Classification of common singular probabilities under rational independence

Classify common singular invariant probabilities for two beta-transformations when 1, beta, and gamma are linearly independent over the rationals.

Background

The main classification theorem applies when 1, beta, and gamma are linearly dependent over the rationals and the relation does not lie on the excluded line a+c = 1. Its proof uses this rational dependence to construct an integer-power invariant irrational rotation, which forces every singular common invariant measure in the covered family to be supported at zero.

The paper explicitly states that the linearly independent case lies outside the theorem. In that setting, the rotation construction is unavailable, and the authors do not provide a classification of common singular invariant probabilities.