TY - JOUR
T1 - Self-embeddings of Bedford-McMullen carpets
AU - Algom, Amir
AU - Hochman, Michael
N1 - Publisher Copyright:
© 2017 Cambridge University Press.
PY - 2019/3/1
Y1 - 2019/3/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85060890782&partnerID=8YFLogxK
U2 - 10.1017/etds.2017.46
DO - 10.1017/etds.2017.46
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AN - SCOPUS:85060890782
SN - 0143-3857
VL - 39
SP - 577
EP - 603
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
IS - 3
ER -