UBC Ergodic Theory and Dynamical Systems Seminar: Doug Lind

Abstract: In the classical setting of x2 and x3 acting on the circle group, the original (and unsolved) Furstenberg Conjecture is that the only continuous probability measure simultaneously invariant under both is Lebesgue measure. A stronger conjecture, attributed to Furstenberg, is that if a continuous measure is x2 invariant, then its iterates under x3 converge weak* to Lebesgue measure. Recently Badea and Grivaux used Baire category arguments together with the weak* density of periodic point measures and Fourier coefficients to prove the existence of counterexamples to this conjecture and more general versions. 
In joint work with Mike Boyle, we first give a streamlined account of the Badea-Grivaux proof the classical setting, and indicate the crucial role that periodic point measures play in the argument. By isolating the Baire Category part, we can avoid the use of Fourier coefficients. Recent work of Yaari introduces a new and more flexible form of specification to create periodic points that should be sufficient to extend our work to a very general setting.