Equidistribution of singular measures on nilmanifolds and skew products

@article{Polo2011EquidistributionOS,
  title={Equidistribution of singular measures on nilmanifolds and skew products},
  author={Fabrizio Polo},
  journal={Ergodic Theory and Dynamical Systems},
  year={2011},
  volume={31},
  pages={1785 - 1817}
}
  • F. Polo
  • Published 16 March 2010
  • Mathematics
  • Ergodic Theory and Dynamical Systems
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… 
Closed Ideals in the Stone-Cech Compactification of a Countable Semigroup and Some Applications to Ergodic Theory and Topological Dynamics
We study the relationship between algebra in the Stone-Čech compactification βS of a countable semigroup and dynamics. In particular, we establish a correspondence between the closed subsets of βS
ETS volume 31 issue 6 Cover and Back matter
  • Ergodic Theory and Dynamical Systems
  • 2011

References

SHOWING 1-10 OF 14 REFERENCES
Strict Ergodicity and Transformation of the Torus
Introduction. If T is a measure preserving transformation ofl a probability space Q with measure Iu, the ergodic theorem assures the existence N-1 almost everywhere with respect to /i of the average
Ergodic Theory via Joinings
Introduction General group actions: Topological dynamics Dynamical systems on Lebesgue spaces Ergodicity and mixing properties Invariant measures on topological systems Spectral theory Joinings Some
Quasi-factors in ergodic theory
Motivated by the notion of quasi-factor in topological dynamics, we introduce an analogous notion in the context of ergodic theory. For two processes,X andY , we haveX∡Y if and only ifY has a factor
Quasifactors of ergodic systems with positive entropy
The relation between the two notions, quasifactors and joinings, is investigated and the notion of a joining quasifactor is introduced. We clarify the close connection between quasifactors and
An application of number theory to ergodic theory and the construction of uniquely ergodic models
Using a combinatorial result of N. Hindman one can extend Jewett’s method for proving that a weakly mixing measure preserving transformation has a uniquely ergodic model to the general ergodic case.
Introduction to Ergodic Theory
Ergodic theory concerns with the study of the long-time behavior of a dynamical system. An interesting result known as Birkhoff’s ergodic theorem states that under certain conditions, the time
Topics in Optimal Transportation
Introduction The Kantorovich duality Geometry of optimal transportation Brenier's polar factorization theorem The Monge-Ampere equation Displacement interpolation and displacement convexity Geometric
On a class of homogeneous spaces of compact Lie groups
ERGODIC PROPERTIES OF AFFINE TRANSFORMATIONS AND FLOWS ON NILMANIFOLDS.
...
...