A CANTOR DYNAMICAL SYSTEM IS SLOW IF AND ONLY IF ALL ITS FINITE ORBITS ARE ATTRACTING

The paper proves the equivalence: a Cantor system embeds into the real line with a differentiable model whose derivative vanishes on the invariant Cantor set if and only if all finite orbits are attracting (Theorem 1.2) . Necessity follows from a differentiability-at-a-periodic-orbit argument (Lemma 4.1 and Corollary 4.2) ; sufficiency is achieved by constructing refining marked clopen partitions/graph coverings, enforcing quantitative shrinking via interval realizations, and then applying Jarník’s C1-extension (Theorem 4.3) , with explicit contraction rates and nested-interval control (Section 4.1.1–4.1.2) . The candidate solution reproduces the same two directions: (a)⇒(b) via local contraction of an iterate near periodic points; (b)⇒(a) via a refining sequence of clopen partitions, a Cantor set embedding by nested intervals with controlled lengths, verification of Whitney-flatness (derivative 0) on the embedded Cantor set, and a C1 extension to R. Differences are expository (direct "marked partitions + Whitney/Jarník" vs. the paper’s explicit graph-covering/marker machinery), not substantive. Hence both are correct with substantially the same proof strategy.