Local structure of self-affine sets

TL;DR

This paper investigates the local structure of planar self-affine sets, revealing that near most points, these sets resemble products of an interval and a Cantor set, contrasting with self-similar sets.

Contribution

It demonstrates that self-affine sets locally resemble product sets of an interval and a Cantor set, highlighting a fundamental difference from self-similar sets.

Findings

Near almost every point, self-affine sets are approximated by product sets of an interval and a Cantor set.

Despite being totally disconnected, the local structure exhibits product-like behavior with interval fibers.

The results connect the local geometry of self-affine sets to attractors in dynamical systems.

Abstract

The structure of a self-similar set with open set condition does not change under magnification. For self-affine sets the situation is completely different. We consider planar self-affine Cantor sets E of the type studied by Bedford, McMullen, Gatzouras and Lalley, for which the projection onto the horizontal axis is an interval. We show that within small square neighborhoods of almost each point x in E, with respect to many product measures on address space, E is well approximated by product sets of an interval and a Cantor set. Even though E is totally disconnected, the limit sets have the product structure with interval fibres, reminiscent to the view of attractors of chaotic differentiable dynamical systems.