Geodesic
Multiple Mixing and Rank One Group Actions
Canadian journal of mathematics, Tome 52 (2000) no. 2, pp. 332-347

Voir la notice de l'article provenant de la source Cambridge University Press

Suppose $G$ is a countable, Abelian group with an element of infinite order and let $\text{ }\!\!\chi\!\!\text{ }$ be a mixing rank one action of $G$ on a probability space. Suppose further that the Følner sequence $\{{{F}_{n}}\}$ indexing the towers of $\text{ }\!\!\chi\!\!\text{ }$ satisfies a “bounded intersection property”: there is a constant $p$ such that each $\{{{F}_{n}}\}$ can intersect no more than $p$ disjoint translates of $\{{{F}_{n}}\}$ . Then $\text{ }\!\!\chi\!\!\text{ }$ is mixing of all orders. When $G\,=\,\mathbf{Z}$ , this extends the results of Kalikow and Ryzhikov to a large class of “funny” rank one transformations. We follow Ryzhikov’s joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of $k$ copies of $\text{ }\!\!\chi\!\!\text{ }$ is necessarily product measure. This method generalizes Ryzhikov’s technique.
DOI : 10.4153/CJM-2000-015-0
Mots-clés : 28D15 
Junco, Andrés del; Yassawi, Reem. Multiple Mixing and Rank One Group Actions. Canadian journal of mathematics, Tome 52 (2000) no. 2, pp. 332-347. doi: 10.4153/CJM-2000-015-0
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[1] [1] Blum, J. R. and Hanson, D. L., On the mean ergodic theorem for subsequences. Bull. Amer. Math. Soc. 66(1960), 308–311. Google Scholar

[2] [2] Ferenczi, S., Systèmes de rang un gauche. Ann. Inst. Henri Poincaré (2) 21(1985), 177–186. Google Scholar

[3] [3] Host, B., Mixing of all orders and pairwise independent joinings of systems with singular spectrum. Israel J. Math 76(1991), 289–298. Google Scholar

[4] [4] del Junco, A., A weakly mixing simple map with no prime factors. Israel J. Math., to appear. Google Scholar

[5] [5] del Junco, A., A Transformation with Simple Spectrum which is not Rank One. Canad. J. Math. (3) 29(1977), 655–663. Google Scholar

[6] [6] del Junco, A. and Rudolph, D., On ergodic actions whose self-joinings are graphs. Ergodic Theory Dynamical Systems 7(1987), 531–557. Google Scholar

[7] [7] Kalikow, S. A., Twofold mixing implies threefold mixing for rank one transformations. Ergodic Theory Dynamical Systems 4(1984), 237–259. Google Scholar

[8] [8] Ornstein, D. and Weiss, B., The Shannon-McMillan-Breiman theorem for amenable groups. Israel J. Math. (1) 44(1983), 53–60. Google Scholar

[9] [9] Rokhlin, V. A., On ergodic compact Abelian groups. Izv. Akad. Nauk. SSSR Ser. Mat. 13(1949), 323–340.(Russian). Google Scholar

[10] [10] Ryzhikov, V. V., Mixing, Rank, andMinimal Self-Joining of Actions with an Invariant Measure. Russian Acad. Sci. Sb. Math. (2) 75(1993), 405–427. Google Scholar

[11] [11] Ryzhikov, V. V., Joinings and multiple mixing of the actions of finite rank. Funct. Anal. Appl. 27(1993), 128–140. Google Scholar

[12] [12] Tempel’man, A. A., Ergodic Theorems for General Dynamical Systems. English transl.: Soviet Math. Doklady (5) 8(1967), 1213–1216. Google Scholar

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