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This thesis studies orbit equivalence in the almost continuous setting. Recently A. del Junco and A. Sahin obtained an almost continuous version of Dye’s theorem. They proved that any two ergodic measure-preserving homeomorphisms of Polish spaces are almost continuously orbit equivalent. One purpose of this thesis is to extend their result to all free actions of countable amenable groups. We also show that the cocycles associated with the constructed orbit equivalence are almost continuous. In the second part of the thesis we obtain an analogue of Dye’s reconstruction theorem for etale equivalence relations in the almost continuous setting. We introduce topological full groups of etale equivalence relations and show that if the topological full groups are isomorphic, then the equivalence relations are almost continuously orbit equivalent.

This thesis studies orbit equivalence in the almost continuous setting. Recently A. del Junco and A. Sahin obtained an almost continuous version of Dye’s theorem. They proved that any two ergodic measure-preserving homeomorphisms of Polish spaces are almost continuously orbit equivalent. One purpose of this thesis is to extend their result to all free actions of countable amenable groups. We also show that the cocycles associated with the constructed orbit equivalence are almost continuous. In the second part of the thesis we obtain an analogue of Dye’s reconstruction theorem for etale equivalence relations in the almost continuous setting. We introduce topological full groups of etale equivalence relations and show that if the topological full groups are isomorphic, then the equivalence relations are almost continuously orbit equivalent.

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Thesis (Ph. D.)--University of New South Wales, 1997. / Also available online.

Thesis (Ph. D.)--University of New South Wales, 1997. / Completed at: University of New South Wales, School of Mathematics. Title from electronic deposit form.

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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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