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Ergodic Theorems for Homogeneous Dilations
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
(English)Article in journal (Refereed) Submitted
Abstract [en]

In this paper we prove a general ergodic theorem for ergodic and measure preserving actions of R^d on standard Borel spaces. In particular, we cover R.L. Jones ergodic theorem on spheres. Our main theorem is concerned with ergodic averages with respect to homogeneous dilations of Rajchman measures on Rd . We establish mean convergence in Hilbert spaces for general Rajchman measures, and give a criterion in terms of the Fourier dimension of the measure when almost everywhere pointwise convergence holds. Applications include averages over smooth submanifolds and polynomial curves.

Keyword [en]
Dynamical Systems (math.DS); Classical Analysis and ODEs (math.CA)
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-13946OAI: oai:DiVA.org:kth-13946DiVA: diva2:328526
Note
QC 20100705Available from: 2010-07-05 Created: 2010-07-05 Last updated: 2010-07-20Bibliographically approved
In thesis
1. Limit Theorems for Ergodic Group Actions and Random Walks
Open this publication in new window or tab >>Limit Theorems for Ergodic Group Actions and Random Walks
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of an introduction, a summary and 7 papers. The papers are devoted to problems in ergodic theory, equidistribution on compact manifolds and random walks on groups.

In Papers A and B, we generalize two classical ergodic theorems for actions of abelian groups. The main result is a generalization of Kingman’s subadditive ergodic theorem to ergodic actions of the group Zd.

In Papers C,D and E, we consider equidistribution problems on nilmanifolds. In Paper C we study the asymptotic behavior of dilations of probability measures on nilmanifolds, supported on singular sets, and prove, under some technical assumptions, effective convergences to Haar measure. In Paper D, we give a new geometric proof of an old result by Koksma on almost sure equidistribution of expansive sequences. In paper E we give necessary and sufficient conditions on a probability measure on a homogeneous Riemannian manifold to be non–atomic.

Papers F and G are concerned with the asymptotic behavior of random walks on groups. In Paper F we consider homogeneous random walks on Gromov hyperbolic groups and establish a central limit theorem for random walks satisfying some technical moment conditions. Paper G is devoted to certain Bernoulli convolutions and the regularity of their value distributions.

Place, publisher, year, edition, pages
Stockholm: KTH, 2009. vii, 18 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 09:06
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-10451 (URN)
Public defence
2009-05-26, Sal F3, KTH, Lindstedtsvägen 26, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC 20100705Available from: 2009-05-18 Created: 2009-05-15 Last updated: 2012-03-27Bibliographically approved

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Citation style
  • apa
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More styles
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  • de-DE
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  • Other locale
More languages
Output format
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