H\"older differentiability of self-conformal devil's staircases

In this paper we consider the probability distribution function of a Gibbs measure supported on a self-conformal set given by an iterated function system (devil's staircase). We use thermodynamic multifractal formalism to calculate the Hausdorff dimension of the sets $S^{\alpha}_{0}$, $S^{\alpha}_{\infty}$ and $S^{\alpha}$, the set of points at which this function has, respectively, H\"older derivative 0, $\infty$ or no derivative in the general sense. This extends recent work by Darst, Dekking, Falconer, Kesseb\"ohmer and Stratmann and Yao, Zhang and Li.Comment: 16 pages, 4 figure

On the Minkowski content of self-similar random homogeneous iterated function systems

The Minkowski content of a compact set is a fine measure of its geometric scaling. For Lebesgue null sets it measures the decay of the Lebesgue measure of epsilon neighbourhoods of the set. It is well known that self-similar sets, satisfying reasonable separation conditions and non-log comensurable contraction ratios, have a well-defined Minkowski content. When dropping the contraction conditions, the more general notion of average Minkowsk content still exists. For random recursive self-similar sets the Minkowski content also exists almost surely, whereas for random homogeneous self-similar sets it was recently shown by Z\"ahle that the Minkowski content exists in expectation. In this short note we show that the upper Minkowski content, as well as the upper average Minkowski content of random homogeneous self-similar sets is infinite, almost surely, answering a conjecture posed by Z\"ahle. Additionally, we show that in the random homogeneous equicontractive self-similar setting the lower Minkowski content is zero and the lower average Minkowski content is also infinite. These results are in stark contrast to the random recursive model or the mean behaviour of random homogeneous attractors.Comment: 15 page