We prove a conjecture of Medvedev and Scanlon for endomorphisms of connected commutative linear algebraic groups G defined over an algebraically closed field k of characteristic 0. That is, if Φ is a dominant endomorphism of G, we prove that one of the following holds: either there exists a non-constant rational function f ∈ k(G) preserved by Φ, or there exists a point x ∈ G(k) whose Φ-orbit is Zariski dense in G.

The first author was partially supported by a Discovery Grant from the National Sciences and Engineering Research Council of Canada. The second author was partially supported by a UBC-PIMS Postdoctoral Fellowship.