Self-embeddings of Bedford–McMullen carpets

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 zero and one. 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.