On the dynamics of a rational semigroup on a convolution measure algebra

TL;DR

This paper explores the dynamical behavior of a rational semigroup acting on convolution measure algebras over abelian compact groups, analyzing stability, invariance, and asymptotics, with applications to finite abelian groups.

Contribution

It introduces and studies the properties of a new rational semigroup on convolution measure algebras, linking it to the Choquet-Deny equation and providing a complete description for finite abelian groups.

Findings

Analysis of the semigroup's stability and invariant sets

Connection with the Choquet-Deny equation

Complete characterization for finite abelian groups

Abstract

We are going to study the dynamical properties of the rational semigroup $Q_{t}(\mu)$ where $Q_{t}(\mu)= (1-t) \mu * (1- t \mu)^{-1},$ for $t \in [0,1)$, that is defined for $\mu \in \mathcal{P}(G)$, the set of Borel probabilities over $(G, \cdot)$ an abelian compact topological group where we define the \textbf{convolution}, $\nu * \mu \in \mathcal{P}(G)$, as usual for a group $\int f d(\nu * \mu)= \int \int f(xy) d\nu(x) d\mu(y),$ then $(\mathcal{P}(G), *)$ became a $\textbf{convolution measure algebra}$ (CM-algebra). We investigate several properties for this semigroup (as the Stable Manifold Theorem, Asymptotic behavior, invariant sets, differential properties, stationary points, etc) and how they are related with the Choquet-Deny equation. As an application we give a complete description of this semigroup for finite abelian groups.