The Hutchinson-Barnsley theory for iterated function systems with general measures

The candidate reproduces the Hutchinson–Barnsley theory for IFSm under the paper’s mild assumptions W1, CP1, H2, H3, H4 with s + r M t < 1. The paper proves: (i) existence/uniqueness of the topological attractor AR (Theorem 5.1 and Remark 5.3) and its Hausdorff attraction, (ii) a contraction property for the M-step Markov/transfer operator leading to a unique invariant measure µR and convergence in Monge–Kantorovich distance (Theorem 6.4 with Lemmas 6.5–6.7; Corollary 6.8), and (iii) equality supp(µR) = AR (Theorem 7.1). The candidate’s step-by-step argument—F^M strict contraction on K*(X), a telescoping Lipschitz bound Lip(B^M) ≤ s + r M t, uniqueness/convergence for T^M and hence for T, and the support-equals-attractor argument using H4—is the same structure and logic as the paper’s proofs, modulo presentation. Minor paper/editorial detail: the constant in Theorem 6.4 should agree with the M-dependent bound used in Corollary 6.8; the candidate correctly uses s + r M t. Overall, the two are aligned and correct.