Ergodic components of an extension by a nilmanifold. (English) Zbl 1180.22022
Let \((T,X,\mu)\) be an ergodic dynamical system. It is shown that all ergodic components of an extension of this system by translations on a nilmanifold \(M\) are isomorphic to extensions of this system by translations on submanifolds of \(M\). It is worthwhile noting that this result is connected to a lemma by W. Parry which claims that a shift-transformation of a compact connected nilmanifold \(M\) is ergodic if and only if it is ergodic on the maximal factor-torus of \(M\).
Reviewer: Michael L. Blank (Moskva)
References:
| [1] | A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold , Ergodic Theory Dynam. Systems 25 (2005), 201–213. · Zbl 1080.37003 · doi:10.1017/S0143385704000215 |
| [2] | A. Leibman, Orbits on a nilmanifold under the action of a polynomial sequences of translations , Ergodic Theory Dynam. Systems 27 (2007), 1239–1252. · Zbl 1121.37005 · doi:10.1017/S0143385706000940 |
| [3] | E. Lesigne, Sur une nil-variété, les parties minimales assocées à une translation sont uniquement ergodiques , Ergodic Theory Dynam. Systems 11 (1991), 379–391. · Zbl 0709.28012 · doi:10.1017/S0143385700006209 |
| [4] | W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds , Amer. J. Math. 91 (1969), 757–771. JSTOR: · Zbl 0183.51503 · doi:10.2307/2373350 |
| [5] | W. Parry, Dynamical systems on nilmanifolds , Bull. Lond. Math. Soc. 2 (1970), 7–40. · Zbl 0194.05601 · doi:10.1112/blms/2.1.37 |
| [6] | 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. · Zbl 0951.22006 |
| [7] | R. Zimmer, Extensions of ergodic group actions , Illinois J. Math. 20 (1976), 373–409. · Zbl 0334.28015 |
| [8] | R. Zimmer, Compact nilmanifold extensions of ergodic actions , Trans. Amer. Math. Soc. 223 (1976), 397–406. · Zbl 0353.28011 · doi:10.2307/1997537 |
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