Quasi-factors of zero-entropy systems

For minimal systems $(X,T)$ of zero topological entropy we demonstrate the sharp difference between the behavior, regarding entropy, of the systems $(M(X),T)$ and $({2^X},T)$ induced by $T$ on the spaces $M(X)$ of probability measures on $X$ and ${2^X}$ of closed subsets of $X$. It is shown that the system $(M(X),T)$ has itself zero topological entropy. Two proofs of this theorem are given. The first uses ergodic theoretic ideas. The second relies on the different behavior of the Banach spaces $l_1^n$ and $l_\infty ^n$ with respect to the existence of almost Hilbertian central sections of the unit ball. In contrast to this theorem we construct a minimal system $(X,T)$ of zero entropy with a minimal subsystem $(Y,T)$ of $({2^X},T)$ whose entropy is positive.