Algebraic Relations amongst Periodic Points of a Lifting of Frobenius

Let p be a prime number and f ∈ Zp[x] a polynomial over the p-adic integers lifting the Frobenius in the sense that f(x) ≡ xp (mod p) and deg(f) = p. Let Πf := {ζ ∈ Q p : f◦n(ζ) = ζ for some n ∈ Z+} be the set of f -periodic points. We show that an irreducible algebraic varieties X ⊆ Am which meet Γ×m f in a Zariski dense set only in the case that X is defined by equations of the form xi = a for some f -periodic point a and xi = f ◦n(xj) for some natural number n.