Let $f_1,\ldots,f_n\in\bar{\mathbb{Q}}[x]$ be polynomials of degree $d>1$ such that no $f_i$ is conjugate to $x^d$ or $\pm C_d(x)$ where $C_d(x)$ is the Chebyshev polynomial of degree $d$. Let $\varphi:\ \mathbb{A}^n\rightarrow \mathbb{A}^n$ be given by $\varphi(x_1,\ldots,x_n)=(f_1(x_1),\ldots,f_n(x_n))$. We prove a dynamical version of the Pink-Zilber conjecture for subvarieties $V$ of $\mathbb{A}^n$ with respect to the dynamical system $(\mathbb{A}^n,\varphi)$ when $\dim(V)\in\{1,n-1,n-2\}$. Our work relies on the Medvedev-Scanlon description of $\varphi$-periodic subvarieties and various results on the arithmetic dynamics of $(\mathbb{A}^n,\varphi)$. This is joint work with Dragos Ghioca.