Self-embedding similitudes of Bedford-McMullen carpets with dependent ratios
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913058439299072 |
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| author | Xiao, Jian-Ci |
| author_facet | Xiao, Jian-Ci |
| contents | 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 Sierpiński 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 Sierpiński carpet satisfies the strong separation condition and permits an oblique rotational self-embedding similitude, then the tangent of the rotation angle takes values $\pm 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02123 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-embedding similitudes of Bedford-McMullen carpets with dependent ratios Xiao, Jian-Ci Classical Analysis and ODEs Metric Geometry Primary 28A80, Secondary 28A78 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 Sierpiński 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 Sierpiński carpet satisfies the strong separation condition and permits an oblique rotational self-embedding similitude, then the tangent of the rotation angle takes values $\pm 1$. |
| title | Self-embedding similitudes of Bedford-McMullen carpets with dependent ratios |
| topic | Classical Analysis and ODEs Metric Geometry Primary 28A80, Secondary 28A78 |
| url | https://arxiv.org/abs/2412.02123 |