Supports of locally linearly independent M-refinable functions, attractors of iterated function systems and tilings

This paper is devoted to a study of supports of locally linearly independent M-refinable functions by means of attractors of iterated function systems, where M is an integer greater than (or equal to) 2. For this purpose, the local linear independence of shifts of M-refinable functions is required. So we give a complete characterization for this local linear independence property by finite matrix products, strictly in terms of the mask. We do this in a more general setting, the vector refinement equations. A connection between self-affine tilings and L-2 solutions of refinement equations without satisfying the basic sum rule is pointed out, which leads to many further problems. Several examples are provided to illustrate the general theory.