  We show that a class of dynamical systems induces an associated operator
system in Hilbert space. The dynamical systems are defined from a fixed
finite-to-one mapping in a compact metric space, and the induced operators form
a covariant system in a Hilbert space of L^2-martingales. Our martingale
construction depends on a prescribed set of transition probabilities, given by
a non-negative function. Our main theorem describes the induced martingale
systems completely. The applications of our theorem include wavelets, the
dynamics defined by iterations of rational functions, and sub-shifts in
symbolic dynamics.
  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. 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.
