Jeremy Tyson
University of Illinois Urbana-Champaign & National Science Foundation, USA

Fraser and Yu (2018) introduced a one-parameter family of dimensional values $\dim_A^\theta(X)$, 0<$\theta$<1, which capture the quantitative and scale-invariant covering properties of a metric space $(X,d)$ with respect to a pair of geometrically related scales 0<$R$<1 and $r = R^{1/\theta}$. This family of dimensions, termed the Assouad spectrum, interpolates between the box-counting and Assouad dimensions. As an application, Fraser--Yu exhibited homeomorphic spaces for which standard notions of dimension (e.g. Hausdorff, box-counting, and Assouad dimension) coincide, but which have different Assouad spectra. Thus Assouad spectra can witness the bi-Lipschitz inequivalence of spaces in situations where classical dimensional notions are insufficient for this purpose. Assouad spectra have by now been computed or estimated for a wide variety of examples, including (both deterministic and random) self-affine fractals, Kleinian limit sets, and sets obtained via certain Diophantine constructions. We will discuss the distortion of Assouad spectra and Assouad dimension under mappings in various regularity classes, including Hölder, Sobolev, and quasiconformal mappings.