Sinai factors of nonsingular systems: Bernoulli shifts and Anosov flows

We show that a totally dissipative system has all nonsingular systems as factors, but that this is no longer true when the factor maps are required to be finitary. In particular, if a nonsingular Bernoulli shift has a limiting marginal distribution p, then it cannot have, as a finitary factor, an independent and identically distributed (iid) system of entropy larger than $ H(p) $; on the other hand, we show that iid systems with entropy strictly lower than $ H(p) $ can be obtained as finitary factors of these Bernoulli shifts, extending Keane and Smorodinsky's finitary version of Sinai's factor theorem to the nonsingular setting. As a consequence of our results we also obtain that every transitive twice continuously differentiable Anosov diffeomorphism on a compact manifold, endowed with volume measure, has iid factors, and if the factor is required to be finitary, then the iid factor cannot have Kolmogorov–Sinai entropy greater than the measure-theoretic entropy of a Sinai–Ruelle–Bowen measure associated with the Anosov diffeomorphism.