  There are two main subjects in this paper.
  1) For a topological dynamical system $(X,T)$ we study the topological
entropy of its "functional envelopes" (the action of $T$ by left composition on
the space of all continuous self-maps or on the space of all
self-homeomorphisms of $X$). In particular we prove that for zero-dimensional
spaces $X$ both entropies are infinite except when $T$ is equicontinuous (then
both equal zero).
  2) We call $Slovak$ $space$ any compact metric space whose homeomorphism
group is cyclic and generated by a minimal homeomorphism. Using Slovak spaces
we provide examples of (minimal) systems $(X,T)$ with positive entropy, yet,
whose functional envelope on homeomorphisms has entropy zero (answering a
question posed by Kolyada and Semikina). Finally, also using Slovak spaces, we
resolve a long standing open problem whether the circle is a unique
non-degenerate continuum admitting minimal continuous transformations but only
invertible: No, some Slovak spaces are such, as well.
