The dynamical Ax–Schanuel conjecture for Böttcher coordinates

Let $f$ be a non-exceptional polynomial of degree $d\ge 2$. Let $Z\subset \mathbb{D}_R^n\times U_\infty(f)^n$ be an analytic subset of dimension $0\le m\le n$ defined by the parametrization

$$\\{(h_0(x), \dots,h_{n-1}(x), \Psi_f(h_0(x)), \dots, \Psi_f(h_{n-1}(x))): x \in D \subset \mathbb{C}^m\\}.$$

Here $D$ is a non-empty open subset of $\mathbb{C}^m$ for some $m\ge 1$, and $h_0,\dots,h_{n-1}$ are holomorphic functions on $D$. Suppose that

$$(\Psi_f(h_0(x)), \dots,\Psi_f(h_{n-1}(x)))$$

is not contained in a proper $f$-special subvariety of $\mathbb{C}^n$. The dynamical Ax–Schanuel conjecture. One has

$$\dim_{\mathbb{C}}\left(\overline{Z}^{\operatorname{Zar}}\right)\ge m+n.$$

This is proposed by analogy with the Ax–Schanuel theorem and gives a transcendence-degree lower bound for parametrized analytic sets of Böttcher coordinates; the source gives no resolution status.