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

TL;DR

This work advances the understanding of self-embedding mappings for Bedford–McMullen carpets by proving non-obliqueness for non-degenerate carpets even when the horizontal and vertical scales are dependent, and establishing a logarithmic commensurability between contraction ratios and the ambient grid. It extends prior results of Algom–Hochman to the dependent-ratio regime and, in the self-similar setting, shows oblique embeddings can occur but are severely restricted under a strong separation condition for rotational cases. The analysis combines slicing arguments, projections of fractal sets, and a logarithmic commensurability framework that leverages deleted-digit set structure and Cantor-type dynamics. The paper also constructs a generalized Sierpiński carpet with oblique self-embedding and provides a sharp constraint on oblique rotational embeddings, contributing to the broader understanding of how fractal geometry interacts with affine self-embeddings and symmetry.

Abstract

We prove that any non-degenerate Bedford-McMullen carpet does not allow 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. We also establish a logarithmic commensurability result on the contraction ratios of such embeddings. This completes a previous study of Algom and Hochman [Ergod. Th. & Dynam. Sys. 39 (2019), 577--603] on Bedford-McMullen carpets generated by multiplicatively independent exponents, together with a new proof on their non-obliqueness statement. For the self-similar case, however, we construct a generalized Sierpinski carpet that is symmetric with respect to an appropriate oblique line and hence allows a reflectional oblique self-embedding. As a complement, we prove that if a generalized Sierpinski carpet satisfies the strong separation condition and permits an oblique rotational self-embedding similitude, then the tangent of the rotation angle takes values $\pm 1$.

Figures (7)

• Figure 1: An illustration of the local behavior of Case 2
• Figure 2: The evolution from $R_k$ to $f(R_k)$ to $\varphi_{Q_k}^{-1}f(R_k)$ (color online)
• Figure 3: $\varphi_{Q_k}^{-1}f(\ell^k)+\xi_0$ is contained in the hole $U+e$ (color online)
• Figure 4: Length of horizontal slices of $\varphi_{Q_k}^{-1}f(R_k)$
• Figure 5: An illustration of $\widetilde{a},\widetilde{b},g(R_t)$ and $g(R'_t)$, where $(K^z+c)\times\mathbb R$ is supported in the shaded region. (color online)

Theorems & Definitions (48)

• Definition 1.1
• Definition 1.2: Obliqueness
• Theorem 1.3: AH19
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Proposition 2.1
• proof
• Proposition 2.2
• proof