Abstract: A Lipschitz map of R² admits a horizontal Lipschitz lift to the Heisenberg group H¹ if and only if it is symplectic, i.e., has constant Jacobian. We give a short and self-contained proof of this well-known result in the case when H¹ is endowed with the so-called Heisenberg (or Korànyi) metric. We apply this theorem to study horizontal iterated function systems (IFS's) in the Heisenberg group. The invariant sets associated with such IFS's are horizontal fractals in H¹; they have tangent planes everywhere coincident with the canonical horizontal plane. We study the dimensions of horizontal invariant sets in both the Heisenberg and Euclidean metrics, as well as the fiber structure. As an application of the latter results, we show that H¹ admits nontrivial horizontal BV surfaces. In contrast, Ambrosio and Kirchheim have recently proved that H¹ admits no nontrivial horizontal Lipschitz surfaces.