  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.
