# A Hilbert space approach to multiple recurrence in ergodic theory

Abstract (Summary)

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.
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University of Pretoria etd – Beyers, F J C (2005)
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.
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University of Pretoria etd – Beyers, F J C (2005)
Bibliographical Information:

Advisor:

School:University of Pretoria/Universiteit van Pretoria

School Location:South Africa

Source Type:Master's Thesis

Keywords:ergodic theory hilbert space recurrent sequences mathematics

ISBN:

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