Self-affine sets with fibered tangents

TL;DR

This paper investigates the tangent sets of self-affine fractals in the plane, showing that under certain conditions, tangent sets are either fibered with porous sets or are line segments, revealing their local geometric structure.

Contribution

It characterizes the local tangent structures of self-affine sets with the strong separation condition, identifying conditions under which tangent sets are fibered or linear.

Findings

Tangent sets are either fibered with porous sets or line segments.

Results apply to self-affine sets satisfying the strong separation condition.

Provides a geometric classification of tangent sets at generic points.

Abstract

We study tangent sets of strictly self-affine sets in the plane. If a set in this class satisfies the strong separation condition and projects to a line segment for sufficiently many directions, then for each generic point there exists a rotation $\mathcal O$ such that all tangent sets at that point are either of the form $\mathcal O((\mathbb R \times C) \cap B(0,1))$, where $C$ is a closed porous set, or of the form $\mathcal O((\ell \times \{ 0 \}) \cap B(0,1))$, where $\ell$ is an interval.