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Preprint Number 165
165. Alice Medvedev and Thomas Scanlon
Submission date: 23 January 2009.
Abstract: We study algebraic dynamical systems (and, more generally, σ-varieties) given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials. Our main result is an explicit description of the skew-invariant varieties, that is, for a field automorphism σ of the complex numbers we describe those affine algebraic varieties which are mapped to their σ-transforms. In particular, taking σ to be the identity function, we characterize the invariant varieties. As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits, a strong form of the dynamical Manin-Mumford conjecture for liftings of the Frobenius, and an answer to the question of definability of nonorthogonality for minimal types in models of ACFA_0 extending the formula σ(x) = f(x) for f a polynomial.
Mathematics Subject Classification: Primary: 37F10; Secondary: 03C60, 11C08, 11G, 14G
Keywords and phrases: dynamics, polynomical compositions, ACFA, orthogonality, difference fields, periodic points
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