The capacity and equivocation of a transducer and a connection with Billingsley dimension

In this paper the study of a noiseless transducer with finite memory, begun by Conner [Ann. Math. Statist.41, No. 6 (1970)], is set forth. The uniqueness and the weak Bernoulli property of the maximal-output dynamical system are proved. A convergence theorem about equivocation, which is analogous to McMillan's theorem about entropy, is obtained. An application of this theorem yields a lower estimation of the Billingsley dimension of the ambiguity set of almost all output messages.