arXiv subjects

Jan Boroński

Publications and source records attributed to Jan Boroński.

4 recordsLinked to original sources

arXiv2026

Continuously homogeneous hereditarily indecomposable continua are tree-like

A topological space $X$ is continuously homogeneous if for any $x,y\in X$ there exists a continuous surjection $f:X\to X$ with $f(x)=y$. We show that continuously homogeneous hereditarily indecomposable continua are tree-like, therefore, extending results of Bing and Rogers for homeomorphism and a result of Sturm for the pseudo-circle and pseudo-solenoids. This also provides a partial answer to the question of Lewis whether all continuously homogeneous hereditarily indecomposable continua are homogeneous.

math.GN↗

arXiv2025

On roundness of rotation sets

Motivated by the question whether a round disk can be realized as the rotation set of a torus diffeomorphism, we study the roundness of rotation sets of a parametric family of torus diffeomorphisms $F_ρ$, where the parameter $ρ$ ranges over irrational numbers in $(0,1)$. Each $F_ρ$ is a Kwapisz-like diffeomorphism with a 2-dimensional non-polygonal rotation set $$Λ'_ρ= \operatorname{conv}\left(\left\{(\pm\frac{\lceil mρ\rceil}{m+n+1}, \pm\frac{\lceil nρ\rceil}{m+n+1}): m, n \in \mathbb{N} _0, \lceil mρ\rceil - mρ<ρ,\lceil nρ\rceil - nρ<ρ\right\}\right)$$ whose extreme point set contains exactly four (two-sided) accumulation points. We define the roundness of $Λ'_ρ$ as the ratio $R_ρ=\frac{\operatorname{Area}(Λ'_ρ)}{πρ^2}$, and give its upper and lower bounds in terms of $ρ$. $R_ρ$ is neither monotone nor continuous.

math.DS↗

arXiv2024

Translation algorithms for graph covers

Graph covers are a way to describe continuous maps (and homeomorphisms) of a Cantor set, more generally than e.g.\ Bratteli-Vershik systems. Every continuous map on a zero-dimensional compact set can be expressed by a graph cover (e.g.\ non-minimality or aperiodicty are no restrictions). We give a survey on the construction, properties and some special cases of graph covers.

math.DS↗

arXiv2023

Densely branching trees as models for Hénon-like and Lozi-like attractors

Inspired by a recent work of Crovisier and Pujals on mildly dissipative diffeomorphisms of the plane, we show that Hénon-like and Lozi-like maps on their strange attractors are conjugate to natural extensions (a.k.a. shift homeomorphisms on inverse limits) of maps on metric trees with dense set of branch points. In consequence, these trees very well approximate the topology of the attractors, and the maps on them give good models of the dynamics. To the best of our knowledge, these are the first examples of canonical two-parameter families of attractors in the plane for which one is guaranteed such a 1-dimensional locally connected model tying together topology and dynamics of these attractors. For the Hénon maps this applies to a positive Lebesgue measure parameter set generalizing the Benedicks-Carleson parameters, the Wang-Young parameter set, and sheds more light onto the result of Barge from 1987, who showed that there exist parameter values for which Hénon maps on their attractors are not natural extensions of any maps on branched 1-manifolds. For the Lozi maps the result applies to an open set of parameters given by Misiurewicz in 1980. Our result can be seen as a generalization to the non-uniformly hyperbolic world of a classical result of Williams from 1967. We also show that no simpler 1-dimensional models exist.

math.DS↗