SELF CONSISTENT TRANSFER OPERATORS IN A WEAK COUPLING REGIME. INVARIANT MEASURES, CONVERGENCE TO EQUILIBRIUM, LINEAR REPONSE AND CONTROL OF THE STATISTICAL PROPERTIES.

The paper proves exponential convergence for self-consistent transfer operators under (Con1–Con3) by grouping iterates into blocks, deriving a two-dimensional (strong/weak) linear inequality with a 2×2 matrix M, and choosing N large and δ small so that the maximal eigenvalue is <1 (via a balanced norm), hence obtaining exponential decay; see the statement of (Con1–Con3) and Theorem 3, and the block-matrix Theorem 6 with its use in the proof of Theorem 3 . The candidate solution implements the same core idea: (i) a uniform weak contraction of N-step products via a telescoping comparison to L0 and the Lasota–Yorke inequality (mirroring Lemma 5) ; (ii) a one-step inequality for the nonlinear difference that couples strong/weak norms (as in the paper’s use of Con1–Con2 in the Theorem 3 proof) ; and (iii) a block recursion with a 2×2 matrix M(δ,N) and spectral-radius <1 for N large and δ small (the same mechanism as Theorem 6) . The only extra normalization the model states (Lδ^0 = L0) is implicitly used in the paper’s Lemma 5 when comparing Lδ^μ to L0; with μ in Bw(0,1) this follows from the set-up of weakly nonlinear (δ·μ) dependence in the examples, and the paper effectively assumes it in the Con2-based estimate there . Overall, both arguments are correct and essentially the same.