Adams proved the following dichotomy for ergodic acyclic graphs in the measure-preserving setting: if the underlying connectedness equivalence relation is amenable then almost every component has at most two ends, while if the relation is nonamenable then the set of ends of almost every component is (nonempty and) perfect with no isolated points.
As stated, this dichotomy fails miserably in the general measure-class-preserving (mcp) setting. In joint work with Anush Tserunyan, we show how to extend the Adams dichotomy to the general mcp setting by using "non-vanishing ends" instead of arbitrary ends (in Adams's original setting, all ends are non-vanishing). The general philosophy is to understand the local component-by-component behavior of the Radon-Nikodym cocycle geometrically within the graph, connecting this behavior back to global properties (e.g., amenability of the relation) via tools such as mass-transport.
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