Locally rich compact sets

We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have almost any other compact set of the cube as a tangent at all points or just in a dense sub-set. Here the "almost all compact sets" means that the tangent collection contains a contracted image of any compact set of the cube and that the contraction ratios are uniformly bounded. In the Euclidean space, the distance of sub-sets is measured by the Hausdorff distance. Also the geometric properties and dimensions of such spaces and sets are studied.Comment: 29 pages, 3 figures. Final versio

Weak separation condition, Assouad dimension, and Furstenberg homogeneity

We consider dimensional properties of limit sets of Moran constructions satisfying the finite clustering property. Just to name a few, such limit sets include self-conformal sets satisfying the weak separation condition and certain sub-self-affine sets. In addition to dimension results for the limit set, we manage to express the Assouad dimension of any closed subset of a self-conformal set by means of the Hausdorff dimension. As an interesting consequence of this, we show that a Furstenberg homogeneous self-similar set in the real line satisfies the weak separation condition. We also exhibit a self-similar set which satisfies the open set condition but fails to be Furstenberg homogeneous.Comment: 22 pages, 2 figure

Visible part of dominated self-affine sets in the plane

The dimension of the visible part of self-affine sets, that satisfy domination and a projection condition, is being studied. The main result is that the Assouad dimension of the visible part equals to 1 for all directions outside the set of limit directions of the cylinders of the self-affine set. The result holds regardless of the overlap of the cylinders. The sharpness of the result is also being discussed.Peer reviewe

Visible part of dominated self-affine sets in the plane

The dimension of the visible part of self-affine sets, that satisfy domination and a projection condition, is being studied. The main result is that the Assouad dimension of the visible part equals to 1 for all directions outside the set of limit directions of the cylinders of the self-affine set. The result holds regardless of the overlap of the cylinders. The sharpness of the result is also being discussed.Peer reviewe

Fractal Percolation and Quasisymmetric Mappings

We study the conformal dimension of fractal percolation and show that, almost surely, the conformal dimension of a fractal percolation is strictly smaller than its Hausdorff dimension.Peer reviewe

Fractal Percolation and Quasisymmetric Mappings

We study the conformal dimension of fractal percolation and show that, almost surely, the conformal dimension of a fractal percolation is strictly smaller than its Hausdorff dimension.Peer reviewe

On measures that improve L-q dimension under convolution

The L-q dimensions, for 1 <q <infinity, quantify the degree of smoothness of a measure. We study the following problem on the real line: when does the Lq dimension improve under convolution? This can be seen as a variant of the well-known L-p-improving property. Our main result asserts that uniformly perfect measures (which include Ahlfors-regular measures as a proper subset) have the property that convolving with them results in a strict increase of the L-q dimension. We also study the case q = infinity, which corresponds to the supremum of the Frostman exponents of the measure. We obtain consequences for repeated convolutions and for the box dimension of sumsets. Our results are derived from an inverse theorem for the L-q norms of convolutions due to the second author.Peer reviewe

Local dimensions of measures on infinitely generated self-affine sets

We show the existence of the local dimension of an invariant probability measure on an infinitely generated self-affine set, for almost all translations. This implies that an ergodic probability measure is exactly dimensional. Furthermore the local dimension equals the minimum of the local Lyapunov dimension and the dimension of the space.peerReviewe