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

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