TY - JOUR
T1 - On the dimensions of a family of overlapping self-affine carpets
AU - Fraser, Jonathan M.
AU - Shmerkin, Pablo
PY - 2016/12/1
Y1 - 2016/12/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84937598501&partnerID=8YFLogxK
U2 - 10.1017/etds.2015.21
DO - 10.1017/etds.2015.21
M3 - Article
AN - SCOPUS:84937598501
SN - 0143-3857
VL - 36
SP - 2463
EP - 2481
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
IS - 8
ER -