INTERVAL MAPS WITH DENSE PERIODICITY

The paper’s main theorem (Theorem 3.1) states exactly the target equivalence: a continuous interval map is chain-recurrent iff its periodic points are dense in I (the class CP) . The (CP ⇒ chain-recurrent) direction in the paper is sound: they build an ε-chain at x by borrowing a nearby periodic orbit of a map g ∈ CP within ε/2 of f, yielding a valid ε-chain for f (the construction x0=x, xi=g^i(p), xn=x is explicit) . However, in the converse direction (chain-recurrent ⇒ CP), the write-up constructs a “connect-the-dots” map g from finitely many ε/6-loops, perturbs g to a piecewise affine leo map h with h ∈ CP, and notes that ||f−h||∞<2ε. The proof then concludes: “Since ε is arbitrary we have f ∈ CP” . This last inference is not justified: uniform closeness to a CP map does not imply membership in CP (indeed, CP is not closed, and the paper later emphasizes delicate density/closure phenomena). The missing step would need either a persistence argument for transverse periodic points ensuring Per(f) hits every open set, or a direct interval argument from the ε-loop (as in the model solution). The model solution provides a complete, classical proof: (i) CP ⇒ chain-recurrent by noting CR(f) is closed and contains Per(f), hence I; (ii) chain-recurrent ⇒ CP by using a small ε-loop inside an arbitrary open interval U to build compact intervals J_i with f(J_i)⊂int(J_{i+1}); then f^n maps J_0 into itself, so an intermediate-value fixed point of f^n lies in U. Thus periodic points are dense. This fills the gap cleanly.