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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Assouad dimension of planar self-affine sets


Authors: Balázs Bárány, Antti Käenmäki and Eino Rossi
Journal: Trans. Amer. Math. Soc. 374 (2021), 1297-1326
MSC (2010): Primary 28A80; Secondary 37C45, 37L30
DOI: https://doi.org/10.1090/tran/8224
Published electronically: November 12, 2020
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Abstract: We calculate the Assouad dimension of a planar self-affine set $ X$ satisfying the strong separation condition and the projection condition and show that $ X$ is minimal for the conformal Assouad dimension. Furthermore, we see that such a self-affine set $ X$ adheres to very strong tangential regularity by showing that any two points of $ X$, which are generic with respect to a self-affine measure having simple Lyapunov spectrum, share the same collection of tangent sets.


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Additional Information

Balázs Bárány
Affiliation: Department of Stochastics, MTA-BME Stochastics Research Group, Budapest University of Technology and Economics, P.O. Box 91, 1521 Budapest, Hungary
Email: balubsheep@gmail.com

Antti Käenmäki
Affiliation: Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
Email: antti.kaenmaki@uef.fi

Eino Rossi
Affiliation: Department of Mathematics and Statistics, P.O. Box 68 (Pietari Kalmin katu 5), FI-00014 University of Helsinki, Finland
Email: eino.rossi@gmail.com

DOI: https://doi.org/10.1090/tran/8224
Keywords: Self-affine set, tangent set, Assouad dimension, conformal dimension
Received by editor(s): June 26, 2019
Received by editor(s) in revised form: March 5, 2020, and June 19, 2020
Published electronically: November 12, 2020
Additional Notes: The first author acknowledges support from the grants OTKA K123782, NKFI PD123970, the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The second author thanks the Academy of Finland (project no. 286877) for financial support. The third author was funded by the University of Helsinki via the project “Quantitative rectifiability of sets and measures in Euclidean spaces and Heisenberg groups” (project no. 7516125).
Article copyright: © Copyright 2020 American Mathematical Society