arXiv subjects

Benjamin Vejnar

Publications and source records attributed to Benjamin Vejnar.

24 records · Page 2Linked to original sources

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↗

arXiv2017

Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point

We are dealing with the question whether every group or semigroup action (with some additional property) on a continuum (with some additional property) has a fixed point. One of such results was given in 2009 by Shi and Sun. They proved that every nilpotent group action on a uniquely arcwise connected continuum has a fixed point. We are seeking for this type of results with e.g. commutative, compact or torsion groups and semigroups acting on dendrites, dendroids, $λ$-dendroids and uniquely arcwise connected continua. We prove that every continuous action of a compact or torsion group on a uniquely arcwise connected continuum has a fixed point. We also prove that every continuous action of a compact and commutative semigroup on a uniquely arcwise connected continuum or on a tree-like continuum has a fixed point.

math.DS↗

arXiv2015

Haar meager sets revisited

In the present article we investigate Darji's notion of Haar meager sets from several directions. We consider alternative definitions and show that some of them are equivalent to the original one, while others fail to produce interesting notions. We define Haar meager sets in nonabelian Polish groups and show that many results, including the facts that Haar meager sets are meager and form a $σ$-ideal, are valid in the more general setting as well. The article provides various examples distinguishing Haar meager sets from Haar null sets, including decomposition theorems for some subclasses of Polish groups. As a corollary we obtain, for example, that $\mathbb Z^ω$, $\mathbb R^ω$ or any Banach space can be decomposed into a Haar meager set and a Haar null set. We also establish the stability of non-Haar meagerness under Cartesian product.

math.GN↗

arXiv2014

Classification of the spaces $C_p^*(X)$ within the Borel-Wadge hierarchy for a projective space $X$

We study the complexity of the space $C^*_p(X)$ of bounded continuous functions with the topology of pointwise convergence. We are allowed to use descriptive set theoretical methods, since for a separable metrizable space $X$, the measurable space of Borel sets in $C^*_p(X)$ (and also in the space $C_p(X)$ of all continuous functions) is known to be isomorphic to a subspace of a standard Borel space. It was proved by A. Andretta and A. Marcone that if $X$ is a $σ$-compact metrizable space, then the measurable spaces $C_p(X)$ and $C^*_p(X)$ are standard Borel and if $X$ is a metrizable analytic space which is not $σ$-compact then the spaces of continuous functions are Borel-$Π^1_1$-complete. They also determined under the assumption of projective determinacy (PD) the complexity of $C_p(X)$ for any projective space $X$ and asked whether a similar result holds for $C^*_p(X)$. We provide a positive answer, i.e. assuming PD we prove, that if $n \geq 2$ and if $X$ is a separable metrizable space which is in $Σ^1_n$ but not in $Σ^1_{n-1}$ then the measurable space $C^*_p(X)$ is Borel-$Π^1_n$-complete. This completes under the assumption of PD the classification of Borel-Wadge complexity of $C^*_p(X)$ for $X$ projective.

math.FA↗

arXiv2009

Covering an uncountable square by countably many continuous functions

We prove that there exists a countable family of continuous real functions whose graphs together with their inverses cover an uncountable square, i.e. a set of the form $X\times X$, where $X$ is an uncountable subset of the real line. This extends Sierpiński's theorem from 1919, saying that $S\times S$ can be covered by countably many graphs of functions and inverses of functions if and only if the size of $S$ does not exceed $\aleph_1$. Our result is also motivated by Shelah's study of planar Borel sets without perfect rectangles.

math.LO↗