ON THE SPACE OF ITERATED FUNCTION SYSTEMS AND THEIR TOPOLOGICAL STABILITY

The paper proves exactly the three targets: (1) the space of IFS on X identifies with the hyperspace K(C0(X)) via Λ↦{ω_λ} and its converse (take Λ=K and evaluate), using uniform equicontinuity of partial maps and continuity of ϕ:Λ→C0(X) and of evaluation; see Theorem 1 and its proof . (2) On a smooth compact manifold, topological stability implies concordant shadowing by first obtaining finite shadowing through a local point-moving diffeomorphism construction and a tailored nearby IFS ω̃ with δ-compatible parameters, then passing to a limit (Lemmas 8.3–8.5 and Theorem 2) . (3) Expansiveness plus concordant shadowing yields topological stability by defining h(x) as the unique concordant shadow of the ω̃-orbit and proving continuity via expansivity (Proposition 9.2, Lemma 9.3, Theorem 3) . The candidate solution follows the same blueprint: (1) the same identification of IFS with K(C0(X)); (2) the same local point-moving construction and compactness/diagonal limit; (3) the same uniqueness-by-expansivity and continuity argument. Minor parameter bookkeeping slips (e.g., “shrink δ_TS”) do not affect correctness and are easily fixed by choosing δ_0≤δ_TS/2. Overall, both arguments coincide in substance.