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

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.