Multivariate Böttcher equation for polynomials with non-negative coefficients

Summary.

We give conditions for the multivariate Böttcher equation $\beta(f (x)) = \beta(x)^{\lambda}$ to have a solution, in the case where $f : \mathbb{R}^d \rightarrow \mathbb{R}^d$ is a polynomial with non-negative coefficients. The solution is constructed from the limit of the functional iterates $-\lambda^{-n} \log f^{n}(x)$.