Sharkovskii theorem for infinite dimensional dynamical systems.

The paper’s Theorem 11 establishes the infinite-dimensional Sharkovskii-type result for compact maps on Banach spaces by: (i) reducing to a 1D piecewise-linear model, (ii) invoking the existence of a non-repeating m-loop of O-intervals, (iii) lifting these interval coverings to horizontal covering relations between h-sets with tails via CC1–CC3, and (iv) applying an itinerary lemma for covering relations to obtain an m-periodic point with least period m. The candidate’s solution mirrors this structure step-by-step, including the same construction of S(Ji), the same CC1–CC3 verification idea (orientation-preserving or reversing), and the same conclusion from non-repeating loops. Key ingredients and checks appear explicitly in the paper: the non-repeating m-loop (Theorem 4), the h-set-with-tail covering criterion CC1–CC3 (Lemma 8) and its use in Lemma 12, and the concluding argument for least period m; the proof sketch for finite dimensions (Theorem 9) is also presented and then generalized to Banach spaces in Theorem 11. No material logical gaps are introduced by the model; the only minor imprecision is the model’s brief mention of using the identity coordinate chart, whereas the paper uses a natural affine reparametrization for the unstable (p) coordinate in cS(Ji). Overall, both arguments agree on hypotheses, construction, and logic, and are essentially the same proof route. See the statements and proof sketches in the paper for Theorem 11 and its CC1–CC3 verification and itinerary step as cited here: Theorem 11 statement and setup, the existence of non-repeating loops, the CC1–CC3 criterion and its verification, and the overall proof idea for lifting from 1D to higher dimensions.