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arXiv · 0712.2344

The Dynamical Mordell-Lang Conjecture

Robert L. Benedetto·Dragos Ghioca·Par Kurlberg·Thomas J. Tucker
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Abstract

We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $ϕ$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $ϕ$ are algebraic, we show that the orbit of a point outside the union of proper preperiodic subvarieties of $(\bP^1)^g$ has only finite intersection with any curve contained in $(\bP^1)^g$. We also show that our result holds for indecomposable polynomials $ϕ$ with coefficients in $\bC$. Our proof uses results from $p$-adic dynamics together with an integrality argument. The extension to polynomials defined over $\bC$ uses the method of specializations coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of $(ϕ,ϕ)$ on $\bA^2$.

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Robert L. Benedetto, Dragos Ghioca, Par Kurlberg, Thomas J. Tucker. 2009-02-06. The Dynamical Mordell-Lang Conjecture. https://arxiv.org/abs/0712.2344

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