Geometry, Topology and Dynamics Seminar
Mike Hochman
Princeton
Local entropy averages and projection of fractal measures
Abstract: If X is a compact set in the plane then, by a classical theorem of Marstrand, almost every projection onto a line maps X to a set of the maximal possible Hausdorff dimension,
i.e. the smaller of dim(X) and 1. While in general the set of exceptional direction can be large, in certain situations arising from dynamical, arithmetic or combinatorial contexts,
it is predicted that there should be either no exceptions, or some small explicit set of exceptions. One example of this is an old conjecture of Furstenberg's, predicting that,
if X=A\times B, and A,B are, respectively, subsets of the unit interval invariant under times-2 mod 1 and times-3 mod 1, then the image of X under projection should behave
in this manner for every (not just almost every) projection, the only exceptions being the coordinate projections.
I will explain the background of this problems and my recent work with Pablo Shmerkin in which we resolve this conjecture positively. If time permits I will describe some other
applications of our methods.
Monday November 23, 2009 at 3:00 PM in SEO 612