A measure-preserving endomorphism is one-sided Bernoulli if it is isomorphic to a noninvertible Bernoulli shift. We show that in piecewise smooth settings this property is very strong and far more subtle than the weak Bernoulli property, by extending of results of W. Parry and P. Walters and proving new results based on continuity of the Radon-Nikodym derivative. In particular, we provide tests which work for noninvariant measures if an invariant measure equivalent to a natural measure exists but its density function is not known. Examples of families of interval maps and complex maps on the Riemann sphere illustrate the results.

This work was partly funded by the LMS (Scheme 2 grant 2603) and by EPSRC (grant GR/S91147/01)