Common invariant measures for beta-transformations with rational affine relations

Abstract: We classify common invariant probabilities for two beta-transformations with a rational affine relation between their bases. Let $β>1$ be irrational and $γ=aβ+c>1$, where $a,c\in\mathbb Q$ and $a+c\ne1$. Every finite nonnegative measure that is singular with respect to Lebesgue measure and invariant under both maps is a multiple of the point mass at zero. No entropy, ergodicity or multiplicative independence assumption is needed. The proof constructs a finite signed measure invariant under an irrational rotation and relates its mass to the first moment of the original measure. Combining this result with the known classification of coincident absolutely continuous invariant probabilities gives all common invariant probabilities in this family. The only additional measures are mixtures with a common absolutely continuous probability for adjacent bases in a known quadratic family. We also determine all common fixed points on the excluded line $a+c=1$.