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A compact minimal space Y such that its square YxY is not minimal

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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.

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. 261-275

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

Version

AM (Accepted Manuscript)

Published in

Advances in Mathematics

Publisher

Elsevier for Academic Press

issn

0001-8708

eissn

1090-2082

Acceptance date

03/07/2018

Copyright date

2018

Available date

14/07/2019

Publisher version

https://www.sciencedirect.com/science/article/pii/S0001870818302597

Notes

The 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.

Language

en

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