Let F ⊆ R2 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) ⊆ 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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Finally, in the case f2 = 0,1, the same analysis applies. The only difference is that (19) implies that b′′ agrees on a growing number of digits with the (unique) expansion of f2. From here the argument is the same, but without taking mod 1. □ Acknowledgements. This research was conducted as part of the first author’s PhD studies at the Hebrew University of Jerusalem. The first author would like to thank Shai Evra for many helpful discussions. The authors are grateful for the hospitality and support received from The Institute for Computational and Experimental Research in Mathematics (ICERM) as part of the Spring 2016 program on dimension and dynamics. Supported by European Research Council grant 306494.
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