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 $$\overline{\dim } _{B}X\leq \dim _{qA}X\leq \dim _{A}X$$ and compute the quasi-Assouad dimension of Moran set.