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Full Description
The Dynamical Mordell-Lang Conjecture is an analogue of the classical Mordell-Lang conjecture in the context of arithmetic dynamics. It predicts the behavior of the orbit of a point $x$ under the action of an endomorphism $f$ of a quasiprojective complex variety $X$. More precisely, it claims that for any point $x$ in $X$ and any subvariety $V$ of $X$, the set of indices $n$ such that the $n$-th iterate of $x$ under $f$ lies in $V$ is a finite union of arithmetic progressions. In this book the authors present all known results about the Dynamical Mordell-Lang Conjecture, focusing mainly on a $p$-adic approach which provides a parametrization of the orbit of a point under an endomorphism of a variety.
Contents
Introduction
Background material
The dynamical Mordell-Lang problem
A geometric Skolem-Mahler-Lech theorem
Linear relations between points in polynomial orbits
Parametrization of orbits
The split case in the dynamical Mordell-Lang conjecture
Heuristics for avoiding ramification
Higher dimensional results
Additional results towards the dynamical Mordell-Lang conjecture
Sparse sets in the dynamical Mordell-Lang conjecture
Denis-Mordell-Lang conjecture
Dynamical Mordell-Lang conjecture in positive characteristic
Related problems in arithmetic dynamics
Future directions
Bibliography
Index
