Minimal direct products

Abstract:

A space is called minimal if it admits a minimal continuous selfmap. We introduce and study the notion of product-minimality. We call a compact metrizable space $Y$ product-minimal if, for every minimal system $(X,T)$ given by a metrizable space $X$ and a continuous selfmap $T$, there is a continuous map $S\colon Y\to Y$ such that the product $(X\times Y,T\times S)$ is minimal. If such a map $S$ always exists in the class of homeomorphisms, we say that $Y$ is a homeo-product-minimal space. We show that many classical examples of minimal spaces, including compact connected metrizable abelian groups, compact connected manifolds without boundary admitting a free action of a nontrivial compact connected Lie group, and many others, are in fact homeo-product-minimal.

Then we give examples of metrizable continua $X$ admitting both minimal homeomorphisms and minimal noninvertible maps, whose squares $X\times X$ are not minimal (i.e., they admit neither minimal homeomorphisms nor minimal noninvertible maps). This shows that the product of two minimal spaces need not be minimal.