Dimension theory and classification of Assouad spectra through homogeneous Moran sets

07.07.2022 15:15 - 17:00

Alex Rutar (University of St Andrews)

Homogeneous Moran sets are a particular family of Borel sets which generalize the usual middle-1/3 Cantor set. Similar to the usual Cantor set, these sets have the maximum amount of homogeneity at a fixed scale, but in contrast to the usual Cantor set, the construction allows inhomogeneity between different scales. In the first part of the talk, I will review some fundamentals of dimension theory using homogeneous Moran sets as a guiding example. The properties of homogeneous Moran sets allow convenient expressions for many of the usual notions of dimension, such as the Hausdorff, box, and Assouad dimensions. A particular focus will be given to a modified version of the Assouad dimension called the Assouad spectrum. Then, in the second part of the talk, I will discuss a new classification result for the possible forms of Assouad spectra. In particular, we will see that homogeneous Moran sets provide a sufficiently rich family to witness all possible behaviours of the Assouad spectrum.

Organiser:

H. Bruin, R. Zweimüller

Location:

BZ 9, 9. OG, OMP1