ultrafilter on the set of natural numbers. An alternative method of defining linear orders on a chainable continuum $X$ uses representation of $X$ as an inverse sequence of arcs and fixed non-principal ultrafilter on $\mathbb{N}$. We compare those two approaches. We prove that there exist exactly $2$ distinct ultrafilter orders on any arc, exactly $4$ distinct ultrafilter orders on the Warsaw sine curve, and exactly $2^{\mathfrak{c}}$ distinct ultrafilter orders on the Knaster continuum. We study the order type of various chainable continua equipped with an ultrafilter order and prove that a chainable continuum $X$ is Suslinean if and only if for every ultrafilter order $\leq_{\mathcal{U}}^{\mathcal{D}}$ on $X$, the space $(X, \leq_{\mathcal{U}}^{\mathcal{D}})$ is order isomorphic to $([0,1],\leq)$. We study also descriptive complexity of ultrafilter orders on chainable continua. We prove that the existence of closed ultrafilter order characterizes the arc and we show that for Suslinean chainable continua, any ultrafilter order is both of type $F_{\sigma}$ and $G_{\delta}$. On the other hand, we prove that there is no analytic and no co-analytic ultrafilter order on the Knaster continuum.

math.GN↗

arXiv2025

Chaos on Peano continua

Generalizing the result of Agronsky and Ceder (1991), we prove that every Peano continuum admits a continuous transformation that is exact Devaney chaotic; that is, it has a dense set of periodic points, and every nonempty open set covers the entire space in finitely many iterations. We identify a natural class of Peano continua, containing all one-dimensional continua and all absolute neighborhood retracts, which allows us to create locally small perturbations. Using this method, we prove that within these specific classes of continua, exact Devaney chaotic systems are dense in all chain transitive systems, mixing systems are generic among chain transitive systems and shadowing is generic among all continuous systems.

math.DS↗

arXiv2024

Generically hereditarily equivalent continua and topological characterization of generic maximal chains of generalized Wa\.zewski dendrites

The notion of hereditarily equivalent continua is classical in continuum theory with only two known nondegenerate examples (arc, and pseudoarc). In this paper we introduce generically hereditarily equivalent continua, i.e. continua which are homeomorphic to comeager many subcontinua. We investigate this notion in the realm of Peano continua and we prove that all the generalized Wa\.zewski dendrites are such. Consequently, we study maximal chains consisting of subcontinua of generalized Wa\.zewski dendrites and we prove that there is always a generic orbit under the homeomorphism group action. As a part of the proof we provide a topological characterization of the generic maximal chain.

math.GN↗

arXiv2024

On the weak$^*$ separability of the space of Lipschitz functions

We conjecture that whenever $M$ is a metric space of density at most continuum, then the space of Lipschitz functions is $w^*$-separable. We prove the conjecture for several classes of metric spaces including all the Banach spaces with a projectional skeleton, Banach spaces with a $w^*$-separable dual unit ball and locally separable complete metric spaces.

math.FA↗

arXiv2024

Classification complexity of chaotic systems

In this paper, we deal with the classification complexity of continuous (Devaney) chaotic systems in dimensions $0,1$ and $\infty$ using the framework of invariant descriptive set theory. We identify the complexity in dimensions $0$ and $\infty$, while in dimension $1$ we get some partial results. More precisely, we prove the topological conjugacy relation of invertible chaotic systems on the Hilbert cube (resp. on all compact metric spaces) has the same complexity as (i.e. is Borel bireducible with) the universal orbit relation induced by a Polish group. As a consequence, this answers a recent question asked by L. Ding. We also prove that the topological conjugacy relation of invertible chaotic systems on the Cantor space has the same complexity as the universal relation induced by the group $S_\infty$. This answers a recent question by M. Foreman. Some non-trivial bounds on the classification complexity of chaotic systems on the interval and on the circle are also obtained. Namely, the lower bound is the Vitali equivalence relation, and the upper bound is the equality of countable sets of reals. This especially implies that the relation is Borel. However, the exact complexity remains unknown.

math.DS↗

arXiv2024

Compact spaces homeomorphic to their respective squares

We deal with topological spaces homeomorphic to their respective squares. Primarily, we investigate the existence of large families of such spaces in some subclasses of compact metrizable spaces. As our main result we show that there is a family of size continuum of pairwise non-homeomorphic compact metrizable zero-dimensional spaces homeomorphic to their respective squares. This answers a question of W. J. Charatonik. We also discuss the situation in the classes of continua, Peano continua and absolute retracts.

