A compact minimal space $Y$ such that its square $Y\times Y$ is not minimal

TL;DR

This paper constructs a minimal compact space Y whose square Y×Y is not minimal, providing a negative answer to a longstanding open problem about minimal homeomorphisms on product spaces.

Contribution

It introduces a novel inverse limit construction inspired by existing techniques to produce minimal spaces with non-minimal squares and explores minimal flows and their extensions.

Findings

Existence of a compact minimal space Y with Y×Y not minimal

Construction of a noninvertible minimal map as an almost 1-1 extension

Negative answer to the open problem about minimal homeomorphisms on product spaces

Abstract

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 $X\times Y$ admit a minimal homeomorphism as well? 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 $\phi\colon M\times\mathbb{R}\to M$ be a continuous, aperiodic minimal flow on the compact, finite--dimensional metric space $M$. Then there is a generic choice of parameters $c\in\mathbb{R}$, such that the homeomorphism $h(x)=\phi(x,c)$ admits a noninvertible minimal map $f\colon M\to M$ as an almost 1-1 extension.