On the dimensions of a family of overlapping self-affine carpets
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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.
Fraser , J M & Shmerkin , P 2016 , ' On the dimensions of a family of overlapping self-affine carpets ' Ergodic Theory and Dynamical Systems , vol 36 , no. 8 , pp. 2463–2481 . DOI: 10.1017/etds.2015.21
Ergodic Theory and Dynamical Systems
© 2015, Cambridge University Press. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at www.cambridge.org / https://doi.org/10.1017/etds.2015.21
The work of J.M.F. was supported by the EPSRC grant EP/J013560/1 whilst at Warwick and an EPSRC doctoral training grant whilst at St Andrews.
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