INTERMEDIATE β-SHIFTS AS GREEDY β-SHIFTS WITH A HOLE

The paper proves the three-part correspondence claimed in Theorem 1.1: (i) every intermediate β-transformation (T_{β,α},[0,1]) is topologically conjugate (almost one-to-one) to a greedy β-transformation with a hole at zero; (ii) the converse fails in general; and (iii) for simple Parry β′ (i.e., T_{β′,0}^n(1)=0) one has a local one-to-one correspondence for sufficiently small t in the bifurcation set E^+_{β′,0} . The candidate solution reproduces these three claims via symbolic coding with lexicographic subshifts, Parry expansions, and a kneading-equation argument, which matches the paper’s framework of conjugating to symbolic systems (via τ^± and commuting diagrams) and then back to interval dynamics , with the kneading-invariant characterization playing the central role . One technical slip in the model’s Part (3) is the claim that β ≤ φ automatically ensures α ∈ [0,2−β]; the correct condition is S ≤ 1/(β(β−1)) for S=∑w_k β^{−k} (and, for small t, this can indeed be enforced without invoking β ≤ φ). The slip is nonfatal and can be repaired by using the exact formula α = β(β−1)S − (β−1) together with the small-t regime (where β(t)↓1 and α(t)→0). Overall, the two arguments are substantively aligned, though the paper’s proof uses a different (combinatorial) route via the BSV characterization (Theorem 2.5) and carefully tracks admissibility sets A_β and B_β in the proof of Theorem 1.1 ; the model’s proof relies directly on Parry expansions and lexicographic subshifts. Given the minor repair needed in Part (3), we judge both to be correct, with different proof styles.