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Self-embeddings of Bedford–McMullen carpets

Published online by Cambridge University Press:  17 August 2017

AMIR ALGOM  and
MICHAEL HOCHMAN
AMIR ALGOM
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 9190401, Israel email amir.algom@mail.huji.ac.il, mhochman@math.huji.ac.il
MICHAEL HOCHMAN
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 9190401, Israel email amir.algom@mail.huji.ac.il, mhochman@math.huji.ac.il
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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 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.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 3 , March 2019 , pp. 577 - 603
Copyright
© Cambridge University Press, 2017 

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