Uniform disconnectedness and Quasi-Assouad Dimension

Abstract

The uniform disconnectedness is an important invariant property under bi-Lipschitz mapping, and the Assouad dimension ${\dim }_{A}X<1$ implies the uniform disconnectedness of $X$. According to quasi-Lipschitz mapping, we introduce the quasi-Assouad dimension ${\dim }_{qA}$ such that ${\dim }_{qA}X<1$ implies its quasi uniform disconnectedness. We obtain ${\overline{\dim }}_{B}X\le {\dim }_{qA}X\le {\dim }_{A}X$ and compute the quasi-Assouad dimension of Moran set.

References in corpus (2)

• Ahlfors-David regular sets and bilipschitz maps
• Assouad dimensions of Moran sets and Cantor-like sets