A note on the dynamics of linear automorphisms of a measure convolution algebra

TL;DR

This paper investigates the dynamics of linear automorphisms in measure convolution algebras over finite groups, providing a direct method to analyze the convergence behavior of convolution powers using subgroup support.

Contribution

It introduces a novel, straightforward approach to understanding the asymptotic behavior of convolution powers in finite groups, extending classical results.

Findings

Convergence of convolution powers is characterized by the subgroup generated by the support.

Provides an alternative to classical convergence results.

Clarifies the asymptotic behavior of automorphisms in measure convolution algebras.

Abstract

In this work we are going to study the dynamics of the linear automorphisms of a measure convolution algebra over a finite group, $T(\mu)=\nu * \mu$. In order to understand an classify the asymptotic behavior of this dynamical system we provide an alternative to classical results, a very direct way to understand convergence of the sequence $\{\nu^{n}\}_{n\in\mathbb{N}}$, where $G$ is a finite group, $\nu\in\mathcal{P}(G)$ and $\nu^n=\underbrace{\nu\ast...\ast\nu}_n$, trough the subgroup generated by his support.