Abstract: We review recent work (much of it joint with Frick or Varchenko) on adic (Bratteli-Vershik) dynamical systems which come from walks or reinforced walks on finite graphs. Identification of the ergodic invariant measures depends on knowing path counts between vertices in the associated diagram, and this leads to interesting combinatorial problems and formulas involving binomial coefficients as well as Eulerian, Stirling,and Delannoy numbers. Among dynamical properties that can be determined are lack of point spectrum, faithful coding by subshifts, topological weak mixing, loosely Bernoulli, and complexity.
