  A gap mapping is a discontinuous interval mapping with two strictly
increasing branches that have a gap between their ranges. They are
one-dimensional dynamical systems, which arise in the study of certain higher
dimensional flows, for example the Lorenz flow and the Cherry flow. In this
paper, we prove hyperbolicity of renormalization acting on $C^3$ dissipative
gap mappings, and show that the topological conjugacy classes of infinitely
renormalizable gap mappings are $C^1$ manifolds.
