Local height arguments toward the dynamical Mordell-Lang conjecture
We consider regular endomorphisms of the complex affine space with a degree gap $k$. They are endomorphisms $f$ of $\mathbb{A}_{\mathbb{C}}^{N}$ of the form $f(x_1,\dots,x_N)=(f_1(x_1,\dots,x_N)+g_1(x_1,\dots,x_N),\dots,f_N(x_1,\dots,x_N)+g_N(x_1,\dots,x_N))$, in which $f_1,\dots,f_N$ are homogeneous polynomials of degree $d$ with no nonzero common zeros and $g_1,\dots,g_N$ are polynomials of degree $\leq d-k$. Such an endomorphism extends to an endomorphism of $\mathbb{P}_{\mathbb{C}}^{N}$. Let $H_{\infty}=\mathbb{P}_{\mathbb{C}}^{N}\setminus\mathbb{A}_{\mathbb{C}}^{N}$ be the infinity hyperplane and we denote $f_{\infty}$ as the induced endomorphism of $H_{\infty}$. Suppose that $k$ is twice greater than the multiplicities of $f_{\infty}$ at the periodic closed points, i.e. $k>2\max\limits_{P\in\mathrm{Per}(f_\infty)}e_{f_{\infty}}(P)$. Then we prove that $f$ satisfies the dynamical Mordell-Lang conjecture for curves. As a by-product of our proof, we show that in this case every periodic curve of $f$ is a "vertical line", i.e. a straight line passing through the origin. There are many examples which satisfy our condition $k>2\max\limits_{P\in\mathrm{Per}(f_\infty)}e_{f_{\infty}}(P)$. Indeed, we prove that for every $d\geq2$, a general endomorphism $f_{\infty}$ of $H_{\infty}\cong\mathbb{P}_{\mathbb{C}}^{N-1}$ of degree $d$ satisfies $\max\limits_{P\in H_{\infty}(\mathbb{C})}e_{f_{\infty}}(P)\leq(N-1)!\cdot2^{N-1}$. So if we take $k=(N-1)!\cdot2^N+1$, then $f$ will satisfy our condition if $f_{\infty}$ is general (of an arbitrary degree $d\geq k$). Moreover, we provide examples to illustrate that this condition is optimal to force every periodic curve to be a vertical line, in the sense that one cannot change "$>$" into "$\geq$".