Quasiregular distortion of dimensions

We investigate the distortion of the Assouad dimension and (regularized) spectrum of sets under planar quasiregular maps. The respective results for the Hausdorff and upper box-counting dimension follow immediately from their quasiconformal counterparts by employing elementary properties of these dimension notions (e.g. countable stability and Lipschitz stability). However, the Assouad dimension and spectrum do not share such properties. We obtain upper bounds on the Assouad dimension and spectrum of images of compact sets under holomorphic and planar quasiregular maps by studying their behavior around their critical points. As an application, the invariance of porosity of compact subsets of the plane under quasiregular maps is established.