# Ultragraph shift spaces and chaos

## TL;DR

This paper explores chaos in ultragraph shift spaces, showing equivalences among different chaos notions and linking them to ultragraph combinatorics, with implications for the structure of scrambled sets.

## Contribution

It establishes the equivalence of various chaos concepts in ultragraph shift spaces and characterizes these conditions via ultragraph combinatorial properties.

## Key findings

- Li-Yorke, Devaney, and distributional chaos are equivalent in ultragraph shift spaces
- Such chaos implies the existence of a perfect, scrambled set
- Results extend understanding beyond labelled edge shifts of infinite graphs

## Abstract

Motivated by C*-algebra theory, ultragraph edge shift spaces generalize shifts of finite type to the infinite alphabet case. In this paper we study several notions of chaos for ultragraph shift spaces. More specifically, we show that Li-Yorke, Devaney and distributional chaos are equivalent conditions for ultragraph shift spaces, and characterize this condition in terms of a combinatorial property of the underlying ultragraph. Furthermore, we prove that such properties imply the existence of a compact, perfect set which is distributionally scrambled of type 1 in the ultragraph shift space (a result that is not known for a labelled edge shift (with the product topology) of an infinite graph).