A dynamical variant of the Pink–Zilber conjecture

Abstract

Let $f_1,\dots ,f_n\in \overline{\mathbb{Q}}\left[x\right]$ be polynomials of degree $d>1$ such that no $f_i$ is conjugate to $x^d$ or to $\pm C_d\left(x\right)$, where $C_d\left(x\right)$ is the Chebyshev polynomial of degree $d$. We let $\phi$ be their coordinatewise action on ${\mathbb{A}}^n$, i.e., $\phi :{\mathbb{A}}^n\to {\mathbb{A}}^n$ is given by $\left(x_1,\dots ,x_n\right)\mapsto \left(f_1\left(x_1\right),\dots ,f_n\left(x_n\right)\right)$. We prove a dynamical version of the Pink–Zilber conjecture for subvarieties $V$ of ${\mathbb{A}}^n$ with respect to the dynamical system $\left({\mathbb{A}}^n,\phi \right)$, if $min\left\{dim\left(V\right),codim\left(V\right)-1\right\}\le 1$.