Tangents of invariant sets

TL;DR

This paper investigates the local geometric structure of invariant sets, showing that for certain attractors, the Assouad dimension equals the Hausdorff dimension of a tangent, especially in self-conformal cases.

Contribution

It establishes a link between Assouad and Hausdorff dimensions of tangent sets for a broad class of invariant sets, including overlapping iterated function systems.

Findings

Assouad dimension equals tangent Hausdorff dimension for general attractors.

Self-conformal attractors have full Hausdorff dimension of tangent sets.

Results apply to overlapping bi-Lipschitz iterated function systems.

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.