Graph maps with zero topological entropy and sequence entropy pairs

TL;DR

This paper characterizes when graph maps with zero topological entropy exhibit Li-Yorke chaos, linking non-separable pairs, IN-pairs, and IT-pairs within solenoidal omega-limit sets, thus completing the understanding of sequence entropy in this context.

Contribution

It provides a complete characterization of zero topological sequence entropy for graph maps, connecting various types of pairs and chaos.

Findings

Graph maps with zero topological entropy are Li-Yorke chaotic iff they have an NS-pair.

Non-diagonal pairs are equivalent to NS-pairs, IN-pairs, and IT-pairs in this setting.

The characterization completes the understanding of sequence entropy for graph maps.

Abstract

We show that graph map with zero topological entropy is Li-Yorke chaotic if and only if it has an NS-pair (a pair of non-separable points containing in a same solenoidal $\omega$-limit set), and a non-diagonal pair is an NS-pair if and only if it is an IN-pair if and only if it is an IT-pair. This completes characterization of zero topological sequence entropy for graph maps.