Algebraic and geometric aspects of rational Γ-inner functions

The set G =(def ){(z + w, zw) : vertical bar z vertical bar < 1, vertical bar w vertical bar < 1} subset of C-2 has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the Mobius band and it has a special subvariety which is the only complex geodesic of G that is invariant under all automorphisms. We exploit the geometry of G to develop an explicit and detailed structure theory for the rational maps from the unit disc to the closure Gamma of G that map the boundary of the disc to the distinguished boundary of Gamma. (C) 2018 The Authors. Published by Elsevier Inc.