Rigid and rigid-like spaces in topological dynamics

A space is rigid if the only continuous selfmaps of this space are the constant maps and the identity. We show that rigid spaces, in fact so called Cook continua, can be used to produce spaces with nontrivial interesting dynamics. We also discuss `rigid-like' spaces in topological dynamics; as an application we get that the class of compact metric spaces admitting minimal maps is not closed with respect to products.