Uniform disconnectedness and Quasi-Assouad Dimension

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

On multi-transitivity with respect to a vector

A topological dynamical system $(X,f)$ is said to be multi-transitive if for every $n\in\mathbb{N}$ the system $(X^{n}, f\times f^{2}\times \dotsb\times f^{n})$ is transitive. We introduce the concept of multi-transitivity with respect to a vector and show that multi-transitivity can be characterized by the hitting time sets of open sets, answering a question proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates, Erg. Th. Dynam. Syst., 32 (2012), 1661--1672]. We also show that multi-transitive systems are Li-Yorke chaotic.Comment: 11 page