The starting point for this work is an old question of H. Furstenberg. Recall that a Borel probability measure on the torus T = R=Z is called p-invariant if p = , where p(x) def = p x (mod 1) and p def = 1 p . Suppose that is non-atomic, and is simultaneously 2and 3-invariant. Assume also that is ergodic under the joint action of 2; 3. Does it follow that is necessarily Lebesgue measure? In his...