It is an open problem to determine for which maps $f$, any compact invariant set $K$ carries an ergodic invariant measure of the same Hausdorff dimension as $K$. If $f$ is conformal and expanding, then it is a known consequence of the thermodynamic formalism that such measures do exist. (We give a proof here under minimal smoothness assumptions.) If $f$ has the form $f(x_1,x_2)=(f_1(x_1),f_2(x_2))$, where $f_1$ and $f_2$ are conformal and expanding maps satisfying $\inf \vert Df_1\vert\geq\sup\vert Df_2\vert$, then for a large class of invariant sets $K$, we show that ergodic invariant measures of dimension arbitrarily close to the dimension of $K$ do exist. The proof is based on approximating $K$ by self-affine sets.

Let µ be an even compactly supported Borel probability measure on the real line. For every N > n consider N independent random vectors X 1 , . . . , X N in R n , with independent coordinates having distribution µ. We establish a sharp threshold for the volume of the random polytope K N := conv X 1 , . . . , X N , provided that the Legendre transform λ of the cumulant generating function of µ satisfies the conditionwhere α is the right endpoint of the support of µ. The method and the result generalize work of Dyer, Füredi and McDiarmid on 0/1 polytopes. We verify ( * ) for a large class of distributions.

No abstract

We formulate and verify an almost-sure lattice renewal theorem for branching random walks, whose non-lattice analogue is originally due to Nerman. We also identify the limit in these renewal theorems (both lattice and non-lattice) as the limit of Kingman's well-known martingale multiplied by a deterministic factor.

Let f n−1 (P) denote the number of facets of a polytope P in R n . We show that there exist 0/1 polytopes P with f n−1 (P) ≥ cn log 2 n n/2 , where c > 0 is an absolute constant. This improves earlier work of Bárány and Pór on a question of Fukuda and Ziegler.