Illinois Journal of Mathematics

Ergodic components of an extension by a nilmanifold

A. Leibman

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We prove that all ergodic components of an extension of an ergodic system by translations on a nilmanifold $X$ are isomorphic to extensions of this system by translations on subnilmanifolds of $X$.

Article information

Illinois J. Math., Volume 52, Number 3 (2008), 957-965.

First available in Project Euclid: 1 October 2009

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Zentralblatt MATH identifier

Primary: 22F10: Measurable group actions [See also 22D40, 28Dxx, 37Axx] 22D40: Ergodic theory on groups [See also 28Dxx]


Leibman, A. Ergodic components of an extension by a nilmanifold. Illinois J. Math. 52 (2008), no. 3, 957--965. doi:10.1215/ijm/1254403724.

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  • A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), 201–213.
  • A. Leibman, Orbits on a nilmanifold under the action of a polynomial sequences of translations, Ergodic Theory Dynam. Systems 27 (2007), 1239–1252.
  • E. Lesigne, Sur une nil-variété, les parties minimales assocées à une translation sont uniquement ergodiques, Ergodic Theory Dynam. Systems 11 (1991), 379–391.
  • W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757–771.
  • W. Parry, Dynamical systems on nilmanifolds, Bull. Lond. Math. Soc. 2 (1970), 7–40.
  • N. Shah, Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements, Lie groups and ergodic theory (Mumbai, 1996), Tata, Bombay, 1998, pp. 229–271.
  • R. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), 373–409.
  • R. Zimmer, Compact nilmanifold extensions of ergodic actions, Trans. Amer. Math. Soc. 223 (1976), 397–406.