Wandering r-tuples for unitary systems

The properties of the set W r ( U ) of all complete wandering r-tuples for a system U of unitary operators acting on a Hilbert space are investigated by parameterizing W r ( U ) in terms of a fixed wandering r-tuple Ψ and the set of all unitary operators which locally commute with U at Ψ. The special case of greatest interest is the system 〈 D , T 〉 of dilation (by 2) and translation (by 1) unitary operators acting on L 2 ( R ) , for which the complete wandering r-tuples are precisely the orthogonal multiwavelets with multiplicity r. We also give some examples for its application.