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

We are going to study the dynamical properties of the rational semigroup $Q_{t}(μ)$ where $Q_{t}(μ)= (1-t) μ* (1- t μ)^{-1},$ for $t \in [0,1)$, that is defined for $μ\in \mathcal{P}(G)$, the set of Borel probabilities over $(G, \cdot)$ an abelian compact topological group where we define the \textbf{convolution}, $ν* μ\in \mathcal{P}(G)$, as usual for a group $\int f d(ν* μ)= \int \int f(xy) dν(x) dμ(y),$ 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.