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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Quasi-factors of zero-entropy systems

Authors: Eli Glasner and Benjamin Weiss
Journal: J. Amer. Math. Soc. 8 (1995), 665-686
MSC: Primary 54H20; Secondary 28D20
MathSciNet review: 1270579
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Abstract: 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.

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Article copyright: © Copyright 1995 American Mathematical Society