Fellgett, Terence Robin (1976) Three problems in ergodic theory. PhD thesis, University of Warwick.

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Abstract

The three problems refered to in the title of this thesis are investigated in three sections, which are entirely independent of each other. Ergodic Theory includes, in our view, Topological Dynamics and, in fact, section two is entirely topological and section three mostly so.
Section1 ■ The concept of a pair of δ-algebras being regular is introduced and, hence, a notion of isomorphism more restrictive than the usual conjugacy of measure preserving automorphisms is defined. This equivalence relation may be interpreted on the endomorphism level (theorem 1) Insubsection 2 an invariant of the relation which is often finer than entropy is introduced.
Section2, Following some work of Gurevic (8) a few simple facts about the topological entropy of sub-shifts of finite type on countably many symbols are derived. This enables us to give an example of a homeomorphism of a zero dimensional space which has both finite and infinite topological entropy, with respect to equivalent metrics.
Section3. This section is divided into five subsections.
1. Following an account of Hahn (12) and Parry's(20) theory of topological group actions with quasi-discrete spectrum we show any transformation to which such an action is transversal is affine (theorem 6).
2. This subsection motivates the next three.
3. A fairly general method of constructing discrete actions of finitely generated abelian groups as affine transformations of finite dimensional tori is given. This method is designed to meet the needs of the proof of Weyl’s theorem in subsection 5
4. Under a mild hypothesis the actions constructed in(3) are shown to be totally ergodic, with respect to Haar measure, (theorem 8) and have quasi-discrete spectrum (theorem 9).we are therefore, in particular, able to give a general theorem (no. 10) about the existence of Zm - actions to which the theory out lined in(l)applies.
5. The results of (3) and (if) are used to give a new proof of Weyl's theorem (28) on the uniform distribution of polynomials of integer variables.
Numbering of Results.
Theorems and propositions are numbered consecutively with in each section. Lemmas are numbered consecutively with in each subsection. When it is necessary to refer to a lemma in a previous subsection, say Y, then it is denoted lemma Y.Z, where Z is the number of the lemma.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Ergodic theory, Topological dynamics, Isomorphisms (Mathematics)
Official Date:	August 1976
Dates:	Date Event August 1976 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Parry, William, 1934-2006
Sponsors:	Science Research Council (Great Britain)
Extent:	[87] leaves
Language:	eng

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