Chain control sets and fiber bundles

Let $S$ be a semigroup of homeomorphisms of a compact metric space $M$ and suppose that ${\cal F}$ is a family of subsets of $S$. This paper gives a characterization of the ${\cal F}$-chain control sets as intersection of control sets for the semigroups generated by the neighborhoods of the subsets in ${\cal F}$. We also study the behavior of ${\cal F}$-chain control sets on principal bundles and their associated bundles.