Bialgebraic geometry of Böttcher coordinates

Becker and Bergweiler showed that if $f$ is a non-exceptional polynomial, then the Böttcher coordinate ${\mathrm{\Psi }}_f\text{\hspace{0.17em}⁣}:{\mathbb{D}}_R\to B_{\mathrm{\infty }}(f)$ associated to $f$ is a transcendental function. In this paper, we study $f$-bialgebraic sets: algebraic subsets of ${\mathbb{D}}_R^n$ whose image under the coordinate-wise action of ${\mathrm{\Psi }}_f$ is contained in an algebraic set of the same dimension. We give a complete dynamical classification of bialgebraic sets under the additional assumption that the Julia set of $f$ is either disconnected, or connected and admits a nondegenerate locally connected model. Inspired by the Ax–Lindemann–Weierstrass theorem and the Ax–Schanuel conjecture, we formulate analogs with ${\mathrm{\Psi }}_f$ in place of the exponential function and prove them in the case where the Julia set $J_f$ is disconnected.