Invariant escaping Fatou components with two rank-one limit functions for automorphisms of ℂ2

We construct automorphisms of ℂ2, and more precisely transcendental Hénon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank one. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms of the form F(z,w)=(g(z,w),z) with g(z,w): ℂ2 → ℂ holomorphic.