On the DML(1) property for regular endomorphisms of affine spaces: multiplicities of points at infinity

Abstract: Let $f$ be a regular endomorphism of $\mathbb{A}<em>{\mathbb{C}}<sup>N$ of algebraic degree $d$ and let $f</em>{\infty}$ be the induced endomorphism of the infinity hyperplane $H_{\infty}$. Suppose that for every periodic point $x_0\in H_{\infty}(\mathbb{C})$ of period $n_0$, the geometric mean of the multiplicities $$e_{f_{\infty}}(x_0),\dots,e_{f_{\infty}}(f_{\infty}<sup>{n_0-1}(x_0))$$ is strictly less than $d$. Then we show that the dynamical Mordell--Lang conjecture for $f$ holds for curves in $\mathbb{A}_{\mathbb{C}}<sup>N$.