We study how the Hausdorff measure is distributed in nonsym-metric narrow cones in R n. As an application, we find an upper bound close to n − k for the Hausdorff dimension of sets with large k-porosity. With k-porous sets we mean sets which have holes in k different directions on every small scale.
We prove that generically, for a self-affine set in R d , removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension. This gives a partial positive answer to a folklore open question.
We expand the ergodic theory developed by Furstenberg and Hochman on dynamical systems that are obtained from magnifications of measures. We prove that any fractal distribution in the sense of Hochman is generated by a uniformly scaling measure, which provides a converse to a regularity theorem on the structure of distributions generated by the scenery… (More)