# Bounded height theorems in arithmetic dynamics

Let $r$ be a positive integer, and let $f_1,..,f_r\in \bar{\mathbb{Q}}[x]$ be polynomials of degrees larger than $1$ such that no $f_i$ is conjugated to a monomial $x^d$ or to $\pm C_d(x)$ (where $C_d(x)$ is the $d$-th Chebyshev polynomial). We denote by $\Phi$ the coordinatewise action of the polynomials $f_1,\dots, f_r$ on $\left(\mathbb{P}^1\right)^r$.

Let $X$ be an irreducible subvariety of $\left(\mathbb{P}^1\right)^r$ of dimension $d$ defined over $\bar{\mathbb{Q}}$. We define the $\Phi$-anomalous locus of $X$ which is related to the $\Phi$-periodic subvarieties of $\left(\mathbb{P}^1\right)^r$. We prove that the $\Phi$-anomalous locus of $X$ is Zariski closed, which is a dynamical analogue of a theorem of Bombieri, Masser, and Zannier. We also prove that the points in the intersection of $X$ with the union of all irreducible $\Phi$-periodic subvarieties of $\left(\mathbb{P}^1\right)^r$ of codimension $d$ have bounded height outside the $\Phi$-anomalous locus of $X$, which is a dynamical analogue of a theorem of Habegger.

This is joint work with Khoa Nguyen.