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Limit Theorems for Ergodic Group Actions and Random Walks
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
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: urn:nbn:se:kth:diva-10451OAI: oai:DiVA.org:kth-10451DiVA: diva2:217655
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
List of papers
1. The asymptotic shape theorem for generalized first passage percolation
Open this publication in new window or tab >>The asymptotic shape theorem for generalized first passage percolation
2010 (English)In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 38, no 2, 632-660 p.Article in journal (Refereed) Published
Abstract [en]

We generalize the asymptotic shape theorem in first passage percolation on Z(d) to cover the case of general semimetrics. We prove a structure theorem for equivariant semimetrics on topological groups and an extended version of the maximal inequality for Z(d)-cocycles of Boivin and Derriennic in the vector-valued case. This inequality will imply a very general form of Kingman's subadditive ergodic theorem. For certain classes of generalized first passage percolation, we prove further structure theorems and provide rates of convergence for the asymptotic shape theorem. We also establish a general form of the multiplicative ergodic theorem of Karlsson and Ledrappier for cocycles with values in separable Banach spaces with the Radon-Nikodym property.

Keyword
First passage percolation, cocycles, subadditive ergodic theory
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-13945 (URN)10.1214/09-AOP491 (DOI)000275938300007 ()2-s2.0-77953553228 (Scopus ID)
Note
QC 20100705Available from: 2010-07-05 Created: 2010-07-05 Last updated: 2017-12-12Bibliographically approved
2. Ergodic Theorems for Homogeneous Dilations
Open this publication in new window or tab >>Ergodic Theorems for Homogeneous Dilations
(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
Dynamical Systems (math.DS); Classical Analysis and ODEs (math.CA)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-13946 (URN)
Note
QC 20100705Available from: 2010-07-05 Created: 2010-07-05 Last updated: 2010-07-20Bibliographically approved
3. Equidistribution of dilations of polynomial curves in nilmanifolds
Open this publication in new window or tab >>Equidistribution of dilations of polynomial curves in nilmanifolds
2009 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 137, no 6, 2111-2123 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we study the asymptotic behaviour under dilations of probability measures supported on polynomial curves in nilmanifolds. We prove, under some mild conditions, the effective equidistribution of such measures to the Haar measure. We also formulate a mean ergodic theorem for R-n-representations on Hilbert spaces, restricted to a moving phase of low dimension. Furthermore, we bound the necessary dilation of a given smooth curve in R-n so that the canonical projection onto T-n is epsilon-dense.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-13947 (URN)10.1090/S0002-9939-09-09836-0 (DOI)000263532900031 ()2-s2.0-77951072821 (Scopus ID)
Note
QC 20100705Available from: 2010-07-05 Created: 2010-07-05 Last updated: 2017-12-12Bibliographically approved
4. Almost sure equidistribution in expansive families
Open this publication in new window or tab >>Almost sure equidistribution in expansive families
2009 (English)In: Indagationes mathematicae, ISSN 0019-3577, E-ISSN 1872-6100, Vol. 20, no 2, 167-177 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we study generic equidistribution in families of sequences of points on tori. We assume that the sequences are parameterized by some subset of a Euclidean space, and we formulate geometric conditions on the dependence so that equidistribution holds almost everywhere with respect to the Lebesgue measure on the parameter space. As a consequence, we can give a new proof of an old result by Koksma.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-13948 (URN)10.1016/S0019-3577(09)00017-2 (DOI)000274561400001 ()2-s2.0-77449137486 (Scopus ID)
Note
QC 20100705Available from: 2010-07-05 Created: 2010-07-05 Last updated: 2017-12-12Bibliographically approved
5. Continuous Measures on Homogeneous Spaces.
Open this publication in new window or tab >>Continuous Measures on Homogeneous Spaces.
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-13949 (URN)
Note
QC 20100705Available from: 2010-07-05 Created: 2010-07-05 Last updated: 2010-07-20Bibliographically approved
6. Central Limit Theorems for Gromov Hyperbolic Groups
Open this publication in new window or tab >>Central Limit Theorems for Gromov Hyperbolic Groups
2010 (English)In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 23, no 3, 871-887 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we study asymptotic properties of symmetric and nondegenerate random walks on transient hyperbolic groups. We prove a central limit theorem and a law of iterated logarithm for the drift of a random walk, extending previous results by S. Sawyer and T. Steger and of F. Ledrappier for certain CAT(-1)-groups. The proofs use a result by A. Ancona on the identification of the Martin boundary of a hyperbolic group with its Gromov boundary. We also give a new interpretation, in terms of Hilbert metrics, of the Green metric, first introduced by S. Brofferio and S. BlachSre.

Keyword
Random walks on groups, Central limit theorems, Martingale approximations, Metric geometry, Ergodic theory
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-13950 (URN)10.1007/s10959-009-0230-x (DOI)000280128700010 ()2-s2.0-77954761264 (Scopus ID)
Note
QC 20100705 Uppdaterad från submitted till published (20110215).Available from: 2010-07-05 Created: 2010-07-05 Last updated: 2017-12-12Bibliographically approved
7. Almost sure absolute continuity of Bernoulli convolutions
Open this publication in new window or tab >>Almost sure absolute continuity of Bernoulli convolutions
2010 (English)In: Ann. Inst. H. Poincaré Probab. Statist., ISSN 0246-0203, Vol. 46, no 3, 888-893 p.Article in journal (Refereed) Published
Abstract [en]

We prove an extension of a result by Peres and Solomyak on almostsure absolute continuity in a class of symmetric Bernoulli convolutions.

Keyword
Bernoulli convolutions; Absolute continuity
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-11647 (URN)10.1214/09-AIHP334 (DOI)000283528100013 ()2-s2.0-77955640920 (Scopus ID)
Funder
Knut and Alice Wallenberg Foundation, KAW 2005.0098
Note
Updated from submitted to published. QC 20120327Available from: 2009-12-03 Created: 2009-11-28 Last updated: 2012-03-27Bibliographically approved

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