Intersection of orbits for polynomials in characteristic $p$

Abstract

In [GTZ08, GTZ12], the following result was established: given polynomials $f,g\in \mathbb{C}[x]$ of degrees larger than $1$, if there exist $\hat{I}\pm ,{\hat{I}}^2\in \mathbb{C}$ such that their corresponding orbits ${\mathcal{O}}_f(\hat{I}\pm )$ and ${\mathcal{O}}_g({\hat{I}}^2)$ (under the action of $f$, respectively of $g$) intersect in infinitely many points, then $f$ and $g$ must share a common iterate, i.e., $f^m=g^n$ for some $m,n\in \mathbb{N}$. If one replaces $\mathbb{C}$ with a field $K$ of characteristic $p$, then the conclusion fails; we provide numerous examples showing the complexity of the problem over a field of positive characteristic. We advance a modified conjecture regarding polynomials $f$ and $g$ which admit two orbits with infinite intersection over a field of characteristic $p$. Then we present various partial results, along with connections with another deep conjecture in the area, the dynamical Mordell-Lang conjecture.

13 pages, comments welcome