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

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.