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

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 × 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.