Mohamed, Abdel-Karim A. (1975) Spectral properties of some ergodic systems. PhD thesis, University of Warwick.

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Abstract

This thesis deals with the spectral properties of some dynamical systems.
In Chapter one of the main tools necessary for this work will be reviewed.This is the spectral theory of unitary operator in Hilbert spaces. Chapter II deals with tensor products of unitary operate (hence direct product of invertible measure preserving transformation, i.m.p.t) In this chapter we develop a technique (the α technique) which enable us to compute a multiplicity pair for the tensor a products of two or more unitary operators. The chapter ends with an application of the main theorem to operators(i.m.p.t's)with simple discrete spectrum.
The other main ttool needed for this work is the theory of Gaussian pprocesses and will be reviewed in Chapter III. Chapter IV deals with some invariant σ-algebras for measure preserving transformation. A generalization αө (T) of a canonical σ-algebras αө (T) defined by .Walters[28] will be given. The properties of the σ-algebras αө(T) will be studied. The sspectral properties of transformation with αө=B will be investigated. Also, after Parry[19],we introduce the concept of representations in ℓө, the class of all the transformations with αө=B. It will be proved that if αө(T)=B than the sequenceT1=Te(1) converges to a limit in the group of all transformations on a Lebesgue space. These to fall such limits will be found to form a group G(T),and G(T) is a conjugacy invariant. The algebras αө(t)T) will be studied in relation to the concept of mixing and in relation to entropy theory. The relation of these o-algebras to group extension and Gaussian processes will be considered.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Ergodic theory, Gaussian processes, Spectral theory (Mathematics)
Official Date:	July 1975
Dates:	Date Event July 1975 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Parry, William, 1934-2006
Sponsors:	Jāmiʻat al-Iskandarīyah
Extent:	91 leaves
Language:	eng

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