Consider a homogeneous polynomial $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ defined by
$$ f(x) = \sum_{|\alpha| = k} b_{\alpha} x^\alpha = \sum_{\alpha_1+\ldots+\alpha_n = k} b_{\alpha} \prod_{j=1}^n x_j^{\alpha_j} $$
where $\alpha \in \mathbb{N}^n$ is a multi-index and $b_{\alpha} \in \mathbb{R}^n$. I am studying the fixed points of $f$ that lie on the standard simplex in $\mathbb{R}^n$, so $x$ is a categorical distribution on $n$ possible outcomes.
Do these fixed points $x = f(x)$ have a closed-form expression? Is there a unique fixed point? Can any properties of fixed points be determined, such as their entropy?