math.GN↗

arXiv2022

The hyperspace of non-blockers of singletons, all the possible examples

Given a metric continuum $X$, a nonempty proper closed subspace $B$ of $X$, does not block a point $p\in X\setminus B$ provided that the union of all subcontinua of $X$ containing $p$ and contained in $X\setminus B$ is a dense subset of $X$. The collection of all nonempty proper closed subspaces $B$ of $X$ such that $B$ does not block any element of $X\setminus B$ is denoted by $NB(F_{1}(X))$. In this paper we prove that for each completely metrizable and separable space $Z$, there exists a continuum $X$ such that $Z$ is homeomorphic to $NB(F_{1}(X))$. This answers a series of questions by Camargo, Capul\'in, Casta\=neda-Alvarado and Maya.

math.GN↗

arXiv2022

Borel Measurable Hahn-Mazurkiewicz Theorem

It is well known due to Hahn and Mazurkiewicz that every Peano continuum is a continuous image of the unit interval. We prove that an assignment, which takes as an input a Peano continuum and produces as an output a continuous mapping whose range is the Peano continuum, can be realized in a Borel measurable way. Similarly, we find a Borel measurable assignment which takes any nonempty compact metric space and assigns a continuous mapping from the Cantor set onto that space. To this end we use the Burgess selection theorem. Finally, a Borel measurable way of assigning an arc joining two selected points in a Peano continuum is found.

math.GN↗

arXiv2022

Topological fractals revisited

We prove that every Peano continuum with uncountably many local cut points is a topological fractal. This extends some recent results and it partially answers a conjecture by Hata. We also discuss the number of mappings which are necessary for witnessing the structure of a topological fractal.

math.GN↗

arXiv2020

The complexity of homeomorphism relations on some classes of compacta with bounded topological dimension

We are dealing with the complexity of the homeomorphism equivalence relation on some classes of metrizable compacta from the viewpoint of invariant descriptive set theory. We prove that the homeomorphism equivalence relation of absolute retracts in the plane is Borel bireducible with the isomorphism equivalence relation of countable graphs. In order to stress the sharpness of this result, we prove that neither the homeomorphism relation of locally connected continua in the plane nor the homeomorphism relation of absolute retracts in $\mathbb R^3$ is Borel reducible to the isomorphism relation of countable graphs. We also improve the recent results of Chang and Gao by constructing a Borel reduction from both the homeomorphism equivalence relation of compact subsets of $\mathbb R^n$ and the ambient homeomorphism equivalence relation of compact subsets of $[0,1]^n$ to the homeomorphism equivalence relation of $n$-dimensional continua in $\mathbb R^{n+1}$.

math.GN↗

arXiv2020

Classification of one dimensional dynamical systems by countable structures

We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of the Hjorth's conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube homeomorphisms is Borel bireducible to the universal orbit equivalence relation.

math.DS↗

arXiv2020

There is no compact metrizable space containing all continua as unique components

We answer a question of Piotr Minc by proving that there is no compact metrizable space whose set of components contains a unique topological copy of every metrizable compactification of a ray (i.e. a half-open interval) with an arc (i.e. closed bounded interval) as the remainder. To this end we use the concept of Borel reductions coming from Invariant descriptive set theory. It follows as a corollary that there is no compact metrizable space such that every continuum is homeomorphic to exactly one component of this space.

math.GN↗

arXiv2018

The complexity of homeomorphism relations on some classes of compacta

We prove that the homeomorphism relation between compact spaces can be continuously reduced to the homeomorphism equivalence relation between absolute retracts which strengthens and simplifies recent results of Chang and Gao, and Cieśla. It follows then that the homeomorphism relation of absolute retracts is Borel bireducible with the universal orbit equivalence relation. We also prove that the homeomorphism relation between regular continua is classifiable by countable structures and hence it is Borel bireducible with the universal orbit equivalence relation of the permutation group on a countable set. On the other hand we prove that the homeomorphism relation between rim-finite metrizable compacta is not classifiable by countable structures.

math.GN↗

arXiv2018

Constant slope, entropy and horseshoes for a map on a tame graph

We study continuous countably (strictly) monotone maps defined on a tame graph, i.e., a special Peano continuum for which the set containing branchpoints and endpoints has a countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map $f$ of a tame graph $G$ is conjugate to a constant slope map $g$ of a countably affine tame graph. In particular, we show that in the case of a Markov map $f$ that corresponds to recurrent transition matrix, the condition is satisfied for constant slope $e^{h_{\operatorname{top}}(f)}$, where $h_{\operatorname{top}}(f)$ is the topological entropy of $f$. Moreover, we show that in our class the topological entropy $h_{\operatorname{top}}(f)$ is achievable through horseshoes of the map $f$.

math.DS↗