DIAMETER DIMINISHING TO ZERO IFSs

The paper proves the equivalence of the five hyperbolic IFS properties (Theorem 4.1) and supplies a correct chain 1⇒3 (via [15]), 3⇒2, 2⇒1 (via [15]), 1⇒4 (with an explicit MB construction), 4⇒5, and 5⇒2; these steps are clearly stated in the PDF with supporting lemmas and references (see Theorem 4.1 and its proof, including 1⇒4 at and the 4⇒5 implications from Section 3, e.g., Propositions 3.1–3.5 at , , , and ). By contrast, the candidate’s 1⇒3 argument incorrectly treats the sets f[ω]n(B) as a decreasing (nested) family, invoking a “nested-singleton ⇒ vanishing diameters” lemma; no forward-invariance of B was assumed, so the sets need not nest. This is a substantive logical gap. The candidate’s other steps largely agree with the paper (e.g., 4⇒5 constructs π and the attractor; 5⇒2 establishes local uniform point-fibredness), but their direct 5⇒1 remetrization claim cites an external converse-of-contraction theorem without checking its hypotheses; the paper instead proves 5⇒2 and then uses 2⇒1 from [15].