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Self-embeddings of Bedford–McMullen carpets



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 zero and one. 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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[1] Bandt, C. and Käenmäki, A.. Local structure of self-affine sets. Ergod. Th. & Dynam. Sys. 33(05) (2013), 13261337.
[2] Bishop, C. J. and Peres, Y.. Fractals in Probability and Analysis (Cambridge Studies in Advanced Mathematics, 162) . Cambridge University Press, Cambridge, UK, 2016.
[3] Elekes, M., Keleti, T. and Máthé, A.. Self-similar and self-affine sets: measure of the intersection of two copies. Ergod. Th. & Dynam. Sys. 30(2) (2010), 399440.
[4] Feng, D.-J., Huang, W. and Rao, H.. Affine embeddings and intersections of Cantor sets. J. Math. Pures Appl. (9) 102(6) (2014), 10621079.
[5] Feng, D.-J. and Wang, Y.. On the structures of generating iterated function systems of Cantor sets. Adv. Math. 222(6) (2009), 19641981.
[6] Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation. Math. Systems Theory 1(1) (1967), 149.
[7] Hochman, M.. Geometric rigidity of × m invariant measures. J. Eur. Math. Soc. (JEMS) 14(5) (2012), 15391563.
[8] Hochman, M. and Shmerkin, P.. Local entropy averages and projections of fractal measures. Ann. of Math. (2) 175(3) (2012), 10011059.
[9] Hochman, M. and Shmerkin, P.. Equidistribution from fractal measures. Invent. Math. 202(1) (2015), 427479.
[10] Hutchinson, J. E.. Fractals and Self Similarity. Department of Mathematics, University of Melbourne, 1979.
[11] Käenmäki, A., Koivusalo, H. and Rossi, E.. Self-affine sets with fibered tangents. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2015.130. Published online 28 January 2016.
[12] Käenmäki, A., Ojala, T. and Rossi, E.. Rigidity of quasisymmetric mappings on self-affine carpets. Int. Math. Res. Not. (2016), to appear.
[13] Peres, Y.. The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure. Math. Proc. Cambridge Philos. Soc. 116(11) (1994), 513526.
[14] Peres, Y. and Shmerkin, P.. Resonance between Cantor sets. Ergod. Th. & Dynam. Sys. 29(1) (2009), 201221.

Self-embeddings of Bedford–McMullen carpets



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