Quasi-factors of Zero-entropy Systems

For minimal systems (X, T) of zero topological entropy we demon-strate the sharp difference between the behavior, regarding entropy, of the sys-tems (M(X) , T) and (2x , T) induced by T on the spaces M(X) of prob-ability measures on X and 2x of closed subsets of X. It is shown that thesystem (M(X) , T) has itself zero topological entropy. Two proofs of this the-orem are given. The first uses ergodic theoretic ideas. The second relies onthe different behavior of the Banach spaces l~ and l~ with respect to theexistence of almost Hilbertian central sections of the unit ball. In contrast tothis theorem we construct a minimal system (X, T) of zero entropy with aminimal subsystem (Y, T) of (2x , T) whose entropy is positive. SCHOOL OF MATHEMATICS, TEL AVIv UNIVERSITY, RAMAT AVIV, ISRAELE-mail address:glasnerGmath.tau.ac.il INSTITUTE OF MATHEMATICS, HEBREW UNIVERSITY, JERUSALEM, ISRAELE-mail address:weissGmath.huji.ac.i1 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use