NonminimalOmnia_AiMEdit_v3.pdf (374.72 kB)
A compact minimal space Y such that its square YxY is not minimal
journal contribution
posted on 2018-08-07, 15:11 authored by J. P. Boroński, Alex Clark, Piotr OprochaThe 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.
Funding
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.
History
Citation
Advances in Mathematics, 335, 2018, pp. 261-275Author affiliation
/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of MathematicsVersion
- AM (Accepted Manuscript)