Information, Geometry, and Physics Seminar

Furstenberg's x2x3 conjecture is an important conjecture in ergodic theory with connections to number-theoretic dynamics. It concerns probability measures on the unit circle which have the property that integrating the x2 or x3 speedup of any function is equivalent to integrating the original function. The conjecture states that every such measure that is also ergodic is either the Lebesgue measure or finitely supported. In this talk, we present joint work with Peter Burton in which we reformulate Furstenberg's conjecture in two alternate settings. The first is a complex analytic setting in which the measures correspond to Carathéodory functions and the second is an operator setting in which the measures correspond to tracial states on a certain group C* algebra.