THE NON-ITERATES ARE DENSE IN THE SPACE OF CONTINUOUS SELF-MAPS

The uploaded preprint states and proves exactly the claim the candidate addressed: for X in {[0,1]^m, R^m, S^1], the complement of the set of iterates is dense in C(X) (Theorem 1; abstract) . The proof rests on a purely set-theoretic obstruction (Theorem 2) that forbids iterative roots of any order n ≥ 2 under fiber-size hypotheses (C1)–(C3) and gives a clean counting/structure argument via the auxiliary sets E0, E−1, E1, E2 around x0 (with the optimal N^3 threshold in (C1)) . The S^1 case is handled by constructing an admissible piecewise-affine map on a fine circular partition and modifying a tiny arc to be constant so that f(x0) ≠ x0, f−1(z) is finite for all z ≠ x0, and f−2(x0) is infinite—hence the obstruction applies . For [0,1]^m and R^m, the paper invokes the same local (piecewise-linear) gadgetry from their earlier ETDS paper and observes that Theorem 2 upgrades those constructions from “no square root” to “no n-th root for any n≥2” . The candidate’s solution reproduces this exact strategy: it states the same (C1)–(C3) obstruction and implements the same localized piecewise-affine modifications on [0,1]^m, outside compacts in R^m, and on S^1 using a circular partition. One minor imprecision is that, in the [0,1]^m sketch, ensuring f(x0) ≠ x0 needs the explicit choice the paper makes in the S^1 proof; but this is a routine adjustment. Overall, the model’s proof aligns closely with the paper’s method and conclusion.