Algebraic Dynamics of Skew-linear Self-maps

Let X be a variety defined over an algebraically closed field k of characteristic 0, let N ∈ N, let g : X99KX be a dominant rational self-map, and let A : AN−→AN be a linear transformation defined over k(X), i.e., for a Zariski open dense subset U ⊂ X, we have that for x ∈ U(k), the specialization A(x) is an N -by-N matrix with entries in k. We let f : X × AN99KX × AN be the rational endomorphism given by (x, y) 7→ (g(x), A(x)y). We prove that if the determinant of A is nonzero and if there exists x ∈ X(k) such that its orbit Og(x) is Zariski dense in X, then either there exists a point z ∈ (X×AN )(k) such that its orbit Of (z) is Zariski dense in X × AN or there exists a nonconstant rational function ψ ∈ k(X ×AN ) such that ψ ◦ f = ψ. Our result provides additional evidence to a conjecture of Medvedev and Scanlon. We also show that if X = A1 and if g is an automorphism, then each irreducible curve C ⊂ X × AN which is not periodic under the action of f intersects any orbit Of (z) in at most finitely many points. Our result yields a new special case of the Dynamical Mordell-Lang Conjecture.