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Assouad type dimensions and homogeneity of fractals

Jonathan Fraser
St. Andrews

Ergodic Theory and Dynamical Systems

Organisers: Andrew Ferguson, Thomas Jordan

ABSTRACT
I will introduce two notions of dimension which are perhaps less well-known than the Hausdorff and packing dimensions, namely the Assouad dimension and its natural dual, the lower dimension.  These dimensions play an important role in the theory of quasi-conformal mappings and embeddability problems, but I will be more interested in using them as a tool in the analysis of fractals.  As such I will review some of their basic properties, which are sometimes quite surprising, and go on to discuss some interesting examples.  The key examples will be the self-affine carpets of Lalley-Gatzouras and Baranski and we will compute the Assouad and lower dimensions for these classes and compare our results with the known results for Hausdorff and packing dimension.