No abstract / Mathematical Sciences / (D. Phil. (Mathematics))

Lam, Ho Yin Theodore. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 59-60). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Shift spaces and Symbolic Dynamical systems --- p.8 / Chapter 2.1 --- Shift spaces --- p.8 / Chapter 2.2 --- Symbolic dynamical systems --- p.10 / Chapter 3 --- β-expansion --- p.12 / Chapter 3.1 --- Expansion of real numbers --- p.12 / Chapter 3.2 --- Ergodic theory of f-expansion with independent digits --- p.16 / Chapter 3.3 --- β-expansion --- p.22 / Chapter 3.3.1 --- Ergodic theory of β-expansion --- p.22 / Chapter 3.3.2 --- Properties of β-expansion --- p.28 / Chapter 3.3.3 --- The normalizing function F(β) --- p.34 / Chapter 4 --- Symbolic Dynamics of β-expansion --- p.40 / Chapter 4.1 --- The symbolic dynamical system Sβ --- p.40 / Chapter 4.2 --- Classification of β and properties of Sβ --- p.42 / Chapter 5 --- Sizes of the classes --- p.49 / Chapter 5.1 --- Sizes of C1 and C2 --- p.49 / Chapter 5.2 --- "Sizes of C3, C4 and C5" --- p.50 / Bibliography --- p.59

A notion of entropy is defined for the non-singular action of finite co-ordinate changes on X - the infinite product of two- point spaces. This quantity - average co-ordinate or AC entropy - is calculated for product measures and G-measures on X, and an equivalence relation is established for which AC entropy is an invariant. The Inverse Vitali Lemma is discussed in a measure preserving context, and it is shown that for a certain class of measures on X known as odometer bounded, the result will still hold for odometer actions. The foundations for a non-singular version of Rudolph's restricted orbit equivalence are established, and a size for non-singular orbit equivalence is introduced. It is shown that provided the Inverse Vitali Lemma still holds, the non-singular orbit equivalence classes can be described using this new size.

In chapters I and II, we show that the group G of invertible, non-singular transformations of a Lebesgue space is perfect, simple, and has no outer automorphisms. Some related results are obtained for the subgroup of measure preserving transformations and for the full group of an ergodic transformation. Further results are given with the underlying Lebesgue space replaced by a homogeneous measure algebra. It is also shown, in chapter III, that ergodic transformations are algebraically distinguishable from non-ergodics. Chapter IV introduces the notion of a fibered ergodic transformation. A fibered analogue of Dye's theorem is proved. In chapter V the family of transformations satisfying Dye's theorem for two fixed ergodics is shown to be dense in the coarse topology. Finally, in chapter VI, the concept of a triangle set in the unit square is introduced. Using this notion, a sufficiency condition for the ergodicity of T x S is obtained.

Thesis (Ph. D.)--University of Washington. / Bibliography: l. 99-100.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.

Inaug. Diss.--Marburg.

Thesis (M.Sc.)(Mathematics)--University of Pretoria, 2004. / Title from opening screen (viewed March 27, 2006). Includes summary. Includes bibliographical references.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 162-164).

No abstract / Mathematical Sciences / (D. Phil. (Mathematics))