Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-24T06:57:46.446Z Has data issue: false hasContentIssue false
  • English
  • Français

Equidistribution of singular measures on nilmanifolds and skew products

Published online by Cambridge University Press:  18 March 2011

FABRIZIO POLO
FABRIZIO POLO*
Affiliation:
Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210-1174, USA (email: polof@math.osu.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that for a minimal rotation T on a two-step nilmanifold and any measure μ, the push-forward Tn⋆μ of μ under Tn tends toward Haar measure if and only if μ projects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density one. These results strengthen Parry’s result [Ergodic properties of affine transformations and flows on nilmanifolds. Amer. J. Math.91 (1968), 757–771] that such systems are uniquely ergodic. Extending the work of Furstenberg [Strict ergodicity and transformations of the torus. Amer. J. Math.83 (1961), 573–601], we get the same result for a large class of iterated skew products. Additionally we prove a multiplicative ergodic theorem for functions taking values in the upper unipotent group. Finally we characterize limits of Tn⋆μ for some skew product transformations with expansive fibers. All results are presented in terms of twisting and weak twisting, properties that strengthen unique ergodicity in a way analogous to that in which mixing and weak mixing strengthen ergodicity for measure-preserving systems.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 6 , December 2011 , pp. 1785 - 1817
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Auslander, L., Green, L. and Hahn, F.. Flows on Homogeneous Spaces (Annals of Mathematics Studies, 53). Princeton University Press, Princeton, NJ, 1963.CrossRefGoogle Scholar
[2]Bellow, A. and Furstenberg, H.. An application of number theory to ergodic theory and the construction of uniquely ergodic models. Israel J. Math. 33(3–4) (1979), 231–240.CrossRefGoogle Scholar
[3]Furstenberg, H.. Strict ergodicity and transformations of the torus. Amer. J. Math. 83 (1961), 573–601.CrossRefGoogle Scholar
[4]Glasner, E.. Quasi-factors in ergodic theory. Israel J. Math. 45(2–3) (1983), 198–208.CrossRefGoogle Scholar
[5]Glasner, E.. Ergodic Theory Via Joinings (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[6]Glasner, S. and Weiss, B.. Quasifactors of ergodic systems with positive entropy. Israel J. Math. 134 (2003), 363–380.CrossRefGoogle Scholar
[7]Malcev, A. I.. On a class of homogeneous spaces. Amer. Math. Soc. Transl. 39 (1951).Google Scholar
[8]Parry, W.. Ergodic properties of affine transformations and flows on nilmanifolds. Amer. J. Math. 91 (1968), 757–771.CrossRefGoogle Scholar
[9]Villani, C.. Topics in Optimal Transportation (Graduate Studies in Mathematics, 58). American Mathematical Society, Providence, RI, 1973.Google Scholar
[10]Walters, P.. Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York–Berlin, 1982.CrossRefGoogle Scholar