Dynamical Anomalous Subvarieties: Structure and Bounded Height Theorems

According to Medvedev and Scanlon [MS14], a polynomial f(x) ∈ Q̄[x] of degree d ≥ 2 is called disintegrated if it is not conjugate to xd or to ±Cd(x) (where Cd is the Chebyshev polynomial of degree d). Let n ∈ N, let f1, . . . , fn ∈ Q̄[x] be disintegrated polynomials of degrees at least 2, and let φ = f1×· · ·×fn be the corresponding coordinate-wise self-map of (P1)n. Let X be an irreducible subvariety of (P1)n of dimension r defined over Q̄. We define the φ-anomalous locus of X which is related to the φ-periodic subvarieties of (P1)n. We prove that the φ-anomalous locus of X is Zariski closed; this is a dynamical analogue of a theorem of Bombieri, Masser, and Zannier [BMZ07]. We also prove that the points in the intersection of X with the union of all irreducible φ-periodic subvarieties of (P1)n of codimension r have bounded height outside the φ-anomalous locus of X; this is a dynamical analogue of Habegger’s theorem [Hab09a] which was previously conjectured in [BMZ07]. The slightly more general self-maps φ = f1 × · · · × fn where each fi ∈ Q̄(x) is a disintegrated rational function are also treated at the end of the paper.