The amenability of the substitution group of formal power series

We study the amenability property for the group $\mathcal{J}(\mathbf{k})$ of formal power series in one variable with coefficients in a commutative ring $\mathbf{k}$ with identity. We show that there exists an invariant mean on the space $C_{\mathrm{u}}^*(\mathcal{J}(\mathbf{k}))$ of uniformly continuous bounded functions on this group. This is equivalent to the fact that every continuous action of $\mathcal{J}(\mathbf{k})$ on every compact space has an invariant probability measure.