Convergence of geometric subdivision schemes

Non-linear RdRd-valued curve subdivision has a high potential of generating limit curves sensitive to the geometry of initial points. A natural condition characterizing geometric subdivision schemes is the commutation of the refinement rules with similarities. In this paper, we introduce this class of geometric subdivision schemes and address the question of convergence. We prove that uniform decay of the edge lengths is necessary and uniform summability thereof is sufficient for convergence. For a special subclass the necessary condition is also sufficient and thus fully characterizes convergence.