## EVENT DETAILS AND ABSTRACT

**Dynamical Systems Seminar**
**Title:** Local entropy averages and projection of fractal measures

**Speaker:** Michael Hochman

**Speaker Info:** Princeton and IAS

**Brief Description:**

**Special Note**:

**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.

**Date:** Tuesday, November 24, 2009

**Time:** 3:00pm

**Where:** Lunt 105

**Contact Person:** Prof. Dave McClendon

**Contact email:** dmm@math.northwestern.edu

**Contact Phone:** 847-467-1928

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