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On the dimensions of a family of overlapping self-affine carpets

Published online by Cambridge University Press:  21 July 2015

JONATHAN M. FRASER  and
PABLO SHMERKIN
JONATHAN M. FRASER
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK email [email protected]
PABLO SHMERKIN
Affiliation:
Department of Mathematics and Statistics, Torcuato Di Tella University, Av. Figueroa Alcorta 7350, Buenos Aires, Argentina email [email protected]
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Abstract

We consider the dimensions of a family of self-affine sets related to the Bedford–McMullen carpets. In particular, we fix a Bedford–McMullen system and then randomize the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford–McMullen set-up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results rely on, and can be seen as a partial extension of, Hochman’s recent work on the dimensions of self-similar sets and measures.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 8 , December 2016 , pp. 2463 - 2481
Copyright
© Cambridge University Press, 2015 

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