THE COINCIDENCE OF RÉNYI-PARRY MEASURES FOR β-TRANSFORMATION

The uploaded paper proves that for two different non-integer parameters β1, β2 > 1, the Rényi–Parry measures coincide if and only if β1 is the root of x^2 − qx − p with p, q ∈ N, 1 ≤ p ≤ q, and β2 = β1 + 1; in particular both are quadratic Pisot numbers. The proof proceeds by: (i) reducing equality of measures to equality of the initial density hβ pointwise (via the normalization Kβ and right continuity), (ii) showing the orbits Oβ are finite and equal except possibly at 0, and (iii) ruling out all finite patterns except the minimal one, which forces β2 = β1 + 1 and the quadratic relation for β1; sufficiency is verified by direct calculation (Theorem 1.1, Proposition 2.1, Propositions 2.4–2.6) . The candidate solution follows the same structure: Parry’s formula for hβ, equality of initial densities, orbit finiteness and matching, and the same minimal pattern leading to β2 = β1 + 1 and β1^2 = qβ1 + p, plus the Pisot corollary. Aside from a minor slip where it says “presence of 1” instead of “presence of 0” in exactly one orbit, the arguments and overall proof strategy are the same as the paper’s necessity/sufficiency proofs .