Bridging the Gap between Reactivity, Contraction, and Finite-Time Lyapunov Exponents

The paper’s Propositions 3 and 4 correctly identify existence/uniqueness and the period-dividing structure when f^p is contractive on the stated invariant sets, and the commutation argument is sound. However, the paper does not actually prove Lyapunov stability for the 1-step dynamics in Proposition 3 (it sketches only convergence via a union-of-orbits argument) and gives no argument for the stability of the periodic orbit claimed in Proposition 4. The candidate solution supplies the missing stability arguments (via uniform continuity/Lipschitz control on compact sets and an invariant-tube construction) and is therefore correct and more complete than the paper on these points. See the statements and proofs of Propositions 3 and 4 in the paper and the equivalence of sup-mean contractivity to ordinary contractivity in the time-invariant case . Proposition 1 in the paper covers stability for the p-iterate map but not the inter-block steps needed for stability of f itself, which is precisely where the model adds the needed details .