## Quasi-factors of zero-entropy systems

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- by Eli Glasner and Benjamin Weiss PDF
- J. Amer. Math. Soc.
**8**(1995), 665-686 Request permission

## 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.## References

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*Entropy pairs for a measure*, preprint.

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**8**(1995), 665-686 - MSC: Primary 54H20; Secondary 28D20
- DOI: https://doi.org/10.1090/S0894-0347-1995-1270579-5
- MathSciNet review: 1270579