Dimension of pluriharmonic measure and polynomial endomorphisms of   $\C^n$
  Let $F$ be a polynomial endomorphism of $\C^n$ which extends holomorphically to $\P^n$. We prove that the dimension of $\mu_F$, the pluriharmonic measure on the boundary of the filled Julia set of $F$, is bounded above by $2n-1$. 
I. Binder and L. DeMarco