The dynamical Mordell-Lang problem for etale maps

Abstract

We prove a dynamical version of the Mordell-Lang conjecture for etale endomorphisms of quasiprojective varieties. We use p-adic methods inspired by the work of Skolem, Mahler, and Lech, combined with methods from algebraic geometry. As special cases of our result we obtain a new proof of the classical Mordell-Lang conjecture for cyclic subgroups of a semiabelian variety, and we also answer positively a question of Keeler/Rogalski/Stafford for critically dense sequences of closed points of a Noetherian integral scheme.

19 pages

Cited by in corpus (21)

• Linear relations between polynomial orbits
• Dynamical Degree and Arithmetic Degree of Endomorphisms on Product Varieties
• A gap principle for dynamics
• Geometric idealizers
• The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker)
• Density of orbits of endomorphisms of commutative linear algebraic groups
• On a Dynamical Mordell-Lang Conjecture for Coherent Sheaves
• A Skolem-Mahler-Lech theorem for iterated automorphisms of $K$-algebras
• On the distribution of orbits in affine varieties
• Arithmetic degrees and dynamical degrees of endomorphisms on surfaces
• On the Arithmetic Dynamics of Monomial Maps
• Examples of dynamical degree equals arithmetic degree
• On the quantitative dynamical Mordell-Lang conjecture
• Periods of rational maps modulo primes
• Dynamical Mordell-Lang and Automorphisms of Blow-ups
• The Dynamical Mordell-Lang Conjecture for skew-linear self-maps
• A Gap Principle for Subvarieties with Finitely Many Periodic Points
• Dynamical Mordell-Lang conjecture for birational polynomial morphisms on ${\mathbb{A}}^2$
• Volume and symplectic structure for l-adic local systems
• Unlikely intersection problems for restricted lifts of p-th power
• Some applications of p-adic uniformization to algebraic dynamics