With an appendix by Umberto Zannier
Abstract
We prove a special case of a dynamical analogue of the classical Mordell–Lang conjecture. Specifically, let φ be a rational function with no periodic critical points other than those that are totally invariant, and consider the diagonal action of φ on \({(\mathbb P^1)^g}\). If the coefficients of φ are algebraic, we show that the orbit of a point outside the union of the proper preperiodic subvarieties of \({(\mathbb P^1)^g}\) has only finite intersection with any curve contained in \({(\mathbb P^1)^g}\). We also show that our result holds for indecomposable polynomials φ with coefficients in \({\mathbb C}\). Our proof uses results from p-adic dynamics together with an integrality argument. The extension to polynomials defined over \({\mathbb C}\) uses the method of specialization coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of (φ, φ) on \({\mathbb A^2}\).
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Benedetto, R.L., Ghioca, D., Kurlberg, P. et al. A case of the dynamical Mordell–Lang conjecture. Math. Ann. 352, 1–26 (2012). https://doi.org/10.1007/s00208-010-0621-4
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DOI: https://doi.org/10.1007/s00208-010-0621-4