On Unique Ergodicity Of Coupled AIMD Flows

The paper proves unique ergodicity for the N-synchronised two-resource AIMD with finite averaging by lifting to X=Σ^{2N}, showing non-expansiveness for all Γ and a strict contraction Γ1 selected with uniformly positive probability, then applying a place-dependent IFS theorem to obtain a unique attractive invariant measure and almost-sure convergence of time-averages (Theorem 6). The candidate solution proves the same result by an equivalent place-dependent IFS route, using a slightly different metric and Elton’s ergodic theorem. The core mechanism—existence of a uniformly positive-probability strict contraction together with non-expansiveness of all other maps—is the same in both arguments. See the paper’s norm, lifting, and Theorem 6 statements and proof, including the definition of probabilities p_{ν,μ} and their positivity lower bound p̂ (>0) for Γ1 (, , , , ).