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

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8 Citations (Scopus)

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 randomise 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, M. Hochman's recent work on the dimensions of self-similar sets and measures.
Original languageEnglish
Pages (from-to)2463–2481
JournalErgodic Theory and Dynamical Systems
Volume 36
Issue number8
Early online date21 Jul 2015
DOIs
Publication statusPublished - Dec 2016

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