Intersection of orbits for polynomials in characteristic p

In GTZ08, GTZ12, the following result was established: given polynomials f, g of degrees larger than 1, if there exist, such that their corresponding orbits Of () and Og () (under the action of f, respectively of g) intersect in infinitely many points, then f and g must share a common iterate, i. e. , fᵐ=gⁿ for some m, n. If one replaces 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.