Dynamical anomalous subvarieties: Structure and bounded height theorems

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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 x d or to ± C d ( x ) (where C d is the Chebyshev polynomial of degree d ). Let n ∈ N , let f 1 , … , f n ∈ Q ¯ [ x ] be disintegrated polynomials of degrees at least 2, and let φ = f 1 × ⋯ × f n be the corresponding coordinate-wise self-map of ( P 1 ) n . Let X be an irreducible subvariety of ( P 1 ) n of dimension r defined over Q ¯ . We define the φ-anomalous locus of X which is related to the φ-periodic subvarieties of ( P 1 ) 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 ( P 1 ) 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 = f 1 × ⋯ × f n where each f i ∈ Q ¯ ( x ) is a disintegrated rational function are also treated at the end of the paper.