## Quasi-factors of zero-entropy systems

HTML articles powered by AMS MathViewer

- by Eli Glasner and Benjamin Weiss
- J. Amer. Math. Soc.
**8**(1995), 665-686 - DOI: https://doi.org/10.1090/S0894-0347-1995-1270579-5
- PDF | 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

- Walter Bauer and Karl Sigmund,
*Topological dynamics of transformations induced on the space of probability measures*, Monatsh. Math.**79**(1975), 81–92. MR**370540**, DOI 10.1007/BF01585664 - F. Blanchard,
*Fully positive topological entropy and topological mixing*, Symbolic dynamics and its applications (New Haven, CT, 1991) Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 95–105. MR**1185082**, DOI 10.1090/conm/135/1185082 - François Blanchard,
*A disjointness theorem involving topological entropy*, Bull. Soc. Math. France**121**(1993), no. 4, 465–478 (English, with English and French summaries). MR**1254749**, DOI 10.24033/bsmf.2216 - F. Blanchard and Y. Lacroix,
*Zero entropy factors of topological flows*, Proc. Amer. Math. Soc.**119**(1993), no. 3, 985–992. MR**1155593**, DOI 10.1090/S0002-9939-1993-1155593-2
F. Blanchard, B. Host, A. Maass, and D. J. Rudolph, - Manfred Denker, Christian Grillenberger, and Karl Sigmund,
*Ergodic theory on compact spaces*, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR**0457675**, DOI 10.1007/BFb0082364 - Nathaniel F. G. Martin and James W. England,
*Mathematical theory of entropy*, Encyclopedia of Mathematics and its Applications, vol. 12, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by James K. Brooks. MR**612318** - T. Figiel, J. Lindenstrauss, and V. D. Milman,
*The dimension of almost spherical sections of convex bodies*, Acta Math.**139**(1977), no. 1-2, 53–94. MR**445274**, DOI 10.1007/BF02392234 - Harry Furstenberg,
*Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation*, Math. Systems Theory**1**(1967), 1–49. MR**213508**, DOI 10.1007/BF01692494 - Harry Furstenberg,
*Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions*, J. Analyse Math.**31**(1977), 204–256. MR**498471**, DOI 10.1007/BF02813304 - H. Furstenberg,
*Recurrence in ergodic theory and combinatorial number theory*, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR**603625**, DOI 10.1515/9781400855162 - S. Glasner,
*Quasifactors in ergodic theory*, Israel J. Math.**45**(1983), no. 2-3, 198–208. MR**719119**, DOI 10.1007/BF02774016 - S. Glasner and B. Weiss,
*Interpolation sets for subalgebras of $l^{\infty }(\textbf {Z})$*, Israel J. Math.**44**(1983), no. 4, 345–360. MR**710239**, DOI 10.1007/BF02761993 - Eli Glasner and Benjamin Weiss,
*Strictly ergodic, uniform positive entropy models*, Bull. Soc. Math. France**122**(1994), no. 3, 399–412 (English, with English and French summaries). MR**1294463**, DOI 10.24033/bsmf.2239 - Eli Glasner and Benjamin Weiss,
*Dynamics and entropy of the space of measures*, C. R. Acad. Sci. Paris Sér. I Math.**317**(1993), no. 3, 239–243 (English, with English and French summaries). MR**1233419** - Eli Glasner and Benjamin Weiss,
*Topological entropy of extensions*, Ergodic theory and its connections with harmonic analysis (Alexandria, 1993) London Math. Soc. Lecture Note Ser., vol. 205, Cambridge Univ. Press, Cambridge, 1995, pp. 299–307. MR**1325706**, DOI 10.1017/CBO9780511574818.011 - M. G. Karpovsky and V. D. Milman,
*Coordinate density of sets of vectors*, Discrete Math.**24**(1978), no. 2, 177–184. MR**522926**, DOI 10.1016/0012-365X(78)90197-8 - A. W. Knapp,
*Functions behaving like almost automorphic functions*, Topological Dynamics (Symposium, Colorado State Univ., Ft. Collins, Colo., 1967) Benjamin, New York, 1968, pp. 299–317. MR**0238294** - William Parry,
*Topics in ergodic theory*, Cambridge Tracts in Mathematics, vol. 75, Cambridge University Press, Cambridge, 2004. Reprint of the 1981 original. MR**2140546** - N. Sauer,
*On the density of families of sets*, J. Combinatorial Theory Ser. A**13**(1972), 145–147. MR**307902**, DOI 10.1016/0097-3165(72)90019-2 - Saharon Shelah,
*A combinatorial problem; stability and order for models and theories in infinitary languages*, Pacific J. Math.**41**(1972), 247–261. MR**307903**, DOI 10.2140/pjm.1972.41.247 - Jean-Paul Thouvenot,
*Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schéma de Bernoulli*, Israel J. Math.**21**(1975), no. 2-3, 177–207 (French, with English summary). MR**399419**, DOI 10.1007/BF02760797 - Robert J. Zimmer,
*Extensions of ergodic group actions*, Illinois J. Math.**20**(1976), no. 3, 373–409. MR**409770** - Robert J. Zimmer,
*Ergodic actions with generalized discrete spectrum*, Illinois J. Math.**20**(1976), no. 4, 555–588. MR**414832**

*Entropy pairs for a measure*, preprint.

## Bibliographic 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