An Improved Fractal Analysis of Directional Densities for Borel Probability Measures

We investigate pointwise rectangular anisotropic and directional densities for Borel probability measures with compact support. Contrary to traditional densities, these new notions have the advantage to weight the side lengths of local boxes differently and take into account long thin boxes. We employ similarities with statistical physics to extract interesting multifractal information using the Hausdorff dimension with respect to the coverings by rectangles with small enough Lebesgue measure (instead of requiring all side lengths to be small). We apply our analysis for invariant measures for some iterated function systems.