Collision of orbits for families of polynomials defined over fields of positive characteristic

Let L be a field of positive characteristic p with a fixed algebraic closure L̄, and let α₁,α₂,β∈ L. For an integer d≥ 2, we consider the family of polynomials f_λ(z) := zᵈ+λ, parameterized by λ̄. Define C(α₁,α₂;β) to be the set of all λ̄ for which there exist m,n such that f_λᵐ(α₁)=f_λⁿ(α₂)=β. In other words, C(α₁,α₂;β) consists of all λ̄ with the property that the orbit of α₁ collides with the orbit of α₂ under the same polynomial f_λ precisely at the point $β$. Assuming α₁,α₂,β are not all contained in a finite subfield of L, we provide explicit necessary and sufficient conditions under which C(α₁,α₂;β) is infinite. We also discuss the remaining case where α₁,α₂,β∈ F̄ₚ and provide ample computational data that suggest a somewhat surprising conjecture. Our problem fits into a long series of questions in the area of unlikely intersections in arithmetic dynamics, which have been primarily studied over fields of characteristic $0$. Working in characteristic p adds significant difficulties, but also reveals the subtlety of our problem, especially when some of the points lie in a finite field or when d is a power of p.