Quasi-factors of zero-entropy systems

Glasner, Eli ; Weiss, Benjamin

Journal of the American Mathematical Society, Tome 08 (1995) no. 3, pp. 665-686

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-1995-1270579-5

Bibliographie

[1] Bauer, Walter, Sigmund, Karl Topological dynamics of transformations induced on the space of probability measures Monatsh. Math. 1975 81 92

[2] Blanchard, F. Fully positive topological entropy and topological mixing 1992 95 105

[3] Blanchard, Franã§Ois A disjointness theorem involving topological entropy Bull. Soc. Math. France 1993 465 478

[4] Blanchard, F., Lacroix, Y. Zero entropy factors of topological flows Proc. Amer. Math. Soc. 1993 985 992

[5] Denker, Manfred, Grillenberger, Christian, Sigmund, Karl Ergodic theory on compact spaces 1976

[6] Martin, Nathaniel F. G., England, James W. Mathematical theory of entropy 1981

[7] Figiel, T., Lindenstrauss, J., Milman, V. D. The dimension of almost spherical sections of convex bodies Acta Math. 1977 53 94

[8] Furstenberg, Harry Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation Math. Systems Theory 1967 1 49

[9] Furstenberg, Harry Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions J. Analyse Math. 1977 204 256

[10] Furstenberg, H. Recurrence in ergodic theory and combinatorial number theory 1981

[11] Glasner, S. Quasifactors in ergodic theory Israel J. Math. 1983 198 208

[12] Glasner, S., Weiss, B. Interpolation sets for subalgebras of 𝑙^{∞}(𝑍) Israel J. Math. 1983 345 360

[13] Glasner, Eli, Weiss, Benjamin Strictly ergodic, uniform positive entropy models Bull. Soc. Math. France 1994 399 412

[14] Glasner, Eli, Weiss, Benjamin Dynamics and entropy of the space of measures C. R. Acad. Sci. Paris Sér. I Math. 1993 239 243

[15] Glasner, Eli, Weiss, Benjamin Topological entropy of extensions 1995 299 307

[16] Karpovsky, M. G., Milman, V. D. Coordinate density of sets of vectors Discrete Math. 1978 177 184

[17] Knapp, A. W. Functions behaving like almost automorphic functions 1968 299 317

[18] Parry, William Topics in ergodic theory 2004

[19] Sauer, N. On the density of families of sets J. Combinatorial Theory Ser. A 1972 145 147

[20] Shelah, Saharon A combinatorial problem Pacific J. Math. 1972 247 261

[21] Thouvenot, Jean-Paul 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. 1975 177 207

[22] Zimmer, Robert J. Extensions of ergodic group actions Illinois J. Math. 1976 373 409

[23] Zimmer, Robert J. Ergodic actions with generalized discrete spectrum Illinois J. Math. 1976 555 588