Common invariant probabilities on finite sets beyond common fixed points

Determine which finite sets, beyond the common fixed points described for beta-transformations satisfying a rational relation between their distances from one, can carry a common invariant probability for the two transformations.

Background

The paper analyzes beta-transformations T_beta(x) = beta x minus floor(beta x) and T_gamma under a rational affine relation between the bases. When the relation satisfies a+c = 1, the paper explicitly classifies the common fixed points and shows that every probability supported on the resulting finite set is invariant under both maps.

The authors emphasize that this fixed-point classification does not settle the broader atomic-measure problem: finite sets may exist on which the two transformations act as nontrivial permutations. The unresolved problem is therefore to characterize all such finite invariant supports and the common invariant probabilities they carry, beyond measures supported on common fixed points.