# weak star density of atomic invariant measures ?

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 $E_3.$

• Probably not, mathoverflow.net/questions/44234/… – Andreas Thom Jan 10 '12 at 9:32
• If the Furstenberg conjecture is true, then the answer is yes: The conjecture states that the invariant measures consist of precisely combinations of atomic measures and Lebesgue measure. Since Lebesgue measure is the weak* limit of the invariant measures $\mu_p=1/(p-1)\sum_{i=1}^{p-1}\delta_{i/p}$ for $p$ prime, we're done. – Anthony Quas Jan 10 '12 at 16:15