Two theorems of Dye in the almost continuous category

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.