Minimal space with non-minimal square

TL;DR

This paper demonstrates that even if a compact metric space admits a minimal homeomorphism, its square may not admit any minimal map, highlighting limitations in the behavior of minimal dynamics under product spaces.

Contribution

It constructs a specific metric continuum with a minimal homeomorphism whose square admits no minimal map, addressing a previously open problem.

Findings

Existence of a metric continuum with a minimal homeomorphism but no minimal map on its square.

Counterexample to the extension of minimality from a space to its product.

Clarification of the limitations of minimal dynamics in product spaces.

Abstract

We completely solve the problem whether the product of two compact metric spaces admitting minimal maps also admits a minimal map. Recently Boro\'nski, Clark and Oprocha gave a negative answer in the particular case when homeomorphisms rather than continuous maps are considered. In the present paper we show that there is a metric continuum $X$ admitting a minimal map, in fact a minimal homeomorphism, such that $X\times X$ does not admit any minimal map.