The Dynamical Mordell-Lang Conjecture

2007/12/14 by Benedetto, Robert L., Ghioca, Dragos, Kurlberg, Par +1 · 2 citations
#14G25 #37F10 #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0712.2344

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 (\bP¹)^g has only finite intersection with any curve contained in (\bP¹)^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 \bA².

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• On simultaneously preperiodic points for one-parameter families of polynomials in characteristic p
• Collision of orbits for families of polynomials defined over fields of positive characteristic

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