Intersections of polynomial orbits, and a dynamical Mordell-Lang conjecture

TL;DR

This paper establishes conditions under which polynomial orbits intersect infinitely often, implying they share a common iterate, and proves a special case of a conjecture linking dynamics and the Mordell-Lang problem.

Contribution

It proves that nonlinear polynomials with infinite orbit intersections share a common iterate and verifies a special case of the dynamical Mordell-Lang conjecture.

Findings

Polynomials with infinite orbit intersection have a common iterate

A special case of the dynamical Mordell-Lang conjecture is proven

Results connect polynomial dynamics with number theory conjectures

Abstract

We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang conjecture.