Negative beta-transformations map the interval by multiplying by a negative non-integer and reducing modulo 1. Different bases produce different dynamics, different symbolic codings, different invariant measures. When do two different bases produce the same invariant measure?
Huang and Sun give the complete answer. Two non-integers β₁ and β₂ share the same invariant measure if and only if β₁ satisfies x² - qx - p = 0 for natural numbers p, q with p ≤ q, and β₂ = β₁ + 1. The condition is algebraic: the pair must be roots of a specific family of quadratic equations, related by a shift of exactly 1.
The constraint is unexpectedly rigid. Sharing an invariant measure — an infinite-dimensional functional object — is determined by a finite algebraic condition on the bases. The dynamical richness of the transformation collapses to a quadratic equation.
The paper also establishes the matching property for generalized multinacci numbers — the negative-beta analogues of the golden ratio and its higher-order relatives. At these special bases, the orbits of the boundary points eventually coincide, producing a clean symbolic dynamics with finite-type subshifts. And the simple negative-beta numbers, whose transformations have finite-type symbolic dynamics, form a dense subset of (1, ∞).
The algebraic structure hiding in a messy-looking number-theoretic setting: two different transformations of the interval, one equation connecting them, and the answer is always “shift by 1.”