The Dynamical Mordell-Lang Conjecture

Details The Dynamical Mordell-Lang Conjecture The Dynamical Mordell-Lang Conjecture

2007-12-14

arxiv.org/abs/0712.2344v2

Mathematics

Number Theory

25 pages. Results strengthened to include the case of indecomposable polynomials with complex coefficients (using some recent

Scientific paper

We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $\phi$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $\phi$ 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 $\phi$ 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 $(\phi,\phi)$ on $\bA^2$.

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