Bratteli diagrams encode the combinatorial structure of dynamical systems on Cantor sets — each level of the diagram corresponds to a partition of the space, and the edges between levels describe how partitions refine. The diagram captures the system's large-scale dynamics while abstracting away metric details.
Separated graphs extend this framework by disentangling two things that Bratteli diagrams conflate: the topological structure of the space and the dynamics of the homeomorphism acting on it. In a separated graph, the space's partition structure is encoded independently from the map's combinatorial action. The separation allows you to read off whether a dynamical system is minimal (every orbit is dense) or essentially minimal (minimal on a large subset) directly from the graph structure, without computing orbits.
The paper demonstrates the construction on four canonical examples: the two-sided shift, the bitwise NOT map, the classical odometer, and the shift on the one-point compactification of the integers. Each produces a visually distinct separated graph that makes the system's dynamical properties immediately readable. The shift's graph has a characteristic branching structure reflecting its exponential orbit growth; the odometer's graph is nearly linear, reflecting its rigid rotation.
The structural claim: dynamics and topology are separable aspects of a dynamical system, and encoding them separately makes both more transparent. The Bratteli diagram's power is that it encodes everything; the separated graph's power is that it encodes things independently. Separating what's combined clarifies what each part contributes.