Bialgebraic geometry of Böttcher coordinates

Abstract

Becker and Bergweiler showed that if $f$ is a non-exceptional polynomial, then the Böttcher coordinate $Ψ_f \colon \mathbb D_R \to B_\infty(f)$ 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 $Ψ_f$ 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 $Ψ_f$ in place of the exponential function and prove them in the case where the Julia set $J_f$ is disconnected.