Let us consider expanding maps $E_m: x\mapsto mx$ on the circle written additively. If we consider the set of all $E_2$ invariant probability Borel measures then the convex hull of atomic measures generated by $E_2$ periodic points are weak star dense in this set.

I want to ask whether this is true for the set of probability Borel measures invariant under both maps $E_2$ and $E_3.$