journal contribution posted on 07.08.2018, 15:11 by J. P. Boroński, Alex Clark, Piotr Oprocha
The following well known open problem is answered in the negative:
Given two compact spaces X and Y that admit minimal homeomorphisms,
must the Cartesian product X × Y admit a minimal homeomorphism
as well? Moreover, it is shown that such spaces can be realized as minimal sets
of torus homeomorphisms homotopic to the identity. A key element of our construction
is an inverse limit approach inspired by combination of a technique of
Aarts & Oversteegen and the construction of Slovak spaces by Downarowicz
& Snoha & Tywoniuk. This approach allows us also to prove the following
result. Let φ: M × R → M be a continuous, aperiodic minimal flow on the
compact, finite–dimensional metric space M. Then there is a generic choice
of parameters c ∈ R, such that the homeomorphism h(x) = φ(x, c) admits a
noninvertible minimal map f : M → M as an almost 1-1 extension.
In part, this work was supported by NPU II project LQ1602
IT4Innovations excellence in science, by Grant IN-2013-045 from the Leverhulme
Trust for an International Network and MSK grant 01211/2016/RRC “Strengthening
international cooperation in science, research and education”, which supported
research visits of the authors.
Research of P. Oprocha was supported by National Science Centre, Poland
(NCN), grant no. 2015/17/B/ST1/01259, and J. Boro´nski’s work was supported
by National Science Centre, Poland (NCN), grant no. 2015/19/D/ST1/01184.
CitationAdvances in Mathematics, 335, 2018, pp. 261-275
Author affiliation/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics
VersionAM (Accepted Manuscript)
Published inAdvances in Mathematics
PublisherElsevier for Academic Press
NotesThe file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.