Self embeddings of Bedford-McMullen carpets

TL;DR

This paper characterizes all similarities mapping Bedford-McMullen carpets into themselves, showing they are essentially axis-aligned isometries, by analyzing tangent sets, projections, and self-similarity properties.

Contribution

It proves that under certain conditions, self-embeddings of Bedford-McMullen carpets are restricted to axis-aligned isometries, advancing understanding of their geometric symmetries.

Findings

Any similarity mapping the carpet into itself is an axis-aligned isometry.

The structure of tangent sets and projection theorems are key to the proof.

Conditions on the carpet's structure restrict possible self-similarities.

Abstract

Let $F \subseteq \mathbb{R}^2$ be a Bedford-McMullen carpet defined by multiplicatively independent exponents, and suppose that either $F$ is not a product set, or it is a product set with marginals of dimension strictly between $0$ and $1$. We prove that any similarity $g$ such that $g(F) \subseteq F$ is an isometry composed of reflections about lines parallel to the axes. Our approach utilizes the structure of tangent sets of $F$, obtained by "zooming in" on points of $F$, projection theorems for products of self-similar sets, and logarithmic commensurability type results for self similar sets in the line.