Flat surface models of ergodic systems

We study connections between translation flows on flat surfaces, adic transformations defined on Bratteli diagrams, and cutting-and-stacking transformations. We do so by introducing a general technique which takes an adic transformation and constructs a flat surface whose vertical translation flow admits a cross section for which the first return map is measurably isomorphic to the adic transformation. We show that any finite entropy, measure-preserving flow on a standard Lebesgue space is measurably isomorphic to the translation flow on a flat surface obtained through our construction. We give a criterion for ergodicity of the translation flow on these surfaces and apply this criterion to several examples, as well as describing specific examples of infinite type flat surfaces on which the translation flow exhibits new phenomena not found for flows on flat surfaces of finite type.