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 (Equation presented) and (Equation presented) 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.