Dynamics on the path space of generalized Bratteli diagrams

The paper proves existence of the Vershik-invariant measure µ with the cylinder formula µ([e(w,v)]) = ξ_v/λ^n under A = F^T irreducible, aperiodic, recurrent, and shows uniqueness in the positive recurrent case via Birkhoff/ratio limits using the PF asymptotics lim_N a^(N)_{wv}/λ^N = ξ_w η_v (Theorems 6.1 and 6.2) . The model constructs the same measure via Carathéodory, checks invariance by equal weights on Rokhlin towers, and proves uniqueness by a tower-weight recursion c(n) = F c(n+1) coupled with the positive-recurrent rank-one limit (Theorem A.7) to force c_v(n) ∝ ξ_v/λ^n. Both reach the same conclusion; the model’s proof is different but substantively correct. One technical step in the model’s argument (applying rank-one convergence to a k-dependent vector) should be justified more carefully; the paper avoids this by a ratio-ergodic argument, and cites Theorem A.7 appropriately . The model’s transpose bookkeeping remark aligns with the paper’s convention A = F^T and use of Aξ = λξ in the cylinder formula . The Vershik-map/tower finiteness used by the model follows from the diagram’s finite in-degree hypothesis (Definition 2.1(iii)) ensuring finite tower heights H_v^{(n)} .