The Dynamical Mordell-Lang Conjecture

TL;DR

This paper proves a special case of a dynamical version of the Mordell-Lang conjecture, showing finiteness of intersections between orbits and curves under certain rational functions and polynomials.

Contribution

It establishes a new result in dynamical systems over algebraic and complex fields, combining $p$-adic dynamics and specialization methods.

Findings

Finite intersection of orbits with curves under specified conditions

Extension of results to indecomposable polynomials over $C$

Use of $p$-adic dynamics and specialization techniques

Abstract

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 $\bA^2$.