Dynamical anomalous subvarieties: Structure and bounded height theorems

According to Medvedev and Scanlon [14], a polynomial f(x)∈Q¯[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∈Q¯[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 Q¯. 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 [4]. 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 [8] which was previously conjectured in [4]. The slightly more general self-maps φ=f1×⋯×fn where each fi∈Q¯(x) is a disintegrated rational function are also treated at the end of the paper.