The Dynamical Mordell-Lang Conjecture for skew-linear self-maps

TL;DR

This paper proves new cases of the Dynamical Mordell-Lang Conjecture for certain rational maps on algebraic varieties, with implications for difference equations and Picard-Vessiot theory.

Contribution

It establishes the conjecture for skew-linear self-maps where the base map is affine linear or of degree greater than one, expanding known cases.

Findings

Irreducible curves intersecting orbits infinitely often are periodic.

Subvarieties with Zariski dense orbit intersections are periodic.

Results have applications to difference equations and Picard-Vessiot extensions.

Abstract

Let k be an algebraically closed field of characteristic 0, let X=P^1\times A^N and let f be a rational endomorphism of X given by (x,y)--->(g(x), A(x)y), where g is a rational function, while A is an N-by-N matrix with entries in k(x). We prove that if g is of the form x--->ax+b, then each irreducible curve C of X which intersects some orbit of f in infinitely many points must be periodic under the action of f. In the case g is an endomorphism of degree greater than 1, then we prove that each irreducible subvariety Y of X intersecting an orbit of f in a Zariski dense set of points must be periodic as well. Our results provide the desired conclusion in the Dynamical Mordell-Lang Conjecture in a couple new instances. Also, our results have interesting consequences towards a conjecture of Rubel and towards a generalized Skolem-Mahler-Lech problem proposed by Wibmer in the context of difference equations. In the appendix it is shown that the results can also be used to construct Picard-Vessiot extensions in the ring of sequences.