Locally rich compact sets

Abstract

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.

29 pages, 3 figures. Final version

References in corpus (2)

• Dynamics of the scenery flow and geometry of measures
• Structure of distributions generated by the scenery flow

Cited by in corpus (4)

• On the Hausdorff dimension of microsets
• Self-affine sets with fibered tangents
• On the Assouad dimension and convergence of metric spaces
• Intersection properties of typical compact sets