On the construction of non-invertible minimal skew products

Let X, Z be infinite compact metric spaces. We show that if the group H(Z) of the homeomorphisms of Z has an arc-wise connected subgroup whose action on Z is minimal then every minimal map f on X (invertible or not) admits a minimal skew product extension F=(f,gx) on X×Z with the fibre maps . In the invertible case this was proved by Glasner and Weiss in 1979. We also contribute to the description of the class Z of those spaces Z which admit a group with the mentioned property. Namely, we show that this class is closed with respect to countable products and contains all countably infinite products of compact connected manifolds, infinitely many of which have nonempty boundary. Further, we show that the subclass of Z formed by all compact metric spaces Z which admit an arc-wise connected group of isometries with a minimal action on Z coincides with the class of all homogeneous spaces of compact connected metrizable groups.