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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On invariant finitely additive measures for automorphism groups acting on tori


Author: S. G. Dani
Journal: Trans. Amer. Math. Soc. 287 (1985), 189-199
MSC: Primary 28D15; Secondary 43A07
DOI: https://doi.org/10.1090/S0002-9947-1985-0766213-0
MathSciNet review: 766213
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Abstract: Consider the natural action of a subgroup $ H$ of $ {\text{GL}}(n,{\mathbf{Z}})$ on $ {{\mathbf{T}}^n}$. We relate the $ H$-invariant finitely additive measures on $ ({{\mathbf{T}}^n},\mathcal{L})$ where $ \mathcal{L}$ is the class of all Lebesgue measurable sets, to invariant subtori $ C$ such that the $ H$-action on either $ C$ or $ {{\mathbf{T}}^n}/C$ factors to an action of an amenable group. In particular, we conclude that if $ H$ is a nonamenable group acting irreducibly on $ {{\mathbf{T}}^n}$ then the normalised Haar measure is the only $ H$-invariant finitely additive probability measure on $ ({{\mathbf{T}}^n},\mathcal{L})$ such that $ \mu (R) = 0$, where $ R$ is the (countable) subgroup consisting of all elements of finite order; this answers a question raised by J. Rosenblatt.

Along the way we analyse $ H$-invariant finitely additive measures defined for all subsets of $ {{\mathbf{T}}^n}$ and deduce, in particular, that the Haar measure extends to an $ H$-invariant finitely additive measure defined on all sets if and only if $ H$ is amenable.


References [Enhancements On Off] (What's this?)

  • [1] F. P. Greenleef, Invariant means on topological groups, Van Nostrand, New York, 1969.
  • [2] A. del Junco and J. Rosenblatt, Counterexamples in ergodic theory and number theory, Math. Ann. 245 (1979), 185-197. MR 553340 (81d:10042)
  • [3] J. Mycielski, Finitely additive invariant measures. I, Colloq. Math. 42 (1979), 309-318. MR 567569 (82g:43003a)
  • [4] J. Rosenblatt, Uniqueness of invariant means for measure-preserving transformations, Trans. Amer. Math. Soc. 265 (1981), 623-636. MR 610970 (83a:28026)
  • [5] J. Tits, Free subgroups of linear groups, J. Algebra 20 (1972), 250-270. MR 0286898 (44:4105)
  • [6] S. Wagon, Invariance properties of finitely additive measures on $ {R^n}$, Illinois J. Math. 25 (1981), 74-86. MR 602898 (82c:28026)
  • [7] P. Walters, Ergodic theory--introductory lectures, Lecture Notes in Math., vol. 458, Springer, New York, 1975. MR 0480949 (58:1096)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0766213-0
Keywords: Invariant finitely additive measures, invariant means
Article copyright: © Copyright 1985 American Mathematical Society

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