Self-embedding similitudes of Bedford-McMullen carpets with dependent ratios

Abstract

We prove that any non-degenerate Bedford-McMullen carpet does not admit oblique self-embedding similitudes; that is, if $f$ is a similitude sending the carpet into itself, then the image of the $x$-axis under $f$ must be parallel to one of the principal axes. This result leads to a logarithmic commensurability result on the contraction ratios of such embeddings, completing a previous study by Algom and Hochman [Ergod. Th. & Dynam. Sys. 39 (2019), 577-603] on Bedford-McMullen carpets generated by multiplicatively independent exponents. Our approach also provides a new proof of their non-obliqueness statement that avoids analyzing the tangent sets. For the self-similar case, however, we construct a generalized Sierpi\'nski carpet that is symmetric with respect to an appropriate oblique line and hence admits a reflectional oblique self-embedding. As a complement, we prove that if a generalized Sierpi\'nski carpet satisfies the strong separation condition and permits an oblique rotational self-embedding similitude, then the tangent of the rotation angle takes values $\pm 1$.