Specification Property for Step Skew Products

The paper proves Theorem 4.1 correctly: under piecewise monotone, continuous, expanding, surjective fibre maps, if the base subshift has specification and admits a periodic word α whose induced composition G is mixing, then the step skew product has specification. The proof hinges on two robust ingredients: (i) a non-shrinking lemma for finite nonautonomous interval systems (Theorem 3.1) ensuring a uniform lower bound on interval lengths under any allowed composition, and (ii) mixing ⇒ locally eventually onto for piecewise monotone interval maps with a uniform power on intervals of a fixed minimal length (Lemma 2.1). Together with a careful use of the base specification metric ρ satisfying property (2), the construction yields a periodic point that ε-traces all prescribed orbit segments with a controlled gap M = mp + 2K, where m depends only on the LEO modulus and K is the base specification gap. These steps are explicit in the paper’s statement and proof of Theorem 4.1 and its supporting lemmas. By contrast, the model’s proof attempts a different route via inverse-branch contractions after inserting α-blocks, but it contains two critical flaws. First, it incorrectly infers that an interval I contained in a monotonicity interval J of G with |I| ≥ δ* (the minimum length among monotonicity intervals) must equal J; this is false when |J| > δ*. The step that concludes G^{N0}(I) = [0,1] from that assertion is therefore unsupported. Second, the proof requires choosing fibre windows U_j so that, for every admissible connector word v of a fixed length, all intermediate images avoid turning points and produce a length threshold; this fails if x_j lies on a preimage of a critical point for some such v (a possibility that must be handled in a general specification proof). The paper circumvents both issues by not relying on monotonicity along connectors and instead using the non-shrinking lemma (Theorem 3.1) and a uniform locally-eventually-onto iterate (Lemma 2.1).