This thesis is broadly concerned with two problems: investigating the ergodic properties of boundary actions, and investigating various properties of Schreier graphs. Our main result concerning the former problem is that, in a variety of situations, the action of an invariant random subgroup of a group G on a boundary of G (e.g. the hyperbolic boundary, or the Poisson boundary) is conservative (there are no wandering sets). This addresses a question asked by Grigorchuk, Kaimanovich, and Nagnibeda and establishes a connection between invariant random subgroups and normal subgroups. We approach the latter problem from a number of directions (in particular, both in the presence and the absence of a probability measure), with an emphasis on what we term Schreier structures (edge-labelings of a given graph which turn it into a Schreier coset graph). One of our main results is that, under mild assumptions, there exists a rich space of invariant Schreier structures over a given unimodular graph structure, in that this space contains uncountably many ergodic measures, many of which we are able to describe explicitly.

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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

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

Inaug. Diss.--Marburg.

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))

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