A monotone Sinai theorem

Anthony Quas, Terry Soo

Published 2013-11-24, updated 2014-09-03Version 2

Sinai proved that a non-atomic ergodic measure-preserving system has any Bernoulli shift of no greater entropy as a factor. Given a Bernoulli shift, we show that any other Bernoulli shift that is of strictly less entropy and is stochastically dominated by the original measure can be obtained as a monotone factor; that is, the factor map has the property that for each point in the domain, its image under the factor map is coordinatewise smaller than or equal to the original point.