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arXiv · 1607.07864

Self embeddings of Bedford-McMullen carpets

Amir Algom·Michael Hochman
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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 $0$ and $1$. 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.

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Amir Algom, Michael Hochman. 2016-07-26. Self embeddings of Bedford-McMullen carpets. https://arxiv.org/abs/1607.07864

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