The Dynamical Mordell-lang Conjecture for Skew-linear Self-maps

Let k be an algebraically closed field of characteristic 0, let N ∈ N, let g : P−→P be a non-constant morphism, and let A : A−→A be a linear transformation defined over k(P), i.e., for a Zariski open dense subset U ⊂ P, we have that for x ∈ U(k), the specialization A(x) is an N -by-N matrix with entries in k. We let f : P×A99KP×A be the rational endomorphism given by (x, y) 7→ (g(x), A(x)y). We prove that if g induces an automorphism of A ⊂ P, then each irreducible curve C ⊂ A × A which intersects some orbit Of (z) in infinitely many points must be periodic under the action of f . Furthermore, in the case g : P−→P is an endomorphism of degree greater than 1, then we prove that each irreducible subvariety Y ⊂ P × A intersecting an orbit Of (z) in a Zariski dense set of points must be periodic. 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.