Rigidity of invariant measures under the action of a multiplicative semigroup of positive logarithmic density on T
by
Alexander Fish
The Ohio State University
Coauthors: Manfred Einsiedler (The Ohio State University)

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.

Date received: April 15, 2008