On multi-transitivity with respect to a vector

TL;DR

This paper introduces a new concept of multi-transitivity with respect to a vector in topological dynamical systems, characterizes it via hitting time sets, and establishes its connection to Li-Yorke chaos.

Contribution

It defines multi-transitivity with respect to a vector, characterizes it through hitting time sets, and links it to Li-Yorke chaos, answering a previously open question.

Findings

Multi-transitivity can be characterized by hitting time sets.

Multi-transitive systems are Li-Yorke chaotic.

The concept generalizes existing notions of transitivity.

Abstract

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.