Lower bound for the maximal number of facets of a 0/1 polytope

We show that there exist 0/1 polytopes in R^n with as many as (cn / (log n)^2)^(n/2) facets (or more), where c>0 is an absolute constant.Comment: 19 page

Non-uniqueness of ergodic measures with full Hausdorff dimension on a Gatzouras-Lalley carpet

In this note, we show that on certain Gatzouras-Lalley carpet, there exist more than one ergodic measures with full Hausdorff dimension. This gives a negative answer to a conjecture of Gatzouras and Peres

Curvature-direction measures of self-similar sets

We obtain fractal Lipschitz-Killing curvature-direction measures for a large class of self-similar sets F in R^d. Such measures jointly describe the distribution of normal vectors and localize curvature by analogues of the higher order mean curvatures of differentiable submanifolds. They decouple as independent products of the unit Hausdorff measure on F and a self-similar fibre measure on the sphere, which can be computed by an integral formula. The corresponding local density approach uses an ergodic dynamical system formed by extending the code space shift by a subgroup of the orthogonal group. We then give a remarkably simple proof for the resulting measure version under minimal assumptions.Comment: 17 pages, 2 figures. Update for author's name chang

The Hausdorff and dynamical dimensions of self-affine sponges : a dimension gap result

We construct a self-affine sponge in R 3 whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge, implying that sponges with a dimension gap represent a nonempty open subset of the parameter space

Multifractal tubes

Tube formulas refer to the study of volumes of $r$ neighbourhoods of sets. For sets satisfying some (possible very weak) convexity conditions, this has a long history. However, within the past 20 years Lapidus has initiated and pioneered a systematic study of tube formulas for fractal sets. Following this, it is natural to ask to what extend it is possible to develop a theory of multifractal tube formulas for multifractal measures. In this paper we propose and develop a framework for such a theory. Firstly, we define multifractal tube formulas and, more generally, multifractal tube measures for general multifractal measures. Secondly, we introduce and develop two approaches for analysing these concepts for self-similar multifractal measures, namely: (1) Multifractal tubes of self-similar measures and renewal theory. Using techniques from renewal theory we give a complete description of the asymptotic behaviour of the multifractal tube formulas and tube measures of self-similar measures satisfying the Open Set Condition. (2) Multifractal tubes of self-similar measures and zeta-functions. Unfortunately, renewal theory techniques do not yield "explicit" expressions for the functions describing the asymptotic behaviour of the multifractal tube formulas and tube measures of self-similar measures. This is clearly undesirable. For this reason, we introduce and develop a second framework for studying multifractal tube formulas of self-similar measures. This approach is based on multifractal zeta-functions and allow us obtain "explicit" expressions for the multifractal tube formulas of self-similar measures, namely, using the Mellin transform and the residue theorem, we are able to express the multifractal tube formulas as sums involving the residues of the zeta-function.Comment: 122 page