Gustav Zickert: Furstenberg's conjecture and measure rigidity for some classes of non-abelian affine actions on tori

In 1967 Furstenberg proved that the set {\({2^n3^mx (mod 1) \ |\ n, m \ integers}\)} is dense in the circle for any irrational x. He also made the following famous measure rigidity conjecture: the only (x2,x3)-ergodic measures are the Lebesgue measure and measures supported on a finite set. In this thesis we discuss both Furstenberg's theorem and his conjecture, as well as the partial solution of the latter given by Rudolph. Following the ideas of Matteus and Avila for a weak version of Rudolph's theorem, we prove a general result on extending measure preservation from a semigroup action to a larger semigroup action. Using this result we obtain restrictions on the set of invariant measures for certain classes of non-abelian affine actions on tori. We also study some general properties of affine abelian and non-abelian actions and we show that analogues of Furstenberg's theorem hold for affine actions on the circle