Dado um processo de Poisson d-dimensional, construímos funções harmônicas na triangulação de Delaunay associada, com comportamento assintótico linear, como limite de um processo de harness sem ruído. Tais funções permitem que construamos uma nova imersão da triangulação de Delaunay, que denominaremos de deformação harmônica. / Given a d-dimensional Poisson point process, we construct harmonic functions on the associated Delaunay triangulation, with linear assymptotic behaviour, as the limit of a noiseless harness process. These mappings allow us to find a new embedding for the Delaunay triangulation. We call it harmonic deformation of the graph.

Dado um processo de Poisson d-dimensional, construímos funções harmônicas na triangulação de Delaunay associada, com comportamento assintótico linear, como limite de um processo de harness sem ruído. Tais funções permitem que construamos uma nova imersão da triangulação de Delaunay, que denominaremos de deformação harmônica. / Given a d-dimensional Poisson point process, we construct harmonic functions on the associated Delaunay triangulation, with linear assymptotic behaviour, as the limit of a noiseless harness process. These mappings allow us to find a new embedding for the Delaunay triangulation. We call it harmonic deformation of the graph.

Le théorème ergodique de Birkhoff nous renseigne sur la convergence de suites de fonctions. Nous nous intéressons alors à étudier la convergence en moyenne et presque partout de ces suites, mais dans le cas où la suite est une suite strictement croissante de nombres entiers positifs. C’est alors que nous définirons les suites uniformes et étudierons la convergence presque partout pour ces suites. Nous regarderons également s’il existe certaines suites pour lesquelles la convergence n’a pas lieu. Nous présenterons alors un résultat dû en partie à Alexandra Bellow qui dit que de telles suites existent. Finalement, nous démontrerons une équivalence entre la notion de transformatiuon fortement mélangeante et la convergence d'une certaine suite qui utilise des “poids” qui satisfont certaines propriétés. / Birkhoff’s ergodic theorem gives us information about the convergence of sequences of functions. We are then interested in studying the mean and pointwise convergence of these sequences, but in the case the sequence is a strictly increasing sequence of positive integers. With that goal in mind, we will define uniform sequences and study the pointwise convergence for these sequences. We will also explore the possibility that there exists some sequences for which the convergence of the sequence does not occur. We will present a result of Alexandra Bellow that says that such sequences exist. Finally, we will prove a result which establishes an equivalence between the notion of a strongly mixing transformation and the convergence of a sequence that uses “weights” which satisfies certain properties.

The use of Hilbert space theory became an important tool for ergodic theoreticians ever since John von Neumann proved the fundamental Mean Ergodic theorem in Hilbert space. Recurrence is one of the corner stones in the study of dynamical systems. In this dissertation some extended ideas besides those of the basic, well-known recurrence results are investigated. Hilbert space theory proves to be a very useful approach towards the solution of multiple recurrence problems in ergodic theory. Another very important use of Hilbert space theory became evident only relatively recently, when it was realized that non-commutative dynamical systems become accessible to the ergodic theorist through the important Gelfand-Naimark-Segal (GNS) representation of C*-algebras as Hilbert spaces. Through this construction we are enabled to invoke the rich catalogue of Hilbert space ergodic results to approach the more general, and usually more involved, non-commutative extensions of classical ergodic-theoretical results. In order to make this text self-contained, the basic, standard, ergodic-theoretical results are included in this text. In many instances Hilbert space counterparts of these basic results are also stated and proved. Chapters 1 and 2 are devoted to the introduction of these basic ergodic-theoretical results such as an introduction to the idea of measure-theoretic dynamical systems, citing some basic examples, Poincairé’s recurrence, the ergodic theorems of Von Neumann and Birkhoff, ergodicity, mixing and weakly mixing. In Chapter 2 several rudimentary results, which are the basic tools used in proofs, are also given. In Chapter 3 we show how a Hilbert space result, i.e. a variant of a result by Van der Corput for uniformly distributed sequences modulo 1, is used to simplify the proofs of some multiple recurrence problems. First we use it to simplify and clarify the proof of a multiple recurrence result by Furstenberg, and also to extend that result to a more general case, using the same Van der Corput lemma. This may be considered the main result of this thesis, since it supplies an original proof of this result. The Van der Corput lemma helps to simplify many of the tedious terms that are found in Furstenberg’s proof. In Chapter 4 we list and discuss a few important results where classical (commutative) ergodic results were extended to the non-commutative case. As stated before, these extensions are mainly due to the accessibility of Hilbert space theory through the GNS construction. The main result in this section is a result proved by Niculescu, Ströh and Zsidó, which is proved here using a similar Van der Corput lemma as in the commutative case. Although we prove a special case of the theorem by Niculescu, Ströh and Zsidó, the same method (Van der Corput) can be used to prove the generalized result. Copyright 2004, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. Please cite as follows: Beters, FJC 2004, A Hilbert space approach to multiple recurrence in ergodic theory, MSc dissertation, University of Pretoria, Pretoria, viewed yymmdd < http://upetd.up.ac.za/thesis/available/etd-02222006-104936 / > / Dissertation (MSc (Applied Mathematics))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted