Self-embeddings of Bedford-McMullen carpets

Let F ⊆ R² 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) ⊆ 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.