arXiv subjects
Publications and source records attributed to Jan P. Boronski.
In this article, we show that R.H. Bing's pseudo-circle admits a minimal non-invertible map. This resolves a problem raised by Bruin, Kolyada and Snoha in the negative. The main tool is the Denjoy-Rees technique, further developed by Béguin-Crovisier-Le Roux, combined with detailed study into the structure of the pseudo-circle.
The result of Boyce and Huneke gives rise to a 1-dimensional continuum, which is the intersection of a descending family of disks, that admits two commuting homeomorphisms without a common fixed point.
We study nonmetric analogues of Vietoris solenoids. Let $Λ$ be an ordered continuum, and let $\vec{p}=\langle p_1,p_2,\dots\rangle$ be a sequence of positive integers. We define a natural inverse limit space $S(Λ,\vec{p})$, where the first factor space is the nonmetric "circle" obtained by identifying the endpoints of $Λ$, and the $n$th factor space, $n>1$, consists of $p_1p_2\cdot\dots \cdot p_{n-1}$ copies of $Λ$ laid end to end in a circle. We prove that for every cardinal $κ\geq 1$, there is an ordered continuum $Λ$ such that $S(Λ,\vec{p})$ is $\frac{1}κ$-homogeneous; for $κ>1$, $Λ$ is built from copies of the long line. Our example with $κ=2$ provides a nonmetric answer to a question of Neumann-Lara, Pellicer-Covarrubias and Puga-Espinosa from 2005, and with $κ=1$ provides an example of a nonmetric homogeneous circle-like indecomposable continuum. Finally, we employ a cohomology argument to prove that for each ordered continuum $Λ$, as $\vec{p}$ varies there are $2^ω$-many nonhomeomorphic spaces $S(Λ,\vec{p})$.
Motivated by a recent result of Ciesielski and Jasinski we study periodic point free Cantor systems that are conjugate to systems with vanishing derivative everywhere, and more generally locally radially shrinking maps. Our study uncovers a whole spectrum of dynamical behaviors attainable for such systems, providing new counterexamples to the Conjecture of Edrei from 1952, first disproved by Williams in 1954.
In 1985 M. Smith constructed a nonmetric pseudo-arc; i.e. a Hausdorff homogeneous, hereditary equivalent and hereditary indecomposable continuum. Taking advantage of a decomposition theorem of W. Lewis, he obtained it as a long inverse limit of metric pseudo-arcs with monotone bonding maps. Extending his approach, and the results of Lewis on lifting homeomorphisms, we construct a nonmetric pseudo-circle, and new examples of homogeneous 1-dimensional continua; e.g. a circle and solenoids of nonmetric pseudo-arcs. Among many corollaries we also obtain an analogue of another theorem of Lewis from 1984: any interval map is semi-conjugate to a homeomorphism of the nonmetric pseudo-arc.