Quasi-Assouad dimension of fractals

Abstract

The Assouad dimension of fractals is not invariant under quasi-Lipschitz mappings, even for Ahlfors–David regular sets. In this manuscript, we shall give a new dimension dim${}_{qA}$ of fractals named the quasi-Assouad dimension, which is invariant under any quasi-Lipschitz mapping, satisfying ${\overline{\dim }}_{B}E\le {\dim }_{qA}E\le {\dim }_{A}E$ for any compact subset $E$ of a metric space. By virtue of the quasi-Assouad dimension, we show that any Bedford–McMullen carpet $F$ is quasi-Lipschitz Assouad-minimal, i.e., dim${}_{A}f(F)\ge {\dim }_{A}F$ for any quasi-Lipschitz mapping $f$.