### 27 mars 2008, kl 13.15-14.15, KTH, sal 3721.

###
Alexander Fish (Ohio State University):
Rigidity of invariant measures under the action of a multiplicative
semigroup of positive logarithmic density on T.

**Abstract.**
(joint work with Manfred Einsiedler) In 1967 H.Furstenberg
proved that any closed, invariant under x2 and x3 action subset of the
torus is either finite or the whole torus. He posed the following
question: Is it true that a (x2,x3)-invariant ergodic Borel probability
measure on the torus is either Lebesgue or has finite support? The best
known result is due to Rudolph: If the entropy of one of the actions with
respect to the measure is positive then the measure is Lebesgue.
We prove that if it is known that ergodic Borel probability measure on
the torus is invariant under "many" T_n actions (T_n(x) = nx mod 1) [the
set of n's should have positive logarithmic density] then it is either has
a finite support or it is Lebesgue measure.