Multi-invariant measures and subsets on nilmanifolds
- 28 Downloads
Given a Zr-action α on a nilmanifold X by automorphisms and an ergodic α-invariant probability measure μ, we show that μ is the uniform measure on X unless, modulo finite index modification, one of the following obstructions occurs for an algebraic factor action
We also deduce a rigidity property for invariant closed subsets.
the factor measure has zero entropy under every element of the action
the factor action is virtually cyclic.
Unable to display preview. Download preview PDF.
- [AR62]L. M. Abraomov and V. A. Rohlin, Rohlin, Entropy of a skew product of mappings with invariant measure (Russian) Vestnik Leningrad. Univ. 17 (1962), 5–13.Google Scholar
- [EL10]M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, Homogeneous Flows, Moduli Spaces and Arithmetic, Amer. Math. Soc., Providence, RI, 2010, pp. 155–241.Google Scholar
- [ELW]M. Einsiedler, E. Lindenstrauss, and Z. Wang, Rigidity properties of abelian actions on tori and solenoids, in preparation.Google Scholar
- [EW11]M. Einseidler and T. Ward, Ergodic Theory with a View Towards Number Theory, Springer-Verlag, London, 2011.Google Scholar
- [Par96]W. Parry, Squaring and cubing the circle—Rudolph’s theorem, Ergodic Theory of Zd actions, Cambridge Univ. Press, Cambridge, 1996, pp. 177–183.Google Scholar
© Hebrew University Magnes Press 2018