Tangents of invariant sets

Abstract

We study the fine scaling properties of sets satisfying various weak forms of invariance. For general attractors of possibly overlapping bi-Lipschitz iterated function systems, we establish that the Assouad dimension is given by the Hausdorff dimension of a tangent at some point in the attractor. Under the additional assumption of self-conformality, we moreover prove that this property holds for a subset of full Hausdorff dimension.

19 pages. v2: This is approximately the first 1/3 of v1. The remainder of v1 will appear as two forthcoming preprints