ASSOUAD-MINIMALITY OF MORAN SETS UNDER QUASI-LIPSCHITZ MAPPINGS

Abstract

The quasi-Lipschitz mappings, weaker than the bi-Lipschitz mappings, preserve Hausdorff, packing and box dimensions, but change Assouad dimension [Formula: see text]. In this paper, for Moran fractals, we investigate the change of their Assouad dimension under the quasi-Lipschitz mappings. We study a class of Moran set which is quasi-Lipschitz Assouad-minimal, i.e. for any [Formula: see text] in the class, [Formula: see text] for all quasi-Lipschitz mappings [Formula: see text] defined on [Formula: see text]. For another class of Moran sets, we prove that for any [Formula: see text] in the class, [Formula: see text] where the infimum is taken over all quasi-Lipschitz mappings [Formula: see text] defined on [Formula: see text], and [Formula: see text] is the quasi-Assouad dimension introduced in [F. Lü and L. F. Xi, Quasi-Assouad dimension of fractals, J. Fractal Geom. 3 (2016) 187–215].