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

Abstract

We are going to study the dynamical properties of the rational semigroup $Q_t(\mu )$ where $Q_t(\mu )=(1-t)\mu \ast (1-t\mu {)}^{-1},$ 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}, $\nu \ast \mu \in \mathcal{P}(G)$, as usual for a group $\int fd(\nu \ast \mu )=\int \int f(xy)d\nu (x)d\mu (y),$ then $(\mathcal{P}(G),\ast )$ became a $\text{convolution measure algebra}$ (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.

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