Intersection of orbits for polynomials in characteristic $p$

TL;DR

This paper explores the intersection properties of polynomial orbits over fields of characteristic p, revealing complexities absent in characteristic zero and proposing a modified conjecture with partial results and connections to the dynamical Mordell-Lang conjecture.

Contribution

It introduces a modified conjecture for polynomial orbit intersections in characteristic p and provides partial results and examples illustrating the problem's complexity.

Findings

Counterexamples to characteristic zero results in characteristic p

Proposal of a new conjecture for infinite orbit intersections

Connections established with the dynamical Mordell-Lang conjecture

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 $\alpha,\beta\in\mathbb{C}$ such that their corresponding orbits $\mathcal{O}_f(\alpha)$ and $\mathcal{O}_g(\beta)$ (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.