arXiv2026
The classical Adams dichotomy [Ada90] characterizes amenability of probability-measure-preserving treed equivalence relations in terms of the ends of a given treeing. We extend this dichotomy to the measure-class-preserving (mcp) setting by analyzing the Radon--Nikodym topography of an acyclic mcp Borel graph, i.e., the interaction between the graph geometry and the associated Radon--Nikodym cocycle. In the general mcp setting one must count not all ends, but only the nonvanishing ends: those ends which have positive upper cocycle-weight. More precisely, we prove that a locally countable acyclic mcp graph is amenable if and only if a.e. component has at most two nonvanishing ends, while it is nowhere amenable exactly when a.e. component has a nonempty perfect closed set of nonvanishing ends. At the heart of our analysis lies the study of acyclic countable-to-one Borel functions. Our critical result is that, outside of the essentially two-ended setting, all back ends in a.e. orbit are vanishing and have cocycle-finite geodesics. We also give several characterizations, including topographical and cocycle-free, of essential one-endedness of such functions, and show that the number of barytropic ends controls the essential number of ends. Our proofs utilize mass transport, end selection, and the notion of the Radon--Nikodym core for acyclic mcp graphs, a new concept that serves as a guiding framework for our topographical analysis.