On the Assouad dimension of self-similar sets with overlaps

Abstract

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the weak separation property is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the weak separation property is not satisfied, in which case the Assouad dimension is maximal (equal to one). In the first case we prove that the self-similar set is Ahlfors regular, and in the second case we use the fact that if the weak separation property is not satisfied, one can approximate the identity arbitrarily well in the group generated by the similarity mappings, and this allows us to build a weak tangent that contains an interval. We also obtain results in higher dimensions and provide illustrative examples showing that the ‘equality/maximal’ dichotomy does not extend to this setting.