Local entropy and projections of dynamically defined fractals

Local entropy and projections of dynamically defined fractals

-
Michael Hochman, Princeton University
Fine Hall 401

If a closed subset $X$ of the plane is projected orthogonally onto a line, then the Hausdorff dimension of the image is no larger than the dimension of $X$ (since the projection is Lipschitz), and also no larger than $1$ (since it is a subset of a line). A classical theorem of Marstrand says that for any such $X$, the projection onto almost every line has the maximal possible dimension given these constraints, i.e. is equal to $min(1,dim(X))$. In general, there can be uncountably many exceptional directions.An old conjecture of Furstenberg is that if $A, B$ are subsets of $[0,1]$ invariant respectively under $x2$ and $x3\mod 1$, then for their product, $X=A\times B$, the only exceptional directions in Marstrand's theorem are the two trivial ones, namely the projections onto the x and y axes. Recently, Y. Peres and P. Shmerkin proved that this is true for certain self-similar fractals, such as regular Cantor sets. I will discuss the proof of the general case, which relies on a method for computing dimension using local entropy estimates. I will also describe some other applications. This is joint work with Pablo Shmerkin.