m at h . C A ] 1 J un 2 00 5 MARTINGALES , ENDOMORPHISMS , AND COVARIANT SYSTEMS OF OPERATORS IN HILBERT SPACE

In the theory of wavelets, in the study of subshifts, in the analysis of Julia sets of rational maps of a complex variable, and, more generally, in the study of dynamical systems, we are faced with the problem of building a unitary operator from a mapping r in a compact metric space X. The space X may be a torus, or the state space of subshift dynamical systems, or a Julia set. Our construction is based on a closer examination of an eigenvalue problem for a transition operator, also called a Perron-Frobenius-Ruelle operator. Under suitable conditions on the given filter functions, our construction takes place in the Hilbert space L 2 (R d). In a variety of examples, for example for frequency localized wavelets, more general filter functions are called for. This then entails basis constructions in Hilbert spaces of L 2-martingales. These martingale Hilbert spaces consist of L 2 functions on certain projective limit spaces X∞ built on a given mapping r : X → X which is onto, and finite-to-one. We study function theory on X∞ in a suitable general framework, as suggested by applications; and we develop our theory in the context of Hilbert space and operator theory. While our motivation derives from some wavelet problems, we have in mind other applications as well; and the issues involving covariant operator systems may be of independent interest.