Ignacio Garcia, Pure Math Department, University of Waterloo
“On generalized dimension of self similar sets with overlaps.”
Self similar sets, such as the ternary Cantor set or the Sierpinski gasket, are well understood from the Geometric Measure Theory viewpoint: its Hausdorff and packing dimension coincide with the similarity dimension, and natural measures for the set are the corresponding Hausdorff or packing measures. This is because the open set condition (OSC) holds for these sets, which roughly means that ’the parts of the set are separated’.
Weak separation condition (WSP) is a generalization of the OSC, where intuitively there may be overlaps in the construction of the set but in a limited number of ways. For self similar sets with this property, Hausdorff and packing measures are also natural measures for the sets, but the dimension here may not be the similarity dimension. In the absence of WSC, the usual Hausdorff or packing measures are no longer natural for the self similar set.
The talk will be focused on a class of self similar sets in the line which are given by iterated function systems that do not satisfy the WSC. These sets were constructed by T. Orponen (after the work of M. Hochman) and have zero packing measure in the corresponding dimension. For these sets it will be considered the Hausdorff and packing dimension in a generalized way (in the sense of dimension functions), where some natural questions arise in this context. A bound for the generalized dimension will be provided.