# On aperiodicity and hypercyclic weighted translation operators

## TL;DR

This paper characterizes aperiodicity in locally compact groups and explores its impact on the existence of hypercyclic weighted translation operators, demonstrating that aperiodic elements lead to chaotic operators with strong dynamical properties.

## Contribution

It provides equivalent characterizations of aperiodicity and shows that aperiodic elements enable the construction of mixing, chaotic, and frequently hypercyclic weighted translation operators.

## Key findings

- Aperiodicity of an element is characterized in multiple equivalent ways.
- Aperiodic elements guarantee the existence of hypercyclic weighted translation operators.
- Constructed operators exhibit mixing, chaos, and frequent hypercyclicity.

## Abstract

We give several equivalent characterization of aperiodicity of an element on locally compact group $G$, and give an intuition for "How strong does the aperiodicity of an element affect the existence of hypercyclic weighted translation operators?". In fact, if $a$ is an aperiodic element in $G$, then there exists a mixing, chaotic and frequently hypercyclic weighted translation $T_{a,w}$ on $L^p(G)$.