Mike Hochman [WS1]: Transverse orbits in dimension zero

Speaker
Mike Hochman, Hebrew University of Jerusalem

Abstract
Furstenberg’s transversality conjecture predicts that when a,b are multiplicatively independent integers and x is irrational, if {a^n x mod 1} has box dimension zero, then {b^n x mod 1} should be dense in [0,1]. Existing results about the conjecture show it is true for “typical” points inside positive dimensional sets (e.g. using the Shmerkin-Wu intersections theorem or Host’s equidistribution theorem), but do not give any information in dimension zero. I will discuss some new results on the problem, which say that in any infinite, zero-dimensional, minimal a-invariant set, and for any non-atomic zero-dimensional a-invariant measure, typical points have a dense b-orbit.