Abstract
Let $k$ be an algebraically closed field of characteristic $0$, let $N\in{\mathbb{N}}$, let $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ be a nonconstant morphism, and let $A:{\mathbb{A}}^N{\longrightarrow } {\mathbb{A}}^N$ be a linear transformation defined over $k({\mathbb{P}}^1)$, that is, for a Zariski-open dense subset $U\subset{\mathbb{P}}^1$, we have that for $x\in U(k)$, the specialization $A(x)$ is an $N$-by-$N$ matrix with entries in $k$. We let $f:{\mathbb{P}}^1\times{\mathbb{A}}^N{\dashrightarrow } {\mathbb{P}}^1\times{\mathbb{A}}^N$ be the rational endomorphism given by $(x,y)\mapsto (\,g(x), A(x)y)$. We prove that if $g$ induces an automorphism of ${\mathbb{A}}^1\subset{\mathbb{P}}^1$, then each irreducible curve $C\subset{\mathbb{A}}^1\times{\mathbb{A}}^N$ that intersects some orbit $\mathcal{O}_f(z)$ in infinitely many points must be periodic under the action of $f$. Furthermore, in the case $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ is an endomorphism of degree greater than $1$, then we prove that each irreducible subvariety $Y\subset{\mathbb{P}}^1\times{\mathbb{A}}^N$ intersecting an orbit $\mathcal{O}_f(z)$ in a Zariski dense set of points must be periodic. Our results provide the desired conclusion in the Dynamical Mordell–Lang Conjecture in a couple new instances. Moreover, our results have interesting consequences toward a conjecture of Rubel and toward a generalized Skolem–Mahler–Lech problem proposed by Wibmer in the context of difference equations. In the appendix it is shown that the results can also be used to construct Picard–Vessiot extensions in the ring of sequences.
Nearly ordinary Galois deformations over arbitrary number fields