Dyadic equivalence to completely positive entropy

Fieldsteel, Adam; Hasfura-Buenaga, J.

doi:10.1090/S0002-9947-98-02115-1

Dyadic equivalence to completely positive entropy
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by Adam Fieldsteel and J. Roberto Hasfura-Buenaga

Trans. Amer. Math. Soc. 350 (1998), 1143-1166

DOI: https://doi.org/10.1090/S0002-9947-98-02115-1

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Abstract:

We show that every free ergodic action of $\bigoplus _1^\infty {\mathbb Z}_2$ of positive entropy is dyadically equivalent to an action with completely positive entropy.

References

Adam Fieldsteel and N. A. Friedman, Restricted orbit changes of ergodic $\textbf {Z}^d$-actions to achieve mixing and completely positive entropy, Ergodic Theory Dynam. Systems 6 (1986), no. 4, 505–528. MR 873429, DOI 10.1017/S0143385700003667
J. R. Hasfura-Buenaga, Mixing for dyadic equivalence, Acta Math. Univ. Comenian. (N.S.) 64 (1995), no. 1, 141–152. MR 1360993
Andres del Junco and Daniel J. Rudolph, Residual behavior of induced maps, Israel J. Math. 93 (1996), 387–398. MR 1380654, DOI 10.1007/BF02761114
Janet Kammeyer, D. J. Rudolph, Restricted orbit equivalence for ergodic $\textbf {Z}^d-$actions, submitted, Ergodic Th. and Dyn. Sys.
Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470
Daniel J. Rudolph, Restricted orbit equivalence, Mem. Amer. Math. Soc. 54 (1985), no. 323, v+150. MR 782648, DOI 10.1090/memo/0323
D. S. Ornstein and M. Smorodinsky, Ergodic flows of positive entropy can be time changed to become $K$-flows, Israel J. Math. 26 (1977), no. 1, 75–83. MR 447526, DOI 10.1007/BF03007657
Donald S. Ornstein and Benjamin Weiss, Every transformation is bilaterally deterministic, Israel J. Math. 21 (1975), no. 2-3, 154–158. MR 382600, DOI 10.1007/BF02760793
Donald Ornstein and Benjamin Weiss, The Shannon-McMillan-Breiman theorem for a class of amenable groups, Israel J. Math. 44 (1983), no. 1, 53–60. MR 693654, DOI 10.1007/BF02763171
Meir Smorodinsky, Ergodic theory, entropy, Lecture Notes in Mathematics, Vol. 214, Springer-Verlag, Berlin-New York, 1971. MR 0422582, DOI 10.1007/BFb0066086
Klaus Schmidt, Cocycles on ergodic transformation groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan Co. of India, Ltd., Delhi, 1977. MR 0578731
A. M. Stepin, The entropy invariant of decreasing squences of measurable partitions. , Funkcional. Anal. i Priložen. 5 (1971), no. 3, 80–84 (Russian). MR 0284568
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
A. M. Veršik, Descending sequences of measurable decompositions, and their applications, Dokl. Akad. Nauk SSSR 193 (1970), 748–751 (Russian). MR 0268360
A. M. Veršik, A continuum of pairwise nonisomorphic dyadic sequences, Funkcional. Anal. i Priložen. 5 (1971), no. 3, 16–18 (Russian). MR 0281879