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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 $E_3.$

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    $\begingroup$ Probably not, mathoverflow.net/questions/44234/… $\endgroup$
    – Andreas Thom
    Jan 10, 2012 at 9:32
  • $\begingroup$ 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. $\endgroup$
    – Anthony Quas
    Jan 10, 2012 at 16:15

